Polyhedral Sets and Integer Rounding

Thomas Niessen and Jaakob Kind

Institute of Statistics, RWTH Aachen, 52056 AachenE-mail: {kind,niessen}@stochastik.rwth-aachen.de

1 Introduction

Let G be a graph with vertex set V and let k be a non-negative integer. A k-multi-coloring ϕ assigns to every vertex of G a subset of {1, 2, . . . , k} such thatadjacent vertices receive disjoint sets. A vector n ∈ Z

V+ is called k-colorable

iff there is a k-multi-coloring ϕ of G such that n(v) = |ϕ(v)| for every vertexv ∈ V , i.e. n counts the number of colors assigned to the vertices of G. Anequivalent formulation uses the graph G ∗ n, i.e., the graph obtained from Gby replacing each vertex v by a clique of order n(v) and joining vertices of twocliques completely by edges if the corresponding vertices of G are adjacent.Then, n is k-colorable iff G∗n is k-colorable in the ordinary sense. By Sk(G) wedenote the set of all vectors being k-colorable with respect to G. Motivated byapplications in mobile telecommunication networks, we looked for alternativedescriptions of Sk(G).

Obviously, S1(G) is the set of all incidence vectors of independent sets in G.Hence conv(S1(G)), i.e. the convex hull of S1(G), is the stable set polytopeSTAB(G). It is easily verified that S2(G) is the set of integral vectors in2 · STAB(G). In general it hold conv(Sk(G)) = k · STAB(G) and Sk(G) ⊆(k · STAB(G)) ∩ Z

V+, where the inclusion can be strict for k ≥ 3. However,

a result of Lovasz states that equality holds for perfect graphs. Additionally,the stable set polytope of perfect graphs is described by the so-called cliqueconstraints, and thus Sk(G) is nicely described for every perfect graph G andevery non-negative integer k.

These observations led to the following definition. A finite set S ⊂ Zn is called

polyhedral iff there is a polyhedron P ⊂ Rn such that S = P ∩Z

n. Of course, Sis polyhedral iff conv(S) is a suitable choice for P , that is, S = conv(S)∩ Z

n.However, our definition has some advantages as we will see below.

1 Partly supported by Deutsche Forschungsgemeinschaft grant Ma1184/4-4

Preprint submitted to Elsevier Preprint 11 May 1999

In the next section we present characterizations of polyhedrality for sets de-fined by a combinatorial optimization problem (like Sk(G)). Thereby we gen-eralize results of Baum and Trotter on integer rounding and integer decom-position properties. In section 3, we give an example for the use of polyhedralsets. Therefore we sketch the proof of the following result: the chromatic num-ber of every proper circular-arc graph can be obtained by rounding up thefractional chromatic number of the graph to the next integer. In the talk, wewill also present a further application: we present all pairs (G, k) such thatSk(G) is described by the clique constraints.

2 Characterizations of polyhedral sets

Let M be a (m × n)-matrix with non-negative integral entries and withoutzero columns. For every n ∈ Z

n+ we consider the covering problem

γ(n) = γ(n,M) = min{1 ·w : wM ≥ n,w ∈ Zm+}

and its fractional analogon

γf(n) = γf(n,M) = min{1 ·w : wM ≥ n,w ≥ 0}.

Furthermore, we let Sk(M) = {n ∈ Zn+ : γ(n) ≤ k} for k ∈ Z+. Note that

S1(M) is the set of all integral vectors n such that 0 ≤ n ≤ m holds for somerow m of M. Note that Sk(G) = Sk(M), if M is a matrix whose columnsare the incidence vectors of independent sets of the graph G. In the followingwe assume that S1(M) is polyhedral. This requirement is always satisfied forS1(G) or for S1(M), respectively, if M is a 0-1 matrix.

Let P (M) = conv(S1(M)). Then, conv(Sk(M)) = k·P (M) and conv(Sk(M))∩Z

n+ = {n ∈ Z

n+ : γf(n) ≤ k} are easy to verify. Thereby, we obtain

Theorem 1. For every k ∈ Z+ the following statements are equivalent.(i) Sk(M) is polyhedral,(ii) for every n ∈ Z

n+: γf(n) ≤ k =⇒ γ(n) ≤ k,

(iii) every integral vector in k ·P (M) is the sum of k integral vectors in P (M).

