An approximability-related parameter on graphs—properties and applications

We introduce a binary parameter on optimisation problems called separation . The parameter is used to relate the approximation ratios of different optimisation problems; in other words, we can convert approximability (and non-approximability) result for one problem into (non)-approximability results for other problems. Our main application is the problem (weighted) maximum H -colourable subgraph (M AX H -C OL ), which is a restriction of the general maximum constraint satisfaction problem (M AX CSP) to a single, binary, and symmetric relation. Using known approximation ratios for M AX k - CUT , we obtain general asymptotic approximability results for M AX H -C OL for an arbitrary graph H . For several classes of graphs, we provide near-optimal results under the unique games conjecture. We also investigate separation as a graph parameter. In this vein, we study its properties on circular complete graphs. Furthermore, we establish a close connection to work by Šámal on cubical colourings of graphs. This connection shows that our parameter is closely related to a special type of chromatic number. We believe that this insight may turn out to be crucial for understanding the behaviour of the parameter, and in the longer term, for understanding the approximability of optimisation problems such as M AX H -C OL .


Introduction
In this article we study an approximation-preserving reducibility called continuous reduction (Simon, 1989).A continuous reduction allows the transfer of constant ratio approximation results from one optimisation problem to another.We introduce a binary parameter on optimisation problems which measures the degradation in the approximation guarantee of a given continuous reduction.We call this parameter the separation of the two problems.
The separation parameter is then used to study a concrete family of optimisation problems called the maximum H-colourable subgraph problems, or MAX H -COL for short.This family includes the problems MAX k-CUT for which good approximation ratios are known (Frieze and Jerrum, 1997).Starting from these ratios, we use the notion of separation to obtain general approximation results for MAX H -COL. Our main contribution in relation to approximation is Theorem 3.10 which gives a constant approximation ratio of 1 − 1 r + 2 ln k k 2 (1 − 1 r + o(1)) for MAX H -COL, when the graph H has clique number r and chromatic number k.
In the setting of MAX H -COL, we view separation as a binary graph parameter.While the initial motivation for introducing this parameter was to study the approximability of optimisation problems, it turns out that separation is a parameter of independent interest in graph theory.We investigate this aspect of separation in the second part of the article.Among the most striking results is the connection between separation and a (generalisation of) cubical colourings and fractional covering by cuts previously studied by Šámal (2005, 2006, 2012).

The Separation Parameter
Before we consider continuous reductions and the separation parameter, we formally define optimisation problems.An optimisation problem M is defined over a set of instances I M ; each instance I ∈ I M has an associated finite set Sol M (I) of feasible solutions.The objective is, given an instance I, to find a feasible solution of optimum value, with respect to some measure (objective function) m M (I, f ), where f ∈ Sol M (I).The optimum of I is denoted by Opt M (I), and is defined as the largest measure of any solution to I. (We will only consider maximisation problems in this article.)We will make the assumption that every instance of every problem considered has some feasible solution and that all feasible solutions have positive rational measure.Then, the following quantity is always defined, where I ∈ I M and f ∈ Sol M (I).
A solution f ∈ Sol M (I) to an instance I of a maximisation problem M is called r-approximate if it satisfies R M (I, f ) ≥ r.
An approximation algorithm for M has approximation ratio r(n) if, given any instance I of M , it outputs an r(|I|)-approximate solution.We say that M can be approximated within r(n) if there exists a polynomial-time algorithm for M with approximation ratio r(n).
All optimisation problems that we consider belong to the class NPO; this class contains the problems for which the instances and solutions can be recognised in polynomial time, the solutions are polynomially bounded in the input size, and the objective function can be computed in polynomial time.An NPO problem is in the class APX if it can be approximated within a constant factor.If, in addition, for any rational value 0 < r < 1, there exists an algorithm which, given an instance, returns an r-approximate solution in time polynomial in the size of the instance, then we say that the problem admits a polynomialtime approximation scheme (PTAS).Note that the dependence of the time on r may be arbitrary.
A reduction from an NPO-problem N to an NPO-problem M is specified by two polynomial-time computable functions, ϕ which maps instances of N to instances of M , and γ which takes an instance I ∈ I N and a solution f ∈ Sol M (ϕ(I)) and returns a solution to I. Completeness in APX can be defined using appropriate reductions and it is known that an APX-complete problem does not admit a PTAS, unless P = NP.For a more comprehensive introduction to these classes and their accompanying reductions, see Ausiello et al. (1999) and Crescenzi (1997).Our main focus will be on the following reducibility.
Definition 1.1 (Simon, 1989;Crescenzi, 1997) A reduction ϕ, γ from N to M is called a continuous reduction if a positive constant α exists such that, for every I ∈ I N and f ∈ Sol N (ϕ(I)), it holds that R N (I, γ(I, f )) ≥ α • R M (ϕ(I), f ). (1) Every continuous reduction preserves membership in APX.More specifically, we have the following.
Proposition 1.2 (Simon, 1989) Assume that there is a continuous reduction from N to M with a constant α.If M can be approximated within a constant ratio r, then N can be approximated within α • r.
Given a fixed continuous reduction ϕ, γ, we ask the following question: Which is the largest constant α that satisfies (1)?
To answer this question we take the supremum of all positive constants satisfying (1) over all I ∈ I N and f ∈ Sol M (ϕ(I)).
Definition 1.3 The separation of M and N , given a continuous reduction ϕ, γ from N to M , is defined as the following quantity.
s(M, N ) := inf Needless to say, the separation is difficult to compute in the general case.Thus, we henceforth concentrate on one particular optimisation problem that is parameterised by graphs.It is, however, important to note that the parameter s can be defined over many different types of optimisation problems and it is by no means restricted to problems parameterised by graphs.

The Maximum H-Colourable Subgraph Problem
Denote by G the set of all non-empty, simple, and undirected graphs.For a graph G ∈ G, let W(G) be the set of weight functions w : E(G) → Q ≥0 such that w is not identically 0. For w ∈ W(G), we let w 1 = e∈E(G) w(e) denote the total weight of G with respect to w.
Let G and H be graphs in G.A graph homomorphism from G to H is a vertex map which carries the edges in G to edges in H.The existence of such a map is denoted by G → H.If both G → H and H → G, then G and H are said to be homomorphically equivalent.This is denoted by G ≡ H.
We now consider the following collection of problems, parameterised by a graph H ∈ G.
Problem 1 The weighted maximum H-colourable subgraph problem, or MAX H -COL for short, is the maximisation problem with Instance: An edge-weighted graph (G, w), where G ∈ G and w ∈ W(G).
Solution: A subgraph G ⊆ G such that G → H.

Measure:
The weight of G with respect to w.
Example 1.Let G be a graph in G. Given a subset of vertices S ⊆ V (G), a cut in G with respect to S is the set of edges from a vertex in S to a vertex in V (G) \ S. The MAX CUT problem asks for the size of a largest cut in G.
More generally, for k ≥ 2, a k-cut in G is the set of edges between S i and S j , i = j, where S 1 , . . ., S k is a partition of V (G).The MAX k-CUT problem asks for the size of a largest k-cut.This problem is readily seen to be equivalent to finding a largest subgraph of G which has a homomorphism to the complete graph K k , i.e. finding a largest k-colourable subgraph of (a uniformly edge-weighted graph) G. Hence, for each k ≥ 2, the problem MAX k-CUT is included in the collection of MAX H -COL problems.It is well known that MAX k-CUT is APX-complete, for k ≥ 2 (Ausiello et al., 1999).
Given an edge-weighted graph (G, w), denote by mc H (G, w) the measure of an optimal solution to the problem MAX H -COL. Denote by mc k (G, w) the (weighted) size of a largest k-cut in (G, w).This notation is justified by the equivalence of the problems MAX k-CUT and MAX K k -COL.The decision version of MAX H -COL, the H-COLOURING problem, has been extensively studied (see the monograph by Hell and Nešetřil (2004) and its many references).Hell and Nešetřil (1990) were the first to show that this problem is in P if H contains a loop or is bipartite, and NP-complete otherwise.
Assuming that M → N , we consider the reduction ϕ 1 , γ 1 defined as follows.The function ϕ 1 maps an instance (G, w) ∈ I N to (G, w) ∈ I M and the function γ 1 maps a solution G ∈ Sol M (G, w) to the solution G ∈ Sol N (G, w).Here, m M (ϕ 1 (I), f ) = m N (I, γ 1 (I, f )), so the separation defined in (2) takes the following simplified form.
Definition 1.4 Let M, N ∈ G.The separation of M and N is defined as the following quantity.
Remark 1.5 Note that we do not require M → N in Definition 1.4.The reason for this is that the definition makes sense independently of its connection to the reduction ϕ 1 , γ 1 .We can then study the separation parameter independently as a graph parameter.
In Section 2 we show that the reduction ϕ 1 , γ 1 is indeed a continuous reduction.Proposition 1.2 therefore implies the following.
Lemma 1.6 Let M → N be two graphs in G.If mc M can be approximated within α, then mc N can be approximated within α • s(M, N ).If it is NP-hard to approximate mc N within β, then mc M is not approximable within β/s(M, N ), unless P = NP.
Example 2. Goemans and Williamson (1995) give an algorithm for MAX CUT which is a 0.87856approximating algorithm for MAX K 2 -COL.In Proposition 2.5 we will see that s(K 2 , C 11 ) = 10/11.
We can now apply Lemma 1.6 with M = K 2 and N = C 11 , and we find that this MAX CUT-algorithm approximates MAX C 11 -COL within 0.87856 • s(K 2 , C 11 ) ≈ 0.79869.

