Set Polytope Of Claw-free Graphs Research Articles

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

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 combinatorial optimization. In spite of the recent development of fast(er) combinatorial algorithms and some progresses in the characterization of the corresponding stable set polytope, the problem of “providing a decent linear description” for this polytope (Grotschel et al. in Geometric algorithms and combinatorial optimization, Springer, New York, 1988) is still open. The main contribution of this paper is to propose an algorithmic answer to that question by providing a polynomial-time and computationally attractive separation routine for the stable set polytope of claw-free graphs, that only requires a combinatorial decomposition algorithm, the solution of (moderate sized) compact linear programs, and Padberg and Rao’s algorithm for separating over the matching polytope. In particular, it is a generalization of the latter and avoids the heavy computational burden of resorting to the ellipsoid method, on which the only poly-time separation routine known so far relied. Besides, our separation routine comes with a ‘small’ (but not polynomial) extended linear programming formulation and a procedure to derive a linear description of the stable set polytope of claw-free graphs in the original space.

The stable set polytope of icosahedral graphs

The problem of finding a complete linear description of the stable set polytope of claw-free graphs has been a major topic in combinatorial optimization in the last decades and it is still not completely solved. While it is known that this linear description contains facet defining inequalities with arbitrarily many and arbitrarily high coefficients, the set of claw-free graphs whose stable set polytope is described only by inequalities with {0,1,2}-valued coefficients is considerably large. In fact Galluccio et al. (2014) proved that this set contains almost all claw-free graphs with stability number greater than three plus some of the building blocks with stability number smaller than or equal to three, identified by Chudnovsky and Seymour in their Structure Theorem.Here we show that another important class of claw-free graphs with stability number three belongs to this set: the class of icosahedral graphs, named S1 by Chudnovsky and Seymour (2008). In particular, we prove that the stable set polytope of icosahedral graphs is described by: rank, lifted 5-wheel and lifted wedge inequalities, and all these linear inequalities have coefficients in {0,1,2}.

The stable set polytope of claw-free graphs with stability number at least four. II. Striped graphs are [formula omitted]-perfect

In [6], Edmonds provided the first complete description of the polyhedron associated with a combinatorial optimization problem: the matching polytope. As the matching problem is equivalent to the stable set problem on line graphs, many researchers tried to generalize Edmonds' result by considering the stable set problem on a superclass of line graphs: the claw-free graphs. However, as testified also by Grötschel, Lovász, and Schrijver [14], “in spite of considerable efforts, no decent system of inequalities describingSTAB(G)for claw-free graphs is known”.Here, we provide an explicit linear description of the stable set polytope of claw-free graphs with stability number at least four and with no 1-join.

The stable set polytope of claw-free graphs with stability number at least four. I. Fuzzy antihat graphs are [formula omitted]-perfect

Fuzzy antihat graphs are graphs obtained as 2-clique-bond compositions of fuzzy line graphs with three different types of three-cliqued graphs. By the decomposition theorem of Chudnovsky and Seymour [2], fuzzy antihat graphs form a large subclass of claw-free, not quasi-line graphs with stability number at least four and with no 1-joins.A graph is W-perfect if its stable set polytope is described by: nonnegativity, rank, and lifted 5-wheel inequalities. By exploiting the polyhedral properties of the 2-clique-bond composition, we prove that fuzzy antihat graphs are W-perfect and we move a crucial step towards the solution of the longstanding open question of finding an explicit linear description of the stable set polytope of claw-free graphs.

The stable set polytope of claw-free graphs with large stability number

Abstract In this paper we give an explicit description of the stable set polytope of a claw-free graph obtained by repeated applications of the strip composition of fuzzy linear interval strips, fuzzy XX-strips, and fuzzy antihat strips. Using a decomposition theorem of Chudnovsky and Seymour, this allows us to describe the stable set polytope of all facet defining claw-free graphs with stability number greater than 3.

On facets of stable set polytopes of claw-free graphs with stability number 3

Obtaining a complete description of the stable set polytopes of claw-free graphs is a long-standing open problem. Eisenbrand et al. recently achieved a breakthrough for the subclass of quasi-line graphs. As a consequence, every non-trivial facet of their stable set polytope has at most two different, but arbitrarily high left hand side coefficients. For the graphs with stability number 2, Cook showed that all their non-trivial facets are 1/2-valued. For claw-free but not quasi-line graphs with stability number at least 4, Stauffer conjectured that the same holds true. In contrast, there are known claw-free graphs with stability number 3 which induce facets with up to eight different left hand side coefficients. We prove that the situation is even worse: for every positive integer b , we exhibit a claw-free graph with stability number 3 inducing a facet with b different left hand side coefficients.

