Nontrivial Facets Research Articles

Geometry of the skeleton graph of the no-signaling polytope for the 2223 Bell scenario and beyond

The study of no-signaling polytopes is addressed under an alternative approach, focusing on the skeleton graph of the polytopes rather than on the Bell inequalities solely. The paper starts with the study of the Bell scenario where one of the parties (Bob) has three output options (2223 scenario). We found that in this polytope the Popescu-Rohrlich (PR) vertices are interconnected, making up two networks around the polytope of local correlations, which is a new feature in relation to the 2222 polytope. We also found that in this scenario each CHSH inequality is violated by five PR vertices: the PR vertex associated to that inequality and its four adjacent PR vertices. Later, we study the facet structure of the corresponding polytope of local correlations. What we obtain is a one-to-one correspondence between some two-dimensional faces of the no-signaling polytope defined by specific loops in the adjacency networks of PR vertices, the lifting Bell functions, and the nontrivial facets of the local polytope. Finally, by studying the skeleton graphs of the polytopes for the general $222{v}_{B}$ scenarios, where ${v}_{B}$ is the number of possible Bob's outcomes, a striking high degeneracy is unveiled. We show that the set of local vertices exhibits a much higher degeneracy than the PR vertices, which in turn are more degenerate than a regular vertex.

The Facets of the Bases Polytope of a Matroid and Two Consequences

Let M be a matroid defined on a finite set E and L ⊂ E . L is locked in M if and are 2-connected, and . In this paper, we prove that the nontrivial facets of the bases polytope of M are described by the locked subsets. We deduce that finding the maximum-weight basis of M is a polynomial time problem for matroids with a polynomial number of locked subsets. This class of matroids is closed under 2-sums and contains the class of uniform matroids, the Vamos matroid and all the excluded minors of 2-sums of uniform matroids. We deduce also a matroid oracle for testing uniformity of matroids after one call of this oracle.

On imposing connectivity constraints in integer programs

In many network applications, one searches for a connected subset of vertices that exhibits other desirable properties. To this end, this paper studies the connected subgraph polytope of a graph, which is the convex hull of subsets of vertices that induce a connected subgraph. Much of our work is devoted to the study of two nontrivial classes of valid inequalities. The first are the a, b-separator inequalities, which have been successfully used to enforce connectivity in previous computational studies. The second are the indegree inequalities, which have previously been shown to induce all nontrivial facets for trees. We determine the precise conditions under which these inequalities induce facets and when each class fully describes the connected subgraph polytope. Both classes of inequalities can be separated in polynomial time and admit compact extended formulations. However, while the a, b-separator inequalities can be lifted in linear time, it is NP-hard to lift the indegree inequalities.

Multipartite causal correlations: Polytopes and inequalities

We consider the most general correlations that can be obtained by a group of parties whose causal relations are well-defined, although possibly probabilistic and dependent on past parties' operations. We show that, for any fixed number of parties and inputs and outputs for each party, the set of such correlations forms a convex polytope, whose vertices correspond to deterministic strategies, and whose (nontrivial) facets define so-called causal inequalities. We completely characterize the simplest tripartite polytope in terms of its facet inequalities, propose generalizations of some inequalities to scenarios with more parties, and show that our tripartite inequalities can be violated within the process matrix formalism, where quantum mechanics is locally valid but no global causal structure is assumed.

Master corner polyhedron: Vertices

We focus on the vertices of the master corner polyhedron (MCP), a fundamental object in the theory of integer linear programming. We introduce two combinatorial operations that transform vertices to their neighbors. This implies that each MCP can be defined by the initial vertices regarding these operations; we call them support vertices. We prove that the class of support vertices of all MCPs over a group is invariant under automorphisms of this group and describe MCP vertex bases. Among other results, we characterize its irreducible points, establish relations between a vertex and the nontrivial facets that pass through it, and prove that this polyhedron is of diameter 2.