Article Outline
The basic properties of separation for the MAX H -COL family are worked out in Section 2. The main result, Theorem 2.1, provides a simplification of (3) which we then use to obtain explicit values and bounds on separation.In particular, this shows that the reduction ϕ 1 , γ 1 defined above is continuous.A linear programming formulation of separation is presented in Section 2.3.
In Section 3, we use separation to study the approximability of MAX H -COL and investigate optimality issues, for several classes of graphs.Comparisons are made to the bounds achieved by the general MAX 2-CSP-algorithm by Håstad (2005).Our investigation covers a spectrum of graphs, ranging from graphs with few edges and/or containing long shortest cycles to an asymptotic result, Theorem 3.10, for arbitrary graphs.We also look at graphs in the G(n, p) random graph model, pioneered by Erdős and Rényi (1960).
In order to apply our method successfully to the MAX H -COL problem but also to get a better understanding of the separation parameter, we want to compute some explicit values of s(M, N ) for various graphs M and N .To this end, we turn to the circular complete graphs in Section 4. We take a close look at 3-colourable circular complete graphs, and amongst other things, find that there are regions of such graphs on which separation is constant.The application of these results to MAX H -COL relies heavily on existing graph homomorphism results, and in this context we will see that a conjecture by Jaeger (1988) has precise and interesting implications (see Section 6.2).
Another way to study separation is to relate it to known graph parameters.In Section 5 we show that our parameter is closely related to a fractional edge-covering problem and an associated "chromatic number", and that we can pass effortlessly between the two views, gaining insights into both.This part is highly inspired by work of Šámal (2005, 2006, 2012) on cubical colourings and fractional covering by cuts.In particular, our connection to Šámal's work brings about a new family of chromatic numbers that provides us with an alternative way of computing our parameter.We also use our knowledge of the behaviour of separation to decide two conjectures concerning cubical colourings.
Finally, we summarise the future prospects and open problems of the method in Section 6.

Basic Properties of the Separation Parameter
In this section we establish a basic theorem on separation, Theorem 2.1, and derive a number of results from it.It follows that the reduction ϕ 1 , γ 1 is continuous.We also give a general bound on the separation parameter, exact values in some special cases, and a linear programming formulation.
Let N ∈ G be a graph.An edge automorphism of N is a permutation of the edges of N that sends edges with a common vertex to edges with a common vertex.The set of all edge automorphisms is called the edge automorphism group of N and its denoted by Aut * (N ).The graph N is called edge-transitive if Aut * (N ) acts transitively on E(N ).Let Ŵ(N ) be the set of weight functions w ∈ W(N ) that satisfy w 1 = 1, and for which w(e) = w(π • e) for all e ∈ E(N ), π ∈ Aut * (N ).That is, w is constant on the orbits of Aut * (N ).
We postpone the proof of Theorem 2.1 to Section 2.2.
Corollary 2.2 Let M → N be graphs in G.The reduction ϕ 1 , γ 1 is continuous with constant s(M, N ).
Proof: For every instance I = (G, w) of MAX N -COL and solution G → N , we have, by definition,

