Matching Polytope Research Articles

The Matching Problem Has No Fully Polynomial Size Linear Programming Relaxation Schemes

Recently, Rothvos established that every linear program (LP) expressing the matching polytope has an exponential number of inequalities (formally, the matching polytope has exponential extension complexity). We generalize this result by deriving strong bounds on the LP inapproximability of the matching problem: for fixed $0 , every $(1 - \varepsilon / n)$ -approximating LP requires an exponential number of inequalities, where $n$ is the number of vertices. This is sharp, given the well-known $\rho $ -approximation of size $O\binom {n}{1/(1 - \rho )}$ provided by the odd-sets of size up to $1/(1 - \rho )$ . Thus, matching is the first problem in $P$ , which does not admit a fully polynomial-size LP relaxation scheme (the LP equivalent of an Fully Polynomial-Time Approximation Scheme), which provides a sharp separation from the polynomial-size LP relaxation scheme obtained, e.g., through constant-sized odd-sets mentioned above. Analyzing the size of LP formulations is equivalent to examining the nonnegative rank of matrices. We study the nonnegative rank through an information–theoretic approach; while it reuses key ideas from Rothvos, the main lower bounding technique is different: we employ the information–theoretic notion of Wyner’s common information used for studying LP formulations. This allows us to analyze the nonnegative rank of perturbations of slack matrices, e.g., the approximations of the matching polytope. It turns out that the high extension complexity for the matching problem stems from the same source of hardness as in the case of the correlation polytope: a direct sum structure.

A characterization of PM-compact Hamiltonian bipartite graphs

The perfect matching polytope of a graph G is the convex hull of the incidence vectors of all perfect matchings in G. A graph is called perfect matching compact (shortly, PM-compact), if its perfect matching polytope has diameter one. This paper gives a complete characterization of simple PM-compact Hamiltonian bipartite graphs. We first define two families of graphs, called the H2C-bipartite graphs and the H23-bipartite graphs, respectively. Then we show that, for a simple Hamiltonian bipartite graph G with |V(G)| ≥ 6, G is PM-compact if and only if G is K3,3, or G is a spanning Hamiltonian subgraph of either an H2C-bipartite graph or an H23-bipartite graph.

Simple extensions of polytopes

We introduce the simple extension complexity of a polytope $$P$$P as the smallest number of facets of any simple (i.e., non-degenerate in the sense of linear programming) polytope which can be projected onto $$P$$P. We devise a combinatorial method to establish lower bounds on the simple extension complexity and show for several polytopes that they have large simple extension complexities. These examples include both the spanning tree and the perfect matching polytopes of complete graphs, uncapacitated flow polytopes for non-trivially decomposable directed acyclic graphs, hypersimplices, and random 0/1-polytopes with vertex numbers within a certain range. On our way to obtain the result on perfect matching polytopes we generalize a result of Padberg and Rao's on the adjacency structures of those polytopes. In addition to the material in the extended abstract (Kaibel and Walter in Integer programming and combinatorial optimization. Lecture Notes in Computer Science, vol 8494. Springer, Berlin, 2014) we include omitted proofs, supporting figures, and an analysis of known upper bounding techniques.

Uncapacitated flow-based extended formulations

An extended formulation of a polytope is a linear description of this polytope using extra variables besides the variables in which the polytope is defined. The interest of extended formulations is due to the fact that many interesting polytopes have extended formulations with a lot fewer inequalities than any linear description in the original space. This motivates the development of methods for, on the one hand, constructing extended formulations and, on the other hand, proving lower bounds on the sizes of extended formulations. Network flows are a central paradigm in discrete optimization, and are widely used to design extended formulations. We prove exponential lower bounds on the sizes of uncapacitated flow-based extended formulations of several polytopes, such as the (bipartite and non-bipartite) perfect matching polytope and TSP polytope. We also give new examples of flow-based extended formulations, e.g., for 0/1-polytopes defined from regular languages.

Hyperbolic surface subgroups of one-ended doubles of free groups

Gromov asked whether every one-ended word-hyperbolic group contains a hyperbolic surface group. We prove that every one-ended double of a free group has a hyperbolic surface subgroup if (1) the free group has rank 2 or (2) every generator is used the same number of times in a minimal automorphic image of the amalgamating words. To prove this, we formulate a stronger statement on Whitehead graphs and prove its specialization by combinatorial induction for (1) and the characterization of perfect matching polytopes by Edmonds for (2).

A CHARACTERIZATION OF PM-COMPACT CLAW-FREE CUBIC GRAPHS

The perfect matching polytope of a graph G is the convex hull of the incidence vectors of all perfect matchings in G. This paper characterizes claw-free cubic graphs whose 1-skeleton graphs of perfect matching polytopes have diameter 1.