Cyclic group blocking polyhedra

Gomory (Linear Algebra Appl 2:451–558, 1969) gave a subadditive characterization of the facets of the group polyhedron. Although there are exponentially many facets (see Gomory and Johnson in Math Program 3:359–389, 1972, Example 4.6) and exponentially many vertices for the group polyhedron for the master cyclic group problem, Gomory’s characterization of the non-trivial facets has polynomially many subadditive inequalities, in fact of order |G|2 for a finite Abelian group G. We reduce this subadditive inequality system to its minimal representation by a triple system of the same order and show the dimensionality of the polytope of non-trivial facets. The system of all triples corresponds to all solution vectors of length three into which every solution vector can be decomposed. Our minimal representation leads to a characterization of the vertices of length three. Gomory et al. (Math Program 96:321–339, 2003) introduced a shooting experiment involving solving the shooting linear program repeatedly to find important facets. We develop a topological network flow model of the dual problem of the shooting linear program in a reverse procedure from the decomposition of solution vectors into triples. Hunsaker (2003) gave a knapsack shooting experiment, which we use to find a simple pattern for the most hit knapsack facets.

Entropic approach to local realism and noncontextuality

For any Bell locality scenario (or Kochen-Specker noncontextuality scenario), the joint Shannon entropies of local (or noncontextual) models define a convex cone for which the non-trivial facets are tight entropic Bell (or contextuality) inequalities. In this paper we explore this entropic approach and derive tight entropic inequalities for various scenarios. One advantage of entropic inequalities is that they easily adapt to situations like bilocality scenarios, which have additional independence requirements that are non-linear on the level of probabilities, but linear on the level of entropies. Another advantage is that, despite the nonlinearity, taking detection inefficiencies into account turns out to be very simple. When joint measurements are conducted by a single detector only, the detector efficiency for witnessing quantum contextuality can be arbitrarily low.

A Characterization of Uniform Matroids

This paper gives a characterization of uniform matroids by means of locked subsets. Locked subsets are 2-connected subsets, their complements are 2-connected in the dual, and the minimum rank of both is 2. Locked subsets give the nontrivial facets of the bases polytope.

Clique-connecting forest and stable set polytopes

Let G=(V,E) be a simple undirected graph. A forest of G is said to be clique-connecting if each tree of F spans a clique of G. This paper adresses the clique-connecting forest polytope. First we give a formulation and a polynomial time separation algorithm. Then we show that the nontrivial nondegenerate facets of the stable set polytope are facets of the clique-connecting polytope. Finally we introduce a family of rank inequalities which are facets, and which generalize the clique inequalities.

On the relative strength of split, triangle and quadrilateral cuts

Integer programs defined by two equations with two free integer variables and nonnegative continuous variables have three types of nontrivial facets: split, triangle or quadrilateral inequalities. In this paper, we compare the strength of these three families of inequalities. In particular we study how well each family approximates the integer hull. We show that, in a well defined sense, triangle inequalities provide a good approximation of the integer hull. The same statement holds for quadrilateral inequalities. On the other hand, the approximation produced by split inequalities may be arbitrarily bad.

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 wheels of the OLS polytope: Facets and separation

Orthogonal Latin squares (OLS) are fundamental combinatorial objects with important theoretical properties and interesting applications. OLS can be represented by integer points satisfying a certain system of equalities. The convex hull of these points is the OLS polytope. This paper adds to the description of the OLS polytope by providing non-trivial facets arising from wheels. Specifically, for each wheel of size five, we identify the variables that can be added to the induced inequality, thus obtaining all distinct families of maximally lifted wheel inequalities. Each of these families induces facets of the OLS polytope which can be efficiently separated in polynomial time.

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.

Polyhedral results on single node variable upper-bound flow models with allowed configurations

In this paper we investigate the convex hull of single node variable upper-bound flow models with allowed configurations. Such a model is defined by a set X ρ ( Z ) = { ( x , z ) ∈ R n × Z | ∑ j = 1 n x j ρ d , 0 ⩽ x j ⩽ u j z j , j = 1 , … , n } , where ρ is one of ⩽ , = or ⩾ , and Z ⊂ { 0 , 1 } n consists of the allowed configurations. We consider the case when Z consists of affinely independent vectors. Under this assumption, a characterization of the non-trivial facets of the convex hull of X ρ ( Z ) for each relation ρ is provided, along with polynomial time separation algorithms. Applications in scheduling and network design are also discussed.