Exact Values and Bounds
Let n(G) and e(G) denote the number of vertices and edges in G, respectively.Let ω(G) denote the clique number of G; the greatest integer r such that K r → G. Let χ(G) denote the chromatic number of G; the least integer n such that G → K n .The Turán graph T (n, r) is a graph formed by partitioning a set of n vertices into r subsets, with sizes as equal as possible, and connecting two vertices whenever they belong to different subsets.The properties of Turán graphs are given by the following theorem.
Using Turán's theorem, we can determine the separation exactly when the second parameter is a complete graph.
Proposition 2.4 Let H be a graph with ω(H) = r and let n be an integer such that χ(H) ≤ n.Then, Proof: The graph K n is edge-transitive.Therefore, by the second part of Theorem 2.1, it suffices to show that mc H (K n , 1/e(K n )) = e(T (n, r))/e(K n ).By definition, T (n, r) is an r-partite subgraph of K n , so Next we consider separation between cycles.The even cycles are all bipartite and therefore homomorphically equivalent to K 2 .The odd cycles, on the other hand, form a chain between K 2 and C 3 = K 3 in the following manner: Note that the chain is infinite on the K 2 -side.The following proposition gives the separation between pairs of graphs in this chain.The value depends only on the target graph.
Proposition 2.5 Let m > k be positive integers.Then, .
Proof: Since C 2k+1 is edge-transitive, it suffices by Theorem 2.1 to show that mc 2 (C 2k+1 ) = 2k = mc C2m+1 (C 2k+1 ).Such cuts clearly exist since after removing one edge from C 2k+1 , the remaining subgraph is isomorphic to a path, and therefore homomorphic to K 2 (and to C 2m+1 ).Moreover, these cuts are the largest possible: Given two graphs M, H ∈ G, it may be difficult to determine s(M, H) directly.However, if we know that H is "homomorphically sandwiched" between M and another graph N , so that M → H → N , then it turns out that we can use s(M, N ) as a lower bound for s(M, H).More generally, we have the following lemma.
Lemma 2.6 The following implications hold.
The second part follows similarly.2 We will see several applications of Lemma 2.6 in the following sections, but first we will use it to get a general lower bound on the separation parameter.
Proposition 2.7 Let M ∈ G be a fixed graph.Then, for any N ∈ G, Proof: The upper bound follows from Theorem 2.1.For the lower bound, let n = χ(N ) be the chromatic number of N , so that N → K n .Then, s(M, N ) ≥ s(M, K n ) = mc M (K n , 1/e(K n )) by Lemma 2.6 and the second part of Theorem 2.1, respectively.
To give a lower bound on mc M (K n , 1/e(K n )) we apply a standard probabilistic argument.Let f : V (K n ) → V (M ) be a function, chosen randomly as follows: for every Every possible function f appears with nonzero probability and each function defines a subgraph of K n by including those edges that are mapped by f to edges in M .We will show that there is at least one function that defines a subgraph K ⊆ K n with the right number of edges.
For e ∈ E(K n ), and e = {u, v} ∈ E(M ), let Y e,e = 1 if f maps e to e , and Y e,e = 0 otherwise.Then, E(Y e,e ) = 2 • deg(u) deg(v)/(2e(M )) 2 .Let X e = 1 if f maps e to some edge in E(M ), and X e = 0 otherwise, so that X e = e ∈E(M ) Y e,e .Then, the total number of edges in K is equal to e∈E(Kn) X e , and by linearity of expectation, Finally, we note that for an arbitrary fixed vertex v m ∈ V (M ), the function defined by f (v) = v m for all v ∈ V (K n ) defines the empty subgraph, and has a non-zero probability.Since K n and M are non-empty we have E(e(K)) > 0, so there must exist at least one f which defines a K with strictly more than the expected total number of edges. 2 From Proposition 2.7, we have s(K m , N ) > m−1 m .It may seem surprising that the value of s(K m , N ) for any graph N can be bounded this close to 1, in particular since we can choose m as large as we like.For large m, it follows from Lemma 1.6 that when K m → N , an algorithm for MAX m-CUT will approximate MAX N -COL almost as well.This seems like a quite strong statement.However, there is a straightforward explanation: This implies that a lower bound for mc M (mc m in our case) immediately yields a lower bound for s(M, N ).Now, a probabilistic argument, analogous to the one in the proof of Proposition 2.7, shows that mc m (G, w/ w 1 ) > m−1 m , for every G and every (non-zero) w.Hence, the fact that s(K m , N ) is close to 1 is merely a reflection of the fact that every (edge-weighted) graph has a large m-cut.
The bipartite density of a graph H is defined as b(H) := mc 2 (H, 1/e(H)), and is a well-studied graph parameter (Alon, 1996;Berman and Zhang, 2003;Bondy and Locke, 1986;Hopkins and Staton, 1982;Locke, 1990).From Theorem 2.1, it follows that b(H) = s(K 2 , H) whenever H is edge-transitive.The following proposition shows that separation is invariant under homomorphic equivalence.Note that this is not true for bipartite density: Let T be a triangle and let H be the graph of two triangles sharing a common edge.In this case, T ≡ H, but b(T ) = 2/3 = 4/5 = b(H).
Proposition 2.8 Let M ≡ K and N ≡ H be two pairs of homomorphically equivalent graphs.Then, s(M, N ) = s(K, H).
Proof: Using Lemma 2.6, H → N , and M → K, we get s(M, N ) ≤ s(M, H) ≤ s(K, H).On the other hand, N → H and K → M , so s(K, H) ≤ s(K, N ) ≤ s(M, N ). 2 2.2 Exploiting Symmetries (Proof of Theorem 2.1) In this section we prove Theorem 2.1.First we need to establish a number of lemmas.The optimum mc H (G, w) is sub-linear with respect to the weight function, as is shown by the following result.
Lemma 2.9 Let G, H ∈ G, α ∈ Q ≥0 and let w, w 1 , . . ., w r ∈ W(G) be weight functions on G.Then, Proof: The first part is trivial.For the second part, let G be an optimal solution to the instance (G, r i=1 w i ) of MAX H -COL.Then, the measure of this solution equals the sum of the measures of G as a (possibly suboptimal) solution to each of the instances (G, w i ). 2 The solutions to a MAX H -COL instance have an alternative description, which is better suited for calculations: for any vertex map f : or, alternatively, when h # is a total function.The set of solutions to an instance (G, w) of MAX H -COL can then be taken to be the set of vertex maps f : w(e). (4) We will predominantly use this description of solutions.Let f : V (G) → V (H) be a solution to the instance (G, w) of MAX H -COL, and define w f ∈ W(H) as follows: Note that w f 1 = 1 iff f is optimal.The following lemma and its corollary establishes half of Theorem 2.1.
Lemma 2.10 Let M, N ∈ G be two graphs.Then, for every G ∈ G, every w ∈ W(G), and any solution Proof: Arbitrarily choose an optimal solution g : Fig. 1: The relation between the graphs and solutions in the proof of Lemma 2.10.
Note that we have w(e) Clearly, the measure of The result follows after division by mc N (G, w). 2 From Lemma 2.10, we have the following corollary, which shows that it is possible to eliminate G ∈ G from the infimum in the definition of s.
Corollary 2.11 Let M, N ∈ G be two graphs.Then, Proof: First, we fix some optimal solution f = f (G, w) for each choice of (G, w).By taking infima over G and w on both sides of the inequality in Lemma 2.10, we then have where the second inequality holds since w f 1 = 1 for any optimal f .For the other direction, we specialise G to N , and restrict w to obtain: This concludes the proof. 2 Proof of Theorem 2.1: From Corollary 2.11, we have that To complete the first part of the theorem, it will be sufficient to prove that for any graph G ∈ G and w ∈ W(G), there is a w ∈ Ŵ(N ) such that the following inequality holds.
Taking infima on both sides of this inequality then shows that Let A = Aut * (N ) be the edge automorphism group of N and let π ∈ A be an arbitrary edge automorphism.If f is an optimal solution to (G, w) as an instance of MAX N -COL, then so is π • f .By Lemma 2.10, inequality (6) is satisfied by w π•f .Summing π in this inequality over A gives where the last inequality follows from Lemma 2.9.The weight function π∈A w π•f is determined as follows: π∈A where Ae denotes the orbit of e under A. We have now shown that the inequality in ( 6) is satisfied by w = π∈A w π•f /|A|, and that w is in Ŵ(N ).The first part of the theorem follows.
For the second part, note that when the edge automorphism group A acts transitively on E(N ), there is only one orbit Ae = E(N ) for all e ∈ E(N ).Then, the weight function w is given by since f is optimal. 2

A Linear Programming Formulation
In this section, we first derive a linear program for the separation parameter based on Corollary 2.11.Later we will see how to reduce the size of this program, but it serves as a good first exercise, and it will also be used for comparison with the linear program studied in Section 5.Each vertex map f : V (N ) → V (M ) induces an edge map f # , which provides a lower bound on the separation: w(e) ≤ s(M, N ). (7) By Corollary 2.11, we want to find the least s such that for some weight function w ∈ W(N ), w 1 = 1, the inequalities (7) hold.Let the variables of the linear program be {w e } e∈E(N ) and s.We then have the following linear program for s(M, N ).
Minimise s Given an optimal solution {w e } e∈E(N ) , s to (8), a weight function which minimises mc M (N, w) is given by w(e) = w e for e ∈ E(N ).The measure of this solution is s = s(M, N ).The program will clearly be very large with |E(N )| + 1 variables and |V (M )| |V (N )| inequalities.Fortunately it can be improved upon.
From Theorem 2.1 it follows that in order to determine s(M, N ), it is sufficient to minimise mc M (N, w) over w ∈ Ŵ(N ).We can use this to describe a smaller linear program for computing s(M, N ).Let A 1 , . . ., A r be the orbits of Aut * (N ).The measure of a solution f when w ∈ Ŵ(N ) is equal to r i=1 f i • w i , where w i is the weight of an edge in A i and f i is the number of edges in A i which are mapped to an edge in M by f .Note that given a w, the measure of a solution f depends only on the vector (f 1 , . . ., f r ) ∈ N r .Therefore, take the solution space to be the set of such vectors.
Let the variables of the linear program be w 1 , . . ., w r and s, where w i represents the weight of each element in the orbit A i and s is an upper bound on the solutions.
Minimise s subject to Given an optimal solution w i , s to this program, a weight function which minimises mc M (N, w) is given by w(e) = w i for e ∈ A i .The measure of this solution is s = s(M, N ).
Example 3. The wheel graph on k vertices, W k , is a graph that contains a cycle of length k − 1 plus a vertex v, which is not in the cycle, such that v is connected to every other vertex.We call the edges of the k − 1-cycle outer edges and the remaining k − 1 edges spokes.It is easy to see that the clique number of W k is equal to 4 when k = 4 (W 4 is isomorphic to K 4 ) and that it is equal to 3 in all other cases.Furthermore, W k is 3-colourable if and only if k is odd, and 4-colourable otherwise.This implies that for odd k, the wheel graphs are homomorphically equivalent to K 3 .
We will determine s(K 3 , W k ) for even k ≥ 6 using the previously described construction of a linear program.The first three wheel graphs for even k are shown in Figure 2. Note that the group action of Aut * (W k ) on E(W k ) has two orbits, one which consists of all outer edges and one which consists of all the spokes.If we remove one outer edge or one spoke from W k , then the resulting graph can be mapped homomorphically to K 3 .Therefore, it suffices to choose F = {f, g} with f = (k − 1, k − 2) and g = (k − 2, k − 1) since all other solutions will have a smaller measure than at least one of these.The program for W k looks as follows: The optimal solution to this program is given by Example 4. In the previous example, the two weights in the optimal solution were equal.Here, we provide another example, where the weights turn out to be different for different orbits.The circular complete graph K 8/3 has vertex set {v 0 , v 1 , . . ., v 7 }, which is placed uniformly around a circle.There is an edge between any two vertices which are at a distance at least 3 from each other.Figure 3 depicts this graph.
We will now calculate s(K 2 , K 8/3 ).Each vertex is at a distance 4 from exactly one other vertex, which means that there are 4 such edges.These edges, which are dashed in the figure, form one orbit under the action of Aut * (K 8/3 ) on E(K 8/3 ).The remaining 8 edges (solid) form a second orbit.Let V (K 2 ) = {u 0 , u 1 }.We can obtain a solution f by mapping f (v i ) = u 0 if i is even, and f (v i ) = u 1 if i is odd.This solution will map all solid edges to K 2 , but none of the dashed, hence f = (0, 8).We obtain a second solution g by mapping g(v i ) = u 0 for 0 ≤ i < 4, and g(v i ) = u 1 for 4 ≤ i < 8.This solution will map all but two of the solid edges in K 8/3 to K 2 , hence g = (4, 6).
Let h be an arbitrary solution.If Otherwise, h 1 > 0, so it maps at least one dashed edge, say the edge between v 0 and v 4 , to K 2 .There are two edge-disjoint even-length paths using solid edges from v 0 to v 4 , one via v 3 and one via v 5 .The solution h sends at most 3 of these solid edges from each path to K 2 .Hence, Therefore, the inequalities given by f and g imply the inequality given by any other solution, and we have the following program for s(K 2 , K 8/3 ): The optimal solution to this program is given by w 1 = 1/20, w 2 = 1/10, and s(K 2 , K 8/3 ) = s = 4/5.

