On the set covering polyhedron of circulant matrices

A well known family of minimally nonideal matrices is the family of the incidence matrices of chordless odd cycles. A natural generalization of these matrices is given by the family of circulant matrices. Ideal and minimally nonideal circulant matrices have been completely identified by Cornuéjols and Novick [G. Cornuéjols, B. Novick, Ideal 0 - 1 matrices, Journal of Combinatorial Theory B 60 (1994) 145–157]. In this work we classify circulant matrices and their blockers in terms of the inequalities involved in their set covering polyhedra. We exploit the results due to Cornuéjols and Novick in the above-cited reference for describing the set covering polyhedron of blockers of circulant matrices. Finally, we point out that the results found on circulant matrices and their blockers present a remarkable analogy with a similar analysis of webs and antiwebs due to Pêcher and Wagler [A. Pêcher, A. Wagler, A construction for non-rank facets of stable set polytopes of webs, European Journal of Combinatorics 27 (2006) 1172–1185; A. Pêcher, A. Wagler, Almost all webs are not rank-perfect, Mathematical Programming Series B 105 (2006) 311–328] and Wagler [A. Wagler, Relaxing perfectness: Which graphs are ‘Almost’ perfect?, in: M. Groetschel (Ed.), The Sharpest Cut, Impact of Manfred Padberg and his work, in: SIAM/MPS Series on Optimization, vol. 4, Philadelphia, 2004; A. Wagler, Antiwebs are rank-perfect, 4OR 2 (2004) 149–152].     

List-Coloring Claw-Free Graphs with Small Clique Number

Chudnovsky and Seymour proved that every connected claw-free graph that contains a stable set of size 3 has chromatic number at most twice its clique number. We improve this for small clique size, showing that every claw-free graph with clique number at most 3 is 4-choosable and every claw-free graph with clique number at most 4 is 7-choosable. These bounds are tight.     

On the complete set packing and set partitioning polytopes: Properties and rank 1 facets

This paper studies two polytopes: the complete set packing and set partitioning polytopes, which are both associated with a binary $n$-row matrix having all possible columns. Cuts of rank 1 for the latter polytope play a central role in recent exact algorithms for many combinatorial problems, such as vehicle routing. We show the precise relation between the two polytopes studied, characterize the multipliers that induce rank 1 clique facets and give several families of multipliers that yield other facets.     

On the Choosability of Claw-Free Perfect Graphs

It has been conjectured that for every claw-free graph G the choice number of G is equal to its chromatic number. We focus on the special case of this conjecture where G is perfect. Claw-free perfect graphs can be decomposed via clique-cutset into two special classes called elementary graphs and peculiar graphs. Based on this decomposition we prove that the conjecture holds true for every claw-free perfect graph with maximum clique size at most 4.     

Claw-free graphs. III. Circular interval graphs

Construct a graph as follows. Take a circle, and a collection of intervals from it, no three of which have union the entire circle; take a finite set of points V from the circle; and make a graph with vertex set V in which two vertices are adjacent if they both belong to one of the intervals. Such graphs are “long circular interval graphs,” and they form an important subclass of the class of all claw-free graphs. In this paper we characterize them by excluded induced subgraphs. This is a step towards the main goal of this series, to find a structural characterization of all claw-free graphs.This paper also gives an analysis of the connected claw-free graphs G with a clique the deletion of which disconnects G into two parts both with at least two vertices.     

Characterizing and bounding the imperfection ratio for some classes of graphs

Perfect graphs constitute a well-studied graph class with a rich structure, which is reflected by many characterizations with respect to different concepts. Perfect graphs are, for instance, precisely those graphs G where the stable set polytope STAB(G) equals the fractional stable set polytope QSTAB(G). The dilation ratio of the two polytopes yields the imperfection ratio of G. It is NP-hard to compute and, for most graph classes, it is even unknown whether it is bounded. For graphs G such that all facets of STAB(G) are rank constraints associated with antiwebs, we characterize the imperfection ratio and bound it by 3/2. Outgoing from this result, we characterize and bound the imperfection ratio for several graph classes, including near-bipartite graphs and their complements, namely quasi-line graphs, by means of induced antiwebs and webs, respectively.      

Combinatorial optimization with 2-joins

A 2-join is an edge cutset that naturally appears in decomposition of several classes of graphs closed under taking induced subgraphs, such as perfect graphs and claw-free graphs. In this paper we construct combinatorial polynomial time algorithms for finding a maximum weighted clique, a maximum weighted stable set and an optimal coloring for a class of perfect graphs decomposable by 2-joins: the class of perfect graphs that do not have a balanced skew partition, a 2-join in the complement, nor a homogeneous pair. The techniques we develop are general enough to be easily applied to finding a maximum weighted stable set for another class of graphs known to be decomposable by 2-joins, namely the class of even-hole-free graphs that do not have a star cutset.We also give a simple class of graphs decomposable by 2-joins into bipartite graphs and line graphs, and for which finding a maximum stable set is NP-hard. This shows that having holes all of the same parity gives essential properties for the use of 2-joins in computing stable sets.     