Polytopes of partitions of numbers

We study the vertices and facets of the polytopes of partitions of numbers. The partition polytope P n is the convex hull of the set of incidence vectors of all partitions n = x 1 + 2 x 2 + ⋯ + n x n . We show that the sequence P 1 , P 2 , … , P n , … can be treated as an embedded chain. The dynamics of behavior of the vertices of P n , as n increases, is established. Some sufficient and some necessary conditions for a point of P n to be its vertex are proved. Representation of the partition polytope as a polytope on a partial algebra—which is a generalization of the group polyhedron in the group theoretic approach to the integer linear programming—allows us to prove subadditive characterization of the nontrivial facets of P n . These facets ∑ i = 1 n p i x i ≥ p 0 correspond to extreme rays of the cone of subadditive functions p : { 1 , 2 , … , n } → R with additional requirements p 0 = p n and p i + p n − i = p n , 1 ≤ i < n . The trivial facets are explicitly indicated. We also show how all vertices and facets of the polytopes of constrained partitions—in which some numbers are forbidden to participate—can be obtained from those of the polytope P n . All vertices and facets of P n for n ≤ 8 and n ≤ 6 , respectively, are presented.

Clique family inequalities for the stable set polytope of quasi-line graphs

In one of fundamental works in combinatorial optimization, Edmonds gave a complete linear description of the matching polytope. Matchings in a graph are equivalent to stable sets in its line graph. Also the neighborhood of any vertex in a line graph partitions into two cliques: graphs with this latter property are called quasi-line graphs. Quasi-line graphs are a subclass of claw-free graphs, and as for claw-free graphs, there exists a polynomial algorithm for finding a maximum weighted stable set in such graphs, but we do not have a complete characterization of their stable set polytope (SSP). In the paper, we introduce a class of inequalities, called clique-family inequalities, which are valid for the SSP of any graph and match the odd set inequalities defined by Edmonds for the matching polytope. This class of inequalities unifies all the known (non-trivial) facet inducing inequalities for the SSP of a quasi-line graph. We, therefore, conjecture that all the non-trivial facets of the SSP of a quasi-line graph belong to this class. We show that the conjecture is indeed correct for the subclasses of quasi-line graphs for which we have a complete description of the SSP. We discuss some approaches for solving the conjecture and a related problem.

On the Two-Level Uncapacitated Facility Location Problem

We study the two-level uncapacitated facility location (TUFL) problem. Given two types of facilities, which we call y-facilities and z-facilities, the problem is to decide which facilities of both types to open, and to which pair of y- and z-facilities each client should be assigned, in order to satisfy the demand at maximum profit. We first present two multi-commodity flow formulations of TUFL and investigate the relationship between these formulations and similar formulations of the one-level uncapacitated facility location (UFL) problem. In particular, we show that all nontrivial facets for UFL define facets for the two-level problem, and derive conditions when facets of TUFL are also facets for UFL. For both formulations of TUFL, we introduce new families of facets and valid inequalities and discuss the associated separation problems. We also characterize the extreme points of the LP-relaxation of the first formulation. While the LP-relaxation of a multi-commodity formulation provides good bounds in general, the number of variables and constraints grows rapidly with the size of the problem instance. An alternative model of TUFL is a single-commodity fixed-charge network flow problem. Rardin and Wolsey showed that by projecting a so-called multi-commodity extended formulation of fixed-charge network flow problems onto the space of flow variables used in the weaker flow formulation, a broad class of valid inequalities can be obtained. We discuss a subclass of these inequalities for TUFL that seems particularly useful for computational purposes.

Compositions of Graphs and Polyhedra III: Graphs with No $W_4 $ Minor

The authors characterize the stable set polytope for graphs that do not have a 4-wheel as a minor. The authors prove that the nontrivial facets are either “edge” inequalities or can be obtained by composing “odd cycles” and “subdivisions of $K_4 $.” By adding some extra variables, it is shown that the stable set problem for these graphs can be formulated as a linear program of polynomial size.

Facet defining inequalities for the dichotomous knapsack problem

Facet defining inequalities for an equivalent form of the dichotomous knapsack problem (DKP) are derived. All nontrivial canonical facets associated with DKP are characterized and for a special case of DKP the class of facets are characterized which will guarantee that the corresponding linear programming relaxation has an integral optimal solution.

On the stable set polytope of a series-parallel graph

We give a short proof of Chvatal's conjecture that the nontrivial facets of the stable set polytope of a series-parallel graph all come from edges and odd holes.