Approximation Bounds for MAX H -COL
In this section we apply the reduction ϕ 1 , γ 1 and use some of the explicit values obtained for s in Section 2 to bound the approximation ratio of MAX H -COL for various families of graphs.First, we would like to remind the reader of some earlier results and also give a hint of what to expect when we start studying the approximability of MAX H -COL.The probabilistic argument in Proposition 2.7 shows that MAX H -COL is in APX.Furthermore, Jonsson et al. ( 2009) have shown that whenever H is loop-free, MAX H -COL does not admit a PTAS, and otherwise the problem is trivial.Let us have a closer look at a concrete, well-known example: the MAX CUT problem.This problem was one of Karp's original 21 NP-complete problems (Karp, 1972) and has a trivial approximation ratio of 1/2, which is obtained by assigning each vertex randomly to either part of the partition.The trivial randomised algorithm is easy to derandomise; Sahni and Gonzalez (1976) gave the first such approximation algorithm.The 1/2 ratio was in fact essentially the best known ratio for MAX CUT until 1995, when Goemans and Williamson (1995), using semidefinite programming (SDP), achieved the ratio 0.87856 mentioned in Example 2. Frieze and Jerrum (1997) determined lower bounds on the approximation ratios for MAX k-CUT using similar SDP techniques.Sharpened results for small values of k have later been obtained by de Klerk et al. (2004).Håstad (2005) has shown that SDP is a universal tool for solving the general MAX 2-CSP problem (where every constraint only involves two variables) over any (finite) domain, in the sense that it establishes non-trivial approximation results for all of those problems.Until recently no other method than SDP was known to yield a non-trivial approximation ratio for MAX CUT.Trevisan (2009) broke this barrier by using techniques from algebraic graph theory to reach an approximation guarantee of 0.531.Soto (2009) later improved this bound to 0.6142 by a more refined analysis.Khot (2002) suggested the unique games conjecture (UGC) as a possible direction for proving inapproximability properties of some important optimisation problems.The conjecture states the following (equivalent form from Khot et al. (2007)): Conjecture 3.1 Given any δ > 0, there is a prime p such that given a set of linear equations x i −x j = c ij (mod p), it is NP-hard to decide which one of the following is true: • There is an assignment to the x i 's which satisfies at least a 1 − δ fraction of the constraints.
• All assignments to the x i 's can satisfy at most a δ fraction of the constraints.
Under the assumption that the UGC holds, Khot et al. (2007) proved the approximation ratio achieved by the Goemans and Williamson algorithm for MAX CUT to be essentially optimal and also provided upper bounds on the approximation ratio for MAX k-CUT, k > 2. The proof for the MAX CUT case crucially relies on Gaussian Analysis.In particular, it uses Borell's Theorem to answer the question of partitioning R n into two sets of equal Gaussian measure so as to minimise the Gaussian noise-sensitivity, thereby transferring a Fourier analytic question to a geometric one.
Recently, Isaksson and Mossel (2012) showed that a similar geometric conjecture have further implications for the approximability of MAX k-CUT.
Conjecture 3.2 The standard k-simplex partition is the most noise-stable balanced partition of R n with n ≥ k − 1.
A partition of R n into k measurable sets A 1 , . . ., A k is called balanced if each A i has Gaussian measure 1/k.The ε-noise sensitivity is defined as the probability that two (1 − 2ε)-correlated n-dimensional standard Gaussian points x, y ∈ R n belong to different sets in the partition.The standard k-simplex partition of R n is obtained by letting R n = R k−1 × R n−k+1 and then partitioning R k−1 into k regular simplicial cones.Assuming this standard simplex conjecture (SSC) and the UGC, Isaksson and Mossel show that the Frieze and Jerrum SDP relaxation obtains the optimal approximation ratio for MAX k-CUT.
Every MAX H -COL problem can be viewed as a MAX CSP(Γ) problem, where Γ, the so called constraint language, is the set containing the single, binary, and symmetric relation given by the edge set of H. Raghavendra (2008) has presented approximation algorithms for every MAX CSP(Γ) problem based on semi-definite programming.Under the UGC, these algorithms optimally approximate MAX CSP(Γ) in polynomial-time, i.e. no other polynomial-time algorithm can approximate the problem substantially better.However, it seems notoriously difficult to determine the approximation ratio implied by this result, for a given constraint language: Raghavendra and Steurer (2009) show that this ratio can in principle be computed, but the algorithm is doubly exponential in the size of the domain.In combination with our results, such ratios could be used to confirm or disprove the UGC.

A General Reduction
Our main tool will be a generalisation of the reduction introduced in Section 1.2.Let M and N be (arbitrary) undirected graphs and consider the following reduction, ϕ 2 , γ 2 , from MAX N -COL to MAX M -COL: The function ϕ 2 maps an instance (G, w) ∈ I N to (G, w) ∈ I M .Let f : V (G) → V (M ) be a solution to (G, w).Let g : V (M ) → V (N ) be an optimal solution in Sol N (M, w f ) (see ( 5)).The function γ 2 maps f to g • f .
Proof: First, we argue that γ 2 can be computed in polynomial time.We must show that g can be found in polynomial time.An optimal solution to (M, w f ) can be obtained by brute force.This takes |V (M )| |V (N )| times the time to evaluate a candidate solution g .The measure of a solution depends on w f , and thereby on f , but for a given candidate g , it can clearly be obtained in polynomial time.Since M and N are fixed, the total time is polynomial in the size of (G, w).
Next, we show that ϕ 2 , γ 2 is continuous with constant s(M, N ) where the final equality follows from Corollary 2.11.From (3) we have the inequality mc N (G, w) ≤ mc M (G, w)/s(M, N ) for all G ∈ G and w ∈ W(G).Consequently, with I = (G, w), Since s(M, N ), s(N, M ) > 0, it follows that the reduction is continuous. 2 As a direct consequence (using Proposition 1.2), we get the following generalisation of Lemma 1.6.
The symmetric nature of this result has some interesting consequences.For example, It is possible to show that 1 − s(M, N ) • s(N, M ) is a metric on the space of graphs taken modulo homomorphic equivalence; cf.Färnqvist et al. (2009).
Our main algorithmic tools will be the following two theorems.
Theorem 3.5 (Goemans and Williamson, 1995) MAX CUT can be approximated within A few logarithms will appear in the upcoming expressions.We fix the notation ln x for the natural logarithm of x, and log y for the base-2 logarithm of y.
Theorem 3.6 (Frieze and Jerrum, 1997) MAX k-CUT can be approximated within ).We note that de Klerk et al. (2004) have presented the sharpest known bounds on α k for small values of k.Table 1 lists α GW together with the first of these lower bounds.Theorem 3.7 (Håstad, 2005) There is an absolute constant c > 0 such that mc H can be approximated within where n = n(H) and t(H) = n 2 − 2e(H).
We will compare the performance of Håstad's algorithm on MAX H -COL with the performance of the algorithms in Theorems 3.5 and 3.6 when analysed using the reduction ϕ 2 , γ 2 and estimates or exact values of the separation parameter.For this purpose, we introduce two functions, F J k and Hå, such that, if H is a graph, then F J k (H) denotes the best bound on the approximation guarantee when Frieze and Jerrum's algorithm for MAX k-CUT is applied to the problem mc H , while Hå(H) is the guarantee when Håstad's algorithm is used to approximate mc H .This comparison is not entirely fair since Håstad's algorithm was not designed with the goal of providing optimal results; the goal was to beat random assignments.However, it is currently the best known algorithm for approximating MAX H -COL, for arbitrary H ∈ G, which also provides an easily computable bound on the guaranteed approximation ratio; this is in contrast with the conjectured optimal algorithms of Raghavendra ( 2008) (see the discussion in Section 6).
Near-optimality of our approximation method will be investigated using results depending on Khot's unique games conjecture (Conjecture 3.1).Hence, we will henceforth assume that the UGC is true, which implies the following inapproximability results.
Theorem 3.8 (Khot et al., 2007) Under the assumption that the UGC is true, the following holds: • For every ε > 0, it is NP-hard to approximate mc 2 within α GW + ε.
Furthermore, assuming the standard simplex conjecture (Conjecture 3.2), we have the following strengthening of Theorem 3.8.Theorem 3.9 (Isaksson and Mossel, 2012) Under the assumption that the UGC and the SSC are true, the following holds: • For every ε > 0, it is NP-hard to approximate mc k within α k + ε.