A fast algorithm for the maximum clique problem

Given a graph, in the maximum clique problem, one desires to find the largest number of vertices, any two of which are adjacent. A branch-and-bound algorithm for the maximum clique problem—which is computationally equivalent to the maximum independent (stable) set problem—is presented with the vertex order taken from a coloring of the vertices and with a new pruning strategy. The algorithm performs successfully for many instances when applied to random graphs and DIMACS benchmark graphs.     

A generalization of antiwebs to independence systems and their canonical facets

We consider independence system polytopes, i.e. polytopes whose extreme points are the incidence vectors of the sets of an independence system. We first give a sufficient condition for recognizing Boolean facets. Then, the notion of antiweb introduced by Trotter for graphs is generalized to independence systems and used for obtaining canonical facets of the associated polytopes. We also point out how our results relate with known ones for knapsack, set covering and matroid polytopes.     

Gal’s conjecture for nestohedra corresponding to complete bipartite graphs

Convex polytopes have interested mathematicians since very ancient times. At present, they occupy a central place in convex geometry, combinatorics, and toric topology and demonstrate the harmony and beauty of mathematics. This paper considers the problem of describing the f-vectors of simple flag polytopes, that is, simple polytopes in which any set of pairwise intersecting facets has nonempty intersection. We show that for each nestohedron corresponding to a connected building set, the h-polynomial is a descent-generating function for some class of permutations; we also prove Gal’s conjecture on the nonnegativity of γ-vectors of flag polytopes for nestohedra constructed over complete bipartite graphs.     

无爪图上团横贯数的界

设 G=(V,E) 为简单图,图 G 的每个至少有两个顶点的极大完全子图称为 G 的一个团. 一个顶点子集 S\subseteq V 称为图 G 的团横贯集, 如果 S 与 G 的所有团都相交,即对于 G 的任意的团 C 有 S\cap{V(C)}\neq\emptyset. 图 G 的团横贯数是图 G 的最小团横贯集所含顶点的数目,记为~${\large\tau}_{C}(G)$. 证明了棱柱图的补图(除5-圈外)、非奇圈的圆弧区间图和 Hex-连接图这三类无爪图的团横贯数不超过其阶数的一半.     

Edge Coloring of Split Graphs

A split graph is a graph whose vertex set admits a partition into a stable set and a clique. The chromatic indexes for some subsets of split graphs, such as split graphs with odd maximum degree and split-indifference graphs, are known. However, for the general class, the problem remains unsolved. This paper presents new results about the classification problem for split graphs as a contribution in the direction of solving the entire problem for this class.     

A polynomial algorithm to find an independent set of maximum weight in a fork-free graph

The class of fork-free graphs is an extension of claw-free graphs and their subclass of line graphs. The first polynomial-time solution to the maximum weight independent set problem in the class of line graphs, which is equivalent to the maximum matching problem in general graphs, has been proposed by Edmonds in 1965 and then extended to the entire class of claw-free graphs by Minty in 1980. Recently, Alekseev proposed a solution for the larger class of fork-free graphs, but only for the unweighted version of the problem, i.e., finding an independent set of maximum cardinality. In the present paper, we describe the first polynomial-time algorithm to solve the problem for weighted fork-free graphs.     

On stable cutsets in claw-free graphs and planar graphs

A stable cutset in a connected graph is a stable set whose deletion disconnects the graph. Let $K_4$ and $K_{1,3}$ (claw) denote the complete (bipartite) graph on 4 and $1+3$ vertices. It is NP-complete to decide whether a line graph (hence a claw-free graph) with maximum degree five or a $K_4$-free graph admits a stable cutset. Here we describe algorithms deciding in polynomial time whether a claw-free graph with maximum degree at most four or whether a (claw, $K_4$)-free graph admits a stable cutset. As a by-product we obtain that the stable cutset problem is polynomially solvable for claw-free planar graphs, and also for planar line graphs.Thus, the computational complexity of the stable cutset problem is completely determined for claw-free graphs with respect to degree constraint, and for claw-free planar graphs. Moreover, we prove that the stable cutset problem remains NP-complete for $K_4$-free planar graphs with maximum degree five.     

Separation routine and extended formulations for the stable set problem in claw-free graphs

Mathematical Programming - The maximum weighted stable set problem in claw-free graphs is a well-known generalization of the maximum weighted matching problem, and a classical problem in...