Corrollary 2. Let K ∈ Z+ ∪ {∞}. Then the following statements are equiv-alent:(i) Sk(M) is polyhedral for every non-negative integer k ≤ K,(ii) for every n ∈ Z

n+ with γf(n) ≤ K it holds γ(n) = �γf (n)�.

(iii) for every non-negative integer k ≤ K and every integral vector n ∈k · P (M)

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there are k integral vectors in P (M) such that their sum is n.

For K = ∞, the equivalence of (ii) and (iii) is a result of Baum and Trotter [1],when (iii) is called the integer decomposition property. To see this one needsonly to observe that for every integral, full-dimensional polytope P there is amatrix M such that P = P (M) (e.g., M can be chosen as the matrix whosecolumns are the integral vectors in P ).

Our proof is easier than that of [1]. We have similar results for packing prob-lems, which are related in the same way to the integer round-down resultsfrom [1].

3 Coloring of proper circular-arc graphs

A graph is a proper circular-arc graph (pca-graph) if it is the intersection graphof a set of arcs of a circle such that no arc is properly contained in anotherone. For every pca-graph G it holds ω(G) ≤ χ(G) ≤ 3ω(G)/2, where ω(G)denotes the clique number of G. Both bounds are best possible. The fractionalchromatic number χf(G) and the fractional clique number ωf (G) of any graphG satisfy ω(G) ≤ ωf(G) = χf (G) ≤ χ(G). Using the concept of polyhedralsets we proved

Theorem 3. For every pca-graph G we have χ(G) = �χf (G)�.Since G ∗ n is a pca-graph if G is, Theorem 3 states in fact that χ(G ∗ n) =�χf (G ∗ n)� holds for every pca-graph G and every n ∈ Z

V+. This property

is already known for perfect graphs [3], near-perfect graphs [5], series-parallelgraphs [2], and line-graphs of series-parallel graphs [4].

By Corollary 2, it suffices to verify that Sk(G) is polyhedral for every pca-graph G and every non-negative integer k. This is done in two steps. First, weprove that the intersection of Sk(G) with certain hyperplanes Hp is polyhe-dral. This step is essentially based on a generalization of a coloring algorithmof Teng and Tucker [6]. An important detail has to be mentioned. The poly-tope conv(Sk(G) ∪ Hp) is only given as a projection of another polyhedron.Hence, no information on STAB(G) is used and only little information on itis provided.

The second step consists of showing that polyhedrality of the intersections isalready sufficient for Sk(G) being polyhedral. This part of the proof works forevery claw-free graph G, since it is obtained by the well-known recoloring ofalternating chains.

Theorem 3 is best possible in the following sense. It cannot be generalized to

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the classes of circular-arc graphs or circle graphs, respectively, unless P=NP.Both classes contain the class of pca-graphs. However, we do not know anycounterexamples explicitely.

4 An open problem

We are interested in a solution for the following problem: is it true that Sk(G)is polyhedral, if SK(G) is polyhedral for some K ≥ k? An affirmative answerwould allow to combine the statements of Theorem 1 and Corollary 2.

In the talk we will discuss this problem in greater detail.

References

[1] S. Baum and L. E. Trotter, Integer rounding for polymatroid and branchingoptimization problems. SIAM J. Algebraic Disc. Meths. 2 (1981), 416–425.

[2] K. Kilakos and O. Marcotte, Fractional and integral colourings.Math. Programming 76 (1997), 333–347.

[3] L. Lovasz, Normal hypergraphs and the perfect graph conjecture. Disc. Math.2 (1972), 253–267.

[4] P. D. Seymour, Colouring series-parallel graphs. Combinatorica 10 (1990), 379–392.

[5] F. B. Shepherd, Near-perfect matrices. Math. Programming 64 (1994), 295–323.[6] A. Teng and A. Tucker, An O(qn) algorithm to q-color a proper family of

circular arcs. Disc. Math. 55 (1985), 233–243.

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