Asymptotic Performance
Next, we derive a general, asymptotic, result on the performance of our method.
Theorem 3.10 Let H ∈ G be a graph with ω(H) = r and χ(H) = k.Then, , where o(1) is with respect to k.
Furthermore, mc H cannot be approximated within , where o(1) is with respect to r.
Proof: By Proposition 2.4, we have By Lemma 3.4 and Theorem 3.6, we then have and the first inequality follows, since For the second part, we have s(K r , H) ≥ s(K r , K k ) = s(H, K k ) by Lemma 2.6 and Proposition 2.4, so Note the similarity between this upper bound and the expression for s(H, K k ).In fact, without losing any precision in the following calculations, we could replace the O(k −2 )-term by O(r −2 ).By Lemma 3.4, mc H cannot be approximated within α r /s(K r , H).An upper bound for α r /s(K r , H) can now be calculated as in the first part with r and k swapped.In the final expression we drop − 1 k from the last parenthesis since this is absorbed by o(1). 2 To give an upper bound on the performance of Håstad's algorithm, we can proceed as follows: Let n = n(H) and r = ω(H), k = χ(H) as in the proposition.By Theorem 2.3, e(H) We see that our algorithm performs asymptotically better.

Some Specific Graph Classes
Next, we investigate the performance of our method on sequences of graphs "tending to" K 2 (K 3 ) in the sense that the separation of K 2 (K 3 ) and a graph H k from the sequence tends to 1 as k tends to infinity.In several cases, the girth of the graphs plays a central role.The girth of a graph G is the length of a shortest cycle in G.The odd girth of G is the length of a shortest odd cycle in G. Hence, if G has odd girth g, then Proposition 3.11 We have the following bounds.
3. Let H be a planar graph with girth at least g = 20k−2
(2) Lai and Liu (2000) show that if H is a graph with the stated properties, then there exists a homomorphism from H to C 2k+1 .Thus, (3) Borodin et al. (2004) show that there exists a homomorphism from H to C 2k+1 .The result follows as for case (2).
(4) We know from Example 3 that The result again follows by Lemma 3.4. 2 We can compare the results of Proposition 3.11 to the performance of Håstad's algorithm as follows: Let n = n(H).In (1), we have e(H) = n; for (2), Dutton and Brigham (1991) have given an upper bound on e(H) of asymptotic order n 1+2/(g−1) ; in (3), e(H) ≤ 3n − 6, since H is planar; and finally in (4), we have e(H) = 2(n − 1).Now note that by ignoring lower-order terms in the expression for Hå(H) in Theorem 3.7, we get Hå(H) = 2e(H) n 2 (1 + o(1)).Hence, for each case (1)-(4), Hå(H) → 0 as n → ∞.Proposition 3.11(3) can be strengthened and extended in several ways: For K 4 -minor-free graphs, Pan and Zhu (2002) have given odd girth-restrictions for 2k + 1-colourability which is better than the girthrestriction in Proposition 3.11(3).Dvořák et al. (2008) have proved that every planar graph H of odd girth at least 9 is homomorphic to the Petersen graph, P .The Petersen graph is edge-transitive and its bipartite density is known to be 4/5 (cf.Berman and Zhang (2003).)In other words, s(K 2 , P ) = 4/5.Consequently, mc H can be approximated within 4 5 • α GW but not within 5 4 • α GW + ε for any ε > 0. This is an improvement on the bounds in Proposition 3.11(3) for planar graphs with girth strictly less than 13.We can also consider graphs embeddable on higher-genus surfaces.For instance, Proposition 3.11(3) is true for graphs embeddable on the projective plane, and it is also true for graphs of girth strictly greater than 20k−2 3 whenever the graphs are embeddable on the torus or Klein bottle.These bounds are direct consequences of results in Borodin et al. (2004).

Random Graphs
Finally, we look at random graphs.Let G(n, p) denote the random graph on n vertices in which every edge is chosen uniformly at random, and independently, with probability p = p(n).We say that G(n, p) has a property A asymptotically almost surely (a.a.s.) if the probability that it satisfies A tends to 1 as n tends to infinity.Here, we let 0 < p < 1 be a fixed constant.Proposition 3.12 Let H ∈ G(n, p).Then, a.a.s., Proof: For H ∈ G(n, p) it is well known that a.a.s.ω(H) assumes one of at most two values around , 1972;Bollobás and Erdős, 1976).Let r = ω(H).By Theorem 3.10, 2 For a comparison to Håstad's algorithm, note that e(H) = n 2 p 2 (1 + o(1)) a.a.s. for H ∈ G(n, p), so 1).The slow logarithmic growth of the clique number of G(n, p) works against our method in this case.However, we still manage to achieve an approximation ratio tending to 1 unlike Håstad's algorithm which ultimately is restricted by the density of the edges.
We conclude this section by looking at what happens for graphs H ∈ G(n, p) when p is no longer chosen to be a constant, but instead we let np tend to a constant ε < 1 as n → ∞.The following theorem allows us to do this.Theorem 3.13 (Erdős and Rényi, 1960) Let c be a positive constant and p = c n .If c < 1, then a.a.s.no component in G(n, p) contains more than one cycle, and no component has more than ln n c−1−ln c vertices.Now we see that if np → ε when n → ∞, then G(n, p) almost surely consists of components with at most one cycle.Thus, each component resembles a cycle where, possibly, trees are attached to certain vertices of the cycle, and each component is homomorphically equivalent to the cycle it contains.Proposition 2.5 is therefore applicable in this part of the G(n, p) spectrum.