The stable set polytope of quasi-line graphs

It is a long standing open problem to find an explicit description of the stable set polytope of claw-free graphs. Yet more than 20 years after the discovery of a polynomial algorithm for the maximum stable set problem for claw-free graphs, there is even no conjecture at hand today. Such a conjecture exists for the class of quasi-line graphs. This class of graphs is a proper superclass of line graphs and a proper subclass of claw-free graphs for which it is known that not all facets have 0/1 normal vectors. The Ben Rebea conjecture states that the stable set polytope of a quasi-line graph is completely described by clique-family inequalities. Chudnovsky and Seymour recently provided a decomposition result for claw-free graphs and proved that the Ben Rebea conjecture holds, if the quasi-line graph is not a fuzzy circular interval graph. In this paper, we give a proof of the Ben Rebea conjecture by showing that it also holds for fuzzy circular interval graphs. Our result builds upon an algorithm of Bartholdi, Orlin and Ratliff which is concerned with integer programs defined by circular ones matrices.

On facets of stable set polytopes of claw-free graphs with stability number three

Abstract Providing a complete description of the stable set polytopes of claw-free graphs is a long-standing open problem since almost twenty years. Eisenbrandt et al. recently achieved a breakthrough for the subclass of quasi-line graphs. As a consequence, every non-trivial facet of their stable set polytope is of the form k ∑ v ∈ V 1 x v + ( k + 1 ) ∑ v ∈ V 2 x v ⩽ b for some positive integers k and b, and non-empty sets of vertices V 1 and V 2 . Roughly speaking, this states that the facets of the stable set polytope of quasi-line graphs have at most two left coefficients. For stable set polytopes of claw-free graphs with maximum stable set size at least four, Stauffer conjectured in 2005 that this still holds. It is already known that some stable set polytopes of claw-free graphs with maximum stable set size three may have facets with up to 5 left coefficients. We prove that the situation is even worse: for every positive integer b, there is a claw-free graph with stability number three whose stable set polytope has a facet with b left coefficients.

A construction for non-rank facets of stable set polytopes of webs

Graphs with circular symmetry, called webs, are relevant w.r.t. describing the stable set polytopes of two larger graph classes, quasi-line graphs [G. Giles, L.E. Trotter Jr., On stable set polyhedra for K 1 , 3 -free graphs, J. Combin. Theory B 31 (1981) 313–326; G. Oriolo, Clique family inequalities for the stable set polytope for quasi-line graphs, in: Stability Problems, Discrete Appl. Math. 132 (2003) 185–201 (special issue)] and claw-free graphs [A. Galluccio, A. Sassano, The rank facets of the stable set polytope for claw-free graphs, J. Combin. Theory B 69 (1997) 1–38; G. Giles, L.E. Trotter Jr., On stable set polyhedra for K 1 , 3 -free graphs, J. Combin. Theory B 31 (1981) 313–326]. Providing a decent linear description of the stable set polytopes of claw-free graphs is a long-standing problem [M. Grötschel, L. Lovász, A. Schrijver, Geometric Algorithms and Combinatorial Optimization, Springer-Verlag, 1988]. However, even the problem of finding all facets of stable set polytopes of webs is open. So far, it is only known that stable set polytopes of webs with clique number ≤ 3 have rank facets only [G. Dahl, Stable set polytopes for a class of circulant graphs, SIAM J. Optim. 9 (1999) 493–503; L.E. Trotter, Jr., A class of facet producing graphs for vertex packing polyhedra, Discrete Math. 12 (1975) 373–388] while there are examples with clique number ≥ 4 having non-rank facets [J. Kind, Mobilitätsmodelle für zellulare Mobilfunknetze: Produktformen und Blockierung, Ph.D. Thesis, RWTH Aachen, 2000; T.M. Liebling, G. Oriolo, B. Spille, G. Stauffer, On non-rank facets of the stable set polytope of claw-free graphs and circulant graphs, Math. Methods Oper. Res. 59 (2004) 25; G. Oriolo, Clique family inequalities for the stable set polytope for quasi-line graphs, in: Stability Problems, Discrete Appl. Math. 132 (2003) 185–201 (special issue); A. Pêcher, A. Wagler, On non-rank facets of stable set polytopes of webs with clique number four, Discrete Appl. Math. 154 (2006) 1408–1415]. In this paper, we provide a construction for non-rank facets of stable set polytopes of webs. This construction is the main tool to obtain in a companion paper [A. Pêcher, A. Wagler, Almost all webs are not rank-perfect, Math. Program 105 (2006) 311–328], for all fixed values of ω ≥ 5 that there are only finitely many webs with clique number ω whose stable set polytopes admit rank facets only.