Circular Complete Graphs
The successful application of our method relies on the ability to compute s(M, N ) for various graphs M and N .In Section 2.3 we saw how this can be accomplished by the means of linear programming.This insight is put to use here in the context of circular complete graphs.We have already come across examples of such graphs in the form of (ordinary) complete graphs, cycles, and the graph K 8/3 in Figure 3.We will now take a closer look at them.Definition 4.1 Let p and q be positive integers such that p ≥ 2q.The circular complete graph, K p/q , has vertex set {v 0 , v 1 , . . ., v p−1 } and edge set {{v The image to keep in mind is that of the vertices placed uniformly around a circle with an edge connecting two vertices if they are at a distance at least q from each other.Example 5. Some well-known graphs are extreme cases of circular complete graphs: • The complete graph K n , n ≥ 2 is a circular complete graph with p = n and q = 1.
• The cycle graph C 2k+1 , k ≥ 1 is a circular complete graph with p = 2k + 1 and q = k.
These are the only examples of edge-transitive circular complete graphs.
A fundamental property of the circular complete graphs is given by the following theorem.
Theorem 4.2 (see Hell and Nešetřil, 2004) For positive integers p, q, p , and q , K p/q → K p /q ⇔ p/q ≤ p /q Due to this theorem, we may assume that whenever we write K p/q , the positive integers p and q are relatively prime.
One of the main reasons for studying circular complete graphs is that they refine the notion of complete graphs.In particular, that they refine the notion of the chromatic number χ(G).Note that an alternative definition of χ(G) is given by χ With this in mind, the following is a natural extension of proper graph colouring, and the chromatic Definition 4.3 The circular chromatic number, χ c (G), of a graph G is defined as inf{p/q | G → K p/q }.A homomorphism from G to K p/q is called a (circular) p/q-colouring of G.
For more on the circular complete graphs and the circular chromatic number, see the book by Hell and Nešetřil (2004), and the survey by Zhu ( 2001).
We will investigate the separation parameter s(K r , K t ) for rational numbers 2 ≤ r < t ≤ 3.In Section 4.1, we fix r = 2 and choose t so that Aut * (K t ) has few orbits.We find some interesting properties of these numbers which lead us to look at certain "constant regions" in Section 4.2, and at the case r = 2 + 1/k, in Section 4.3.Our method is based on solving a relaxation of the linear program (9) which was presented in Section 2.3, combined with arguments that the chosen relaxation in fact finds the optimum in the original program.Most of the calculations, which involve some rather lengthy ad hoc constructions of solutions, are left out.The complete proofs can be found in the technical report Engström et al. (2009a).

Maps to an Edge
We consider s(K 2 , K t ) for t = 2 + n/k with k > n ≥ 1, where n and k are integers.The number of orbits of Aut * (K t ) then equals (n + 1)/2 .We will denote these orbits by for c = 1, . . ., (n + 1)/2 .Since the number of orbits determine the number of variables of the linear program (9), we choose to begin our study of s(K 2 , K t ) using small values of n.For n = 1 we have seen that the graph K 2+1/k is isomorphic to the cycle C 2k+1 .For n = 2 we can assume that k is odd in order to have 2k + n and k relatively prime.We will write this number as . The map f sends v i to w 0 if i is even and to w 1 if i is odd.Then, f maps all of A 1 to K 2 but none of the edges in A 2 , so f = (4k, 0).The solution g sends a vertex v i to w 0 if 0 ≤ i < 2k and to w 1 if 2k ≤ i < 4k.It is not hard to see that g = (4k − 2, 2k).It remains to argue that these two solutions suffice to determine s.But we see that any map h = (h 1 , h 2 ) with h 2 > 0 must cut at least two edges in the even cycle A 1 .Therefore, h 1 ≤ 4k − 2, so h ≤ g, componentwise.The proposition now follows by solving the relaxation of (9) using only the two inequalities obtained from f and g. 2 Note that the graph K 8/3 from Example 4 is covered by Proposition 4.4.The argument in that example is very similar to the proof of the general case.

Constant Regions
In the previous section we saw that s ) and used it to prove that s . This is a special case of a phenomenon described more generally in the following proposition.
Proposition 4.7 Let k ≥ 1, and let r and t be rational numbers such that Proof: From Theorem 4.2, we have the following chain of homomorphisms. .
By Lemma 2.6, this implies 2k+1 .Two more applications of Lemma 2.6 show that which proves the proposition.2 We find that there are intervals } where the function s r (t) = s(K r , K t ) is constant for any 2 ≤ r < (2k + 1)/k.In Figure 4  It turns out that similar intervals appear throughout the space of circular complete graphs.Indeed, it follows from Proposition 2.4 that for a positive integer n and a rational number r such that 2 ≤ r ≤ n, we have From ( 10) we see that s(K r , K n ) remains constant for rational numbers r in the interval k ≤ r < k +1, where k is any fixed integer k < n.Furthermore, for positive integers k and m, we have When we combine this fact with (10) and Lemma 2.6, we find that s

Maps to Odd Cycles
It was seen in Proposition 4.7 that s(K r , K t ) is constant on the region (r, t) ∈ [2, 2 + 1/k) × I k .In this section, we will study what happens when t remains in I k , but r assumes the value 2 + 1/k.A first observation is that the absolute jump of the function s(K r , K t ) when r goes from being less than 2 + 1/k to r = 2 + 1/k must be largest for t = 2 + 2/(2k − 1).Let V (K 2+2/(2k−1) ) = {v 0 , . . ., v 4k−1 } and V (K 2+1/k ) = {w 0 , . . ., w 2k }, and let the function f map v i to w j , with j = 2k+1 4k • i .Then, f maps all edges except {v 0 , v 2k−1 } from the orbit A 1 to some edge in K r .Since the subgraph A 1 is isomorphic to C 4k , any map to an odd cycle must exclude at least one edge from A 1 .It follows that f alone determines s, and we can solve the linear program (9) to obtain s(K 2+1/k , K 2+2/(2k−1) ) = (4k − 1)/4k.Thus, for r < 2 + 1/k, we have .
Smaller t ∈ I k can be expressed as t = 2 + 1/(k − x), where 0 ≤ x < 1/2.We will write x = m/n for positive integers m and n which implies the form t = 2 + n/(kn − m), with m < n/2.For m = 1, it turns out to be sufficient to keep two inequalities from (9) to get an optimal value of s.From this we get the following result.
Proposition 4.8 Let k, n ≥ 2 be integers.Then, There is still a non-zero jump of s(K r , K t ) when we move from K r < 2 + 1/k to K r = 2 + 1/k, but it is smaller, and tends to 0 as n increases.For m = 2, we have 2(kn − m) + n and kn − m relatively prime only when n is odd.In this case, it turns out that we need to include an increasing number of inequalities to obtain a good relaxation.Furthermore, we are not able to ensure that the obtained value is the optimum of the original (9).We will therefore have to settle for a lower bound on s.Brute force calculations have shown that, for small values of k and n, equality holds in Proposition 4.9.We conjecture this to be true in general.
Proposition 4.9 Let k ≥ 2 be an integer and n ≥ 3 be an odd integer.Then, , , where A k is a constant that does not depend on n.

Fractional Covering by H-cuts
In the following, we generalise the work of Šámal (2005, 2006, 2012) on fractional covering by cuts.We obtain a complete correspondence between a family of chromatic numbers, χ H (G), and s(H, G).These chromatic numbers are generalisations of Šámal's cubical chromatic number χ q (G); the latter corresponds to the case when H = K 2 .Two more expressions for χ H (G) are given in Section 5.2.We believe that these alternative views on the separation parameter can provide great benefits to the understanding of its properties.We transfer a result in the other direction, in Section 5.3, disproving a conjecture by Šámal on χ q , and settle another conjecture by him in the positive, in Section 5.4.