A construction for non-rank facets of stable set polytopes of webs

Graphs with circular symmetry, called webs, are relevant w.r.t, describing the stable set polytopes of two larger graph classes, quasi-line graphs [G. Giles, L.E. Trotter Jr., On stable set polyhedra for K1,3-free graphs, J. Combin. Theory B 31 (1981) 313-326; G. Oriolo, Clique family inequalities for the stable set polytope for quasi-line graphs, in: Stability Problems, Discrete Appl. Math. 132 (2003) 185-201 (special issue)] and claw-free graphs [A. Galluccio, A. Sassano, The rank facets of the stable set polytope for claw-free graphs, J. Combin. Theory B 69 (1997) 1-38; G. Giles, L.E. Trotter Jr., On stable set polyhedra for K1,3-free graphs, J. Combin. Theory B 31 (1981) 313-326]. Providing a decent linear description of the stable set polytopes of claw-free graphs is a long-standing problem [M. Grotschel, L. Lovasz, A. Schrijver, Geometric Algorithms and Combinatorial Optimization, Springer-Verlag, 1988]. However, even the problem of finding all facets of stable set polytopes of webs is open. So far, it is only known that stable set polytopes of webs with clique number ≤ 3 have rank facets only [G. Dahl, Stable set polytopes for a class of circulant graphs, SIAM J. Optim. 9 (1999) 493-503; L.E. Trotter, Jr., A class of facet producing graphs for vertex packing polyhedra, Discrete Math. 12 (1975) 373-388] while there are examples with clique number ≥ 4 having non-rank facets [J. Kind, Mobilitatsmodelle fur zellulare Mobilfunknetze: Produktformen und Blockierung, Ph.D. Thesis, RWTH Aachen, 2000; T.M. Liebling, G. Oriolo, B. Spille, G. Stauffer, On non-rank facets of the stable set polytope of claw-free graphs and circulant graphs, Math. Methods Oper. Res. 59 (2004) 25; G. Oriolo, Clique family inequalities for the stable set polytope for quasi-line graphs, in: Stability Problems, Discrete Appl. Math. 132 (2003) 185-201 (special issue); A. Pecher, A. Wagler, On non-rank facets of stable set polytopes of webs with clique number four, Discrete Appl. Math. 154 (2006) 1408-1415].In this paper, we provide a construction for non-rank facets of stable set polytopes of webs. This construction is the main tool to obtain in a companion paper [A. Pecher, A. Wagler, Almost all webs are not rank-perfect, Math. Program 105 (2006) 311-328], for all fixed values of ω ≥ 5 that there are only finitely many webs with clique number ω whose stable set polytopes admit rank facets only.

Generalized clique family inequalities for claw-free graphs

Abstract Providing a complete description of the stable set polytopes of claw-free graphs is a long-standing open problem [M. Grotschel, L. Lovasz, A. Schrijver, Geometric Algorithms and Combinatorial Optimization, Springer-Verlag, 1988]. For the subclass of quasi-line graphs, Ben Rebea conjectured and Eisenbrandt et al. [F. Eisenbrand, G. Oriolo, G. Stauffer, P. Ventura, Circular Ones Matrices and the Stable Set Polytope of Quasi-Line Graphs, in: Lecture Notes in Computer Science, 3509, 2005, pp. 291–305] recently proved that all non-trivial facets belong to only one class, the so-called clique family inequalities. For general claw-free graphs, however, more complex facets are required to describe the stable set polytope. We introduce a generalization of clique family inequalities, and prove that several facet-defining inequalities for general claw-free graphs are of this type.

On non-rank facets of stable set polytopes of webs with clique number four

Graphs with circular symmetry, called webs, are relevant for describing the stable set polytopes of two larger graph classes, quasi-line graphs and claw-free graphs. Providing a decent linear description of the stable set polytopes of claw-free graphs is a long-standing problem. However, even the problem of finding all facets of stable set polytopes of webs is open. So far, it is only known that stable set polytopes of webs with clique number ⩽ 3 have rank facets only while there are examples with clique number > 4 having non-rank facets. The aim of the present paper is to treat the remaining case with clique number = 4 : we provide an infinite sequence of such webs whose stable set polytopes admit non-rank facets.