Separation as a Chromatic Number
We start by recalling the notion of a fractional colouring of a hypergraph.Let G be a (hyper-) graph with vertex set V (G) and edge set is a subset of J. Let J denote the set of all independent sets of G and for a vertex v ∈ V (G), let J (v) denote all independent sets which contain v. Let J 1 , . . ., J n ∈ J be a collection of independent sets.
Definition 5.1 An n/k independent set cover is a collection J 1 , . . ., J n of independent sets in J such that every vertex of G is in at least k of them.
The fractional chromatic number χ f (G) of G is given by the following expression.
| there exists an n/k independent set cover of G}.
The definition of fractional covering by cuts mimics that of fractional covering by independent sets, but replaces vertices with edges and independent sets with certain cut sets of the edges.Let G and H be undirected simple graphs and f be an arbitrary vertex map from G to H. Recall that the map f induces a partial edge map f # : E(G) → E(H).We will call the preimage of E(H) under f # an H-cut in G.When H is a complete graph K k , this is precisely the standard notion of a k-cut in G. Let C denote the set of H-cuts in G and for an edge e ∈ E(G), let C(e) denote all H-cuts which contain e.The following definition is a generalisation of cut n/k-covers (Šámal, 2006) to arbitrary H-cuts.
The graph parameter χ H is defined as: | there exists an H-cut n/k-cover of G}.Šámal (2006) called the parameter χ K2 (G), the cubical chromatic number of G.Both the fractional chromatic number and the cubical chromatic number also have linear programming formulations.This, in particular, shows that the value in the infimum of the corresponding definition is obtained exactly for some n and k.For our generalisation of the cubical chromatic number, the linear program is the following: Proposition 5.3 The graph parameter χ H (G) is given by the optimum of the linear program in (11).
Proof: The proof is completely analogous to those for the corresponding statements for the fractional chromatic number (cf.Godsil and Royle, 2001) and for the cubical chromatic number (Lemma 5.1.3 in Šámal, 2006).Let C 1 , . . ., C n be an H-cut n/k-cover of G.The solution f (C) = 1/k if C ∈ {C 1 , . . ., C n }, and f (C) = 0 otherwise, has a measure of n/k in (11).Thus, the optimum of the linear program is at most χ H (G).
For the other direction, note that the coefficients of the program (11) are integral.Hence, there is a rational optimal solution f * .Let N be the least common multiple of the divisors of f * (C) for C ∈ C. Assume that the measure of f * is n/k.Construct a collection of H-cuts by including the cut C a total of N • f * (C) times.This collection covers each edge at least N times using C∈C N • f * (C) = N • n/k cuts, i.e. it is an H-cut n/k-cover, so χ H (G) is at most equal to the optimum of (11). 2 We are now ready to work out the correspondence to separation.
Proof: Consider the dual program of (11). Maximise g(e) subject to e∈X g(e) ≤ 1 for all H-cuts X ∈ C , In ( 12) let 1/s = e∈E(G) g(e) and make the variable substitution w = g • s.This leaves the following program.w(e) = 1 Since max s −1 = (min s) −1 , a comparison with (8) establishes the proposition.2

More Guises of Separation
For fractional colourings, it is well-known that an equivalent definition is obtained by taking χ f (G) = inf{n/k | G → K n,k }, where K n,k denotes the Kneser graph, the vertex set of which is the k-subsets of [n] and with an edge between u and v if u ∩ v = ∅.For H = K 2 , a corresponding definition of χ H (G) = χ q (G) was obtained in Šámal (2006)  A parameterised graph family which determines a particular chromatic number in this way is sometimes referred to as a scale.In addition to the previously mentioned fractional chromatic number χ f , where the scale is the set of Kneser graphs, and the cubical chromatic number χ q , where the scale is {Q n/k }, another prominent example is the circular chromatic number (Section 4) for which the scale is given by the family of circular complete graphs K n/k .
We now generalise the family {Q n/k } to produce one scale for each χ H .To this end, let H n k be the graph on vertex set V (H) n and an edge between (u 1 , . . ., u n ) and (v 1 , . . ., v n ) when |{i | {u i , v i } ∈ E(H)}| ≥ k.The proof of the following proposition is straightforward, but instructive.
where b(S) denotes the bipartite density of S. Conjecture 5.5.3 in Šámal (2006) suggested that this inequality, expressed on the form χ q (S) ≥ 1/(min S⊆G b(S)), could be replaced by an equality.We answer this in the negative, using K 11/4 as our counterexample.Lemma 4.5 with k = 1 gives s(K 2 , K 11/4 ) = 17/22.If s(K 2 , K 11/4 ) = b(S) for some S ⊆ K 11/4 it means that S must have at least 22 edges.Since K 11/4 has exactly 22 edges it follows that S = K 11/4 .However, a cut in a cycle must contain an even number of edges.Since the edges of K 11/4 can be partitioned into two cycles, we have that the maximum cut in K 11/4 must be of even size, hence |E(K 11/4 )| • b(K 11/4 ) = 17.This is a contradiction.

Confirmation of a Scale
As a part of his investigation of the cubical chromatic number, Šámal (2006) set out to determine the value of χ q (Q n/k ) for general n and k.For the fractional chromatic number and the circular chromatic number, results for such measuring of the scale exist and provide very appealing formulae: ), we are immediately out of luck as 1/2 < s(K 2 , G) ≤ 1, i.e. 1 ≤ χ q (G) < 2 for all non-empty graphs.For 1 ≤ n/k < 2, however, Šámal gave a conjecture (Conjecture 5.4.2 in Šámal, 2006).We complete the proof of his conjecture to obtain the following result.
Proposition 5.7 Let k, n be integers such that k ≤ n < 2k.Then, If we make sure that k is even, possibly by multiplying both k and n by a factor two, we get the following interesting corollary.
Corollary 5.9 For every rational number r, 1/2 < r ≤ 1, there is a graph G such that s(K 2 , G) = r.Šámal (2006) provides the upper bound for Proposition 5.7 and an approach to the lower bound of using the largest eigenvalue of the Laplacian of a subgraph of Q n/k .The computation of this eigenvalue boils down to an inequality (Conjecture 5.4.6 in Šámal, 2006) involving some binomial coefficients.We first introduce the necessary notation and then prove the remaining inequality in Proposition 5.11, whose second part, for odd k, corresponds to one of the formulations of the conjecture.Proposition 5.7 then follows from Theorem 5.4.7 in Šámal (2006), conditioned on the result of this proposition.
Let k, n be positive integers such that k ≤ n, and let x be an integer such that 1 When x is odd, the function f : S o (2k, k, x) → S e (2k, k, x), given by the complement f Lemma 5.10 Assume that x is odd, with 1 ≤ x < n = 2k − 1.Then, N e (n, k, x) = N e (n, k, x + 1) and To prove this, define the function f : 2 That is, f acts on σ by ignoring the element n − x and renumbering subsequent elements so that the image is a subset of The first part of the lemma now follows from the injectivity of the restrictions f | A2 and f | B2 .The second equality is proved similarly.2 Proposition 5.11 Choose k, n and x so that k ≤ n < 2k and 1 ≤ x ≤ n.Then, Proof: We will proceed by induction over n and x.The base cases are given by x = 1, x = n, and n = k.
where the inequality holds for all n < 2k.For x = n and odd k, we have N e (n, k, x) = 0, and for even k, we have The case for N o (n, k, x) and even k is treated identically.
Finally, let n = 2k − 1.If x is odd, then Lemma 5.10 is applicable, so we can assume that x is even.Now, as before where the first term is evaluated using ( 16).The same inequality can be shown for N o (2k − 1, k, x) and even k, which completes the proof. 2 6 Discussion and Open Problems What started out as a very simple idea has diverged in a number of directions, with plenty of room in each for further investigation and improvements.We single out two main topics, and discuss their respective future prospects and interesting open problems.These two topics relate to the application of our approach to the approximability of the problem MAX H -COL (and more generally to MAX CSP(Γ)), and to the computation and interpretation of the separation parameter.