Almost all webs are not rank-perfect

Graphs with circular symmetry, called webs, are crucial for describing the stable set polytopes of two larger graph classes, quasi-line graphs[8,12] and claw-free graphs [7,8]. Providing a complete linear description of the stable set polytopes of claw-free graphs is a long-standing problem [9]. Ben Rebea conjectured a description for quasi-line graphs, see [12]; Chudnovsky and Seymour [2] verified this conjecture recently for quasi-line graphs not belonging to the subclass of fuzzy circular interval graphs and showed that rank facets are required in this case only. Fuzzy circular interval graphs contain all webs and even the problem of finding all facets of their stable set polytopes is open. So far, it is only known that stable set polytopes of webs with clique number ≤3 have rank facets only [5,17] while there are examples with clique number ≥4 having non-rank facets [10_12,15].In this paper we prove, building on a construction for non-rank facets from [16], that the stable set polytopes of almost all webs with clique number ≥5 admit non-rank facets. This adds support to the belief that these graphs are indeed the core of Ben Rebea's conjecture. Finally, we present a conjecture how to construct all facets of the stable set polytopes of webs.

On non-rank facets of the stable set polytope of claw-free graphs and circulant graphs

We deal with non-rank facets of the stable set polytope of claw-free graphs. We extend results of Giles and Trotter [7] by (i) showing that for any nonnegative integer a there exists a circulant graph whose stable set polytope has a facet-inducing inequality with (a,a+1)-valued coefficients (rank facets have only coefficients 0, 1), and (ii) providing new facets of the stable set polytope with up to five different non-zero coefficients for claw-free graphs. We prove that coefficients have to be consecutive in any facet with exactly two different non-zero coefficients (assuming they are relatively prime). Last but not least, we present a complete description of the stable set polytope for graphs with stability number 2, already observed by Cook [3] and Shepherd [18].

On Non-Rank Facets of Stable Set Polytopes of Webs with Clique Number Four–Extended Abstract–

Abstract Graphs with circular symmetry, called webs, are relevant w.r.t. describing the stable set polytopes of two larger graph classes, quasi-line graphs [6,10] and claw-free graphs [5,6]. Providing a decent linear description of the stable set polytopes of claw-free graphs is a long-standing problem [7]. However, even the problem of finding all facets of stable set polytopes of webs is open. So far, it is only known that stable set polytopes of webs with clique number ≤ 3 have rank facets only [3,12] while there are examples with clique number > 4 having non-rank facets [8,10]. The aim of the present paper is to treat the remaining case with clique number = 4: we provide an infinite sequence of such webs whose stable set polytopes admit non-rank facets.

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Graphs with circular symmetry, called webs, are relevant w.r.t. describing the stable set polytopes of two larger graph classes, quasi-line graphs [G. Giles, L.E. Trotter Jr., On stable set polyhedra for K1,3-free graphs, J. Combin. Theory B 31 (1981) 313–326; G. Oriolo, Clique family inequalities for the stable set polytope for quasi-line graphs, in: Stability Problems, Discrete Appl. Math. 132 (2003) 185–201 (special issue)] and claw-free graphs [A. Galluccio, A. Sassano, The rank facets of the stable set polytope for claw-free graphs, J. Combin. Theory B 69 (1997) 1–38; G. Giles, L.E. Trotter Jr., On stable set polyhedra for K1,3-free graphs, J. Combin. Theory B 31 (1981) 313–326]. Providing a decent linear description of the stable set polytopes of claw-free graphs is a long-standing problem [M. Grötschel, L. Lovász, A. Schrijver, Geometric Algorithms and Combinatorial Optimization, Springer-Verlag, 1988]. However, even the problem of finding all facets of stable set polytopes of webs is open. So far, it is only known that stable set polytopes of webs with clique number ≤3 have rank facets only [G. Dahl, Stable set polytopes for a class of circulant graphs, SIAM J. Optim. 9 (1999) 493–503; L.E. Trotter, Jr., A class of facet producing graphs for vertex packing polyhedra, Discrete Math. 12 (1975) 373–388] while there are examples with clique number ≥4 having non-rank facets [J. Kind, Mobilitätsmodelle für zellulare Mobilfunknetze: Produktformen und Blockierung, Ph.D. Thesis, RWTH Aachen, 2000; T.M. Liebling, G. Oriolo, B. Spille, G. Stauffer, On non-rank facets of the stable set polytope of claw-free graphs and circulant graphs, Math. Methods Oper. Res. 59 (2004) 25; G. Oriolo, Clique family inequalities for the stable set polytope for quasi-line graphs, in: Stability Problems, Discrete Appl. Math. 132 (2003) 185–201 (special issue); A. Pêcher, A. Wagler, On non-rank facets of stable set polytopes of webs with clique number four, Discrete Appl. Math. 154 (2006) 1408–1415].In this paper, we provide a construction for non-rank facets of stable set polytopes of webs. This construction is the main tool to obtain in a companion paper [A. Pêcher, A. Wagler, Almost all webs are not rank-perfect, Math. Program 105 (2006) 311–328], for all fixed values of ω≥5 that there are only finitely many webs with clique number ω whose stable set polytopes admit rank facets only.