Separation and Approximability
Our initial idea of separation applied to the MAX H -COL problem lead us to a binary graph parameter that, in a sense, measures how close one graph is to be homomorphic to another.While not apparent from the original definition in (3), which involves taking an infimum over all possible instances, we have shown that the parameter can be computed effectively by means of linear programming.Given a graph H and known approximability properties for MAX H -COL, this parameter allows us to deduce bounds on the corresponding properties for MAX H -COL for graphs H that are "close" to H. Our approach can be characterised as local; the closer the separation of two graphs is to 1, the more precise are our bounds.
We have shown that given a "base set" containing the complete graphs, our method can be used to derive good bounds on the approximability of MAX H -COL for any graph H.
For the applications in Section 3, we have used the complete graphs as our base set of known problems.We have shown that this set of graphs is sufficient for achieving new, non-trivial bounds on several different classes of graphs.That is, when we apply Frieze and Jerrum's algorithm (Frieze and Jerrum, 1997) to MAX H -COL, we obtain results comparable to, or better than those guaranteed by Håstad's MAX 2-CSP algorithm (Håstad, 2005), for the classes of graphs we have considered.This comparison should however be taken with a grain of salt.The analysis of Håstad's MAX 2-CSP algorithm only aims to prove it better than a random assignment, and may leave room for strengthening of the approximation guarantee.At the same time, we are overestimating the distance for most of the graphs under consideration.It is likely that both results can be improved, within their respective frameworks.When considering inapproximability, we have relied on the unique games conjecture.Weaker inapproximability results, independent of the UGC, exist for both MAX CUT (Håstad, 2001) and MAX k-CUT (Kann et al., 1997), and they are applicable in our setting.We emphasise that our method is not per se dependent on the truth of the UGC.
For the purpose of extending the applicability of our method, a possible direction to take is to find a larger base set of MAX H -COL problems.We suggest two candidates for further investigation: the circular complete graphs, for which we have obtained partial results for the parameter s in Section 4, and the Kneser graphs, see for example Hell and Nešetřil (2004).Both of these classes generalise the complete graphs, and have been subject to substantial previous research.The Kneser graphs contain many examples of graphs with low clique number, but high chromatic number.They could thus prove to be an ideal starting point for studying this phenomenon in relation to our parameter.
We conclude this part of the discussion by considering some possible extensions of our approximability results.We have already noted that MAX H -COL is a special case of the MAX CSP(Γ) problem, parameterised by a finite constraint language Γ.It should be relatively clear that we can define a generalised separation parameter on a pair of general constraint languages.This would constitute a novel method for studying the approximability of MAX CSP-a method that may cast some new light on the performance of Raghavendra's algorithm.As a way of circumventing the hardness result by Khot et al. (2007), Kaporis et al. (2006) show that mc 2 is approximable within 0.952 for any given average degree d, and asymptotically almost all random graphs G in G(n, m = d 2 n ).Here, G(n, m) is the probability space of random graphs on n vertices and m edges, selected uniformly at random.A different approach is taken by Coja-Oghlan et al. (2005) who give an algorithm that approximates mc k within 1 − O(1/ √ np) in expected polynomial time, for graphs from G(n, p).Kim and Williams (2011) give an algorithm for finding a cut with value at least an additive constant k better than α GW times the value of an optimal cut (provided such a cut exists) in a given graph, if you are willing to spend time exponential in k to do so.In a similar vein, Crowston et al. (2011) show how to use time exponential in k to find a cut better than the Edwards-Erdős bound, i.e., with value e(G)/2 + (n(G) − 1)/4 + k in a given graph G or decide that no such cut exists.It would be interesting to study whether separation can be used to extend these results to improved approximability bounds on MAX H -COL.

Separation as a Graph Parameter
For a graph G with a circular chromatic number r close to 2 we can use Lemma 2.6 to bound s(K 2 , G) ≥ s(K 2 , K r ).Due to Proposition 4.4, we have also seen that with this method, we are unable to distinguish between the class of graphs with circular chromatic number 2 + 1/k and the (larger) class of graphs with circular chromatic number 2 + 2/(2k − 1).Nevertheless, the method is quite effective when applied to sequences of graph classes for which the circular chromatic number tends to 2, as was the case in Proposition 3.11(1)-(3).Much of the extensive study conducted in this direction was instigated by the restriction of a conjecture by Jaeger (1988) to planar graphs.This conjecture is equivalent to the claim that every planar graph of girth at least 4k has a circular chromatic number at most 2 + 1/k, for k ≥ 1.
The case k = 1 is Grötzsch's theorem; that every triangle-free planar graph is 3-colourable.Currently, the best lower bound on the girth of a planar graph which implies a circular chromatic number of at most 2 + 1/k is 20k−2 3 , and is due to Borodin et al. (2004).We remark that Jaeger's conjecture implies a weaker statement in our setting.Namely, if G is a planar graph with girth greater than 4k, then G → C k implies s(K 2 , G) ≥ s(K 2 , C k ) = 2k/(2k + 1).Deciding this to be true would certainly provide support for the original conjecture, and would be an interesting result in its own right.Our starting observation shows that the slightly weaker condition G → K 2+2/(2k−1) implies the same result.
For edge-transitive graphs G, it is not surprising that the expression s(K r , G) assumes a finite number of values, as a function of r.Indeed, Theorem 2.1 states that s(K r , G) = mc Kr (G, 1/e(G)), which leaves at most e(G) possible values for s.This produces a number of constant intervals that are partly responsible for the constant regions of Proposition 4.7, and the discussion in Section 4.2.More surprising are the constant intervals that arise from s(K r , K 2+2/(2k−1) ).They give some hope that the behaviour of the separation parameter can be characterised more generally.We propose investigating the existence of more constant regions, and possibly showing that they tile the entire space.
In Section 5 we generalised the notion of covering by cuts due to Šámal.In doing this, we found a dif-ferent interpretation of the separation parameter as an entire family of chromatic numbers.It is our belief that these alternate viewpoints can benefit from each other.The refuted conjecture in Section 5.3 is an immediate example of this.It is tempting to look for a generalisation of Proposition 5.7 with K 2 replaced by an arbitrary graph H.A trivial upper bound of s(H, H n k ) ≤ k/n is obtained from Proposition 5.5, but we have not identified anything corresponding to the parity criterion which appears in the case H = K 2 .This leads us to believe that this bound can be improved upon.The approach of Šámal on the lower bound does not seem to generalise.The reason for this is that it uses bounds on maximal cuts obtained from the Laplacian of (a subgraph of) Q n/k .We know of no such results for maximal k-cuts, with k > 2, much less for general H-cuts.
In recent work, Šámal and coauthors have shown that the cubical chromatic number χ q can be approximated within α GW (Šámal, 2012).This suggests the interesting possibility of a close connection between the approximability of mc H and that of s(H, G), with H fixed. Let M , N , and H be graphs.A function g : E(M ) → E(N ) is said to be H-cut continuous if, for any H-cut C ⊆ E(N ) in N , we have that g −1 (C) ⊆ E(M ) is an H-cut in M .For any homomorphism h, the edge map h # is H-cut continuous for every H. Šámal (2005) used cut continuous maps (H = K 2 ) to show that certain non-homomorphic graphs have the same cubical chromatic number.Here we show how general H-cut continuous maps can be used to generalise the implication in Lemma 2.6.
where w N (e) = e ∈f −1 (e) w(e ).Let g : V (N ) → V (M ) be an optimal solution to (N, w N ).Then, C = (g # ) −1 (E(M )) is an M -cut in N , so f −1 (C) is an M -cut in H. Hence, there exists a solution to (H, w) which contains precisely the edges in f −1 (C).The measure of this solution is given by w(e) = e∈(g # ) −1 (E(M )) e ∈f −1 (e) w(e ) = m M (g).
Since g is optimal, m M (g) = mc M (N, w N ), and inequality (17) holds. 2 The possibility of efficiently computing (bounds on) s(M, N ) have an immediate application: Lemma 2.6 and Lemma 6.1 give necessary conditions for the existence of a homomorphism N → M .As noted by Šámal (2012), this can be used as a no-homomorphism lemma, proving the absence of a homomorphism between two given graphs.Needless to say, establishing such properties is often a non-trivial task.
otherwise.Let x > 1 and consider N e (n, k, x) for odd k and k < n < 2k − 1. Partition the sets σ ∈ S e (n, k, x) into those for which n ∈ σ on the one hand and those for which n ∈ σ on the other hand.These parts contain N o (n − 1, k − 1, x − 1) and N e (n − 1, k, x − 1) sets, respectively.Since k − 1 is even, and since k Lemma 6.1 Let M , N , and H be graphs in G.If there exists an M -cut continuous map from H to N , then s(M, H) ≥ s(M, N ).Proof: Let f : E(H) → E(N ) be an M -cut continuous function.It suffices to show that for any graph H ∈ G and w ∈ W(H), we have mc M (H, w) ≥ mc M (N, w N ), It remains to show that s(M, N ) is positive.For every edge-weighting w of N , there is at least one edge e with w(e) ≥ 1/|E(N )|.Since M is non-empty, the subgraph consisting of only e maps homomorphically to M , so mc M (N, w) ≥ 1/|E(N )|.By Theorem 2.1, it follows that s(M, N ) ≥ 1/|E(N )| > 0.2 by taking the infimum over n/k for n and k such that G → Q n/k .Here, Q n/k is the graph on vertex set {0, 1} n with an edge between u and v if d H (u, v) ≥ k, where d H denotes the Hamming distance.