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.

Extended formulations, nonnegative factorizations, and randomized communication protocols

An extended formulation of a polyhedron $$P$$P is a linear description of a polyhedron $$Q$$Q together with a linear map $$\pi $$? such that $$\pi (Q)=P$$?(Q)=P. These objects are of fundamental importance in polyhedral combinatorics and optimization theory, and the subject of a number of studies. Yannakakis' factorization theorem (Yannakakis in J Comput Syst Sci 43(3):441---466, 1991) provides a surprising connection between extended formulations and communication complexity, showing that the smallest size of an extended formulation of $$P$$P equals the nonnegative rank of its slack matrix $$S$$S. Moreover, Yannakakis also shows that the nonnegative rank of $$S$$S is at most $$2^c$$2c, where $$c$$c is the complexity of any deterministic protocol computing $$S$$S. In this paper, we show that the latter result can be strengthened when we allow protocols to be randomized. In particular, we prove that the base-$$2$$2 logarithm of the nonnegative rank of any nonnegative matrix equals the minimum complexity of a randomized communication protocol computing the matrix in expectation. Using Yannakakis' factorization theorem, this implies that the base-$$2$$2 logarithm of the smallest size of an extended formulation of a polytope $$P$$P equals the minimum complexity of a randomized communication protocol computing the slack matrix of $$P$$P in expectation. We show that allowing randomization in the protocol can be crucial for obtaining small extended formulations. Specifically, we prove that for the spanning tree and perfect matching polytopes, small variance in the protocol forces large size in the extended formulation.

From One Stable Marriage to the Next: How Long Is the Way?

The diameter of the stable matching (stable marriage) polytope is bounded from above by $\left\lfloor\frac{n}{2}\right\rfloor$, where $n$ is the number of men (or women) involved in the matching; this bound is attainable.

Blockers and antiblockers of stable matchings

An implicit linear description of the stable matching polytope is provided in terms of the blocker and antiblocker sets of constraints of the matroid-kernel polytope. The explicit identification of both these sets is based on a partition of the stable pairs in which each agent participates. Here, we expose the relation of such a partition to rotations. We provide a time-optimal algorithm for obtaining such a partition and establish some new related results; most importantly, that this partition is unique.

Three-matching intersection conjecture for perfect matching polytopes of small dimensions

In the study of Fulkerson’s conjecture, Fan and Raspaud (1994) proposed a weaker conjecture: in every bridgeless cubic graph G, there are three perfect matchings M1, M2, and M3 such that M1∩M2∩M3=0̸. In this note we prove that this conjecture is true if the dimension of the perfect matching polytope of G is at most 9. This includes the class of bridgeless cubic graphs with the property that all the vertices of the corresponding perfect matching polytope are affinely independent.

Symmetry Matters for Sizes of Extended Formulations

In 1991, Yannakakis [J. Comput. System Sci., 43 (1991), pp. 441--466] proved that no symmetric extended formulation for the matching polytope of the complete graph $K_n$ with $n$ nodes has a number of variables and constraints that is bounded subexponentially in $n$. Here, symmetric means that the formulation remains invariant under all permutations of the nodes of $K_n$. It was also conjectured by Yannakakis that “asymmetry does not help much,” but no corresponding result for general extended formulations has been found so far. In this paper we show that for the polytopes associated with the matchings in $K_n$ with $\lfloor\log n\rfloor$ edges there are nonsymmetric extended formulations of polynomial size, while nevertheless no symmetric extended formulations of polynomial size exist. We furthermore prove similar statements for the polytopes associated with cycles of length $\lfloor\log n\rfloor$. Thus, with respect to the question for smallest possible extended formulations, in general symmetry requirements may matter a lot. Compared to the extended abstract [Integer Pgrogramming and Combinatiorial Optimization, Lecture Notes in Comput. Sci. 6080, Springer, New York, 2010, pp. 135--148], this paper not only contains proofs that had been omitted there but also presents slightly generalized and sharpened lower bounds.

The Maximum-Weight Stable Matching Problem: Duality and Efficiency

Given a preference system $(G, \prec)$ and an integral weight function defined on the edge set of $G$ (not necessarily bipartite), the maximum-weight stable matching problem is to find a stable matching of $(G, \prec)$ with maximum total weight. In this paper we study this $NP$-hard problem using linear programming and polyhedral approaches. We show that the Rothblum system for defining the fractional stable matching polytope of $(G, \prec)$ is totally dual integral if and only if this polytope is integral if and only if $(G, \prec)$ has a bipartite representation. We also present a combinatorial polynomial-time algorithm for the maximum-weight stable matching problem and its dual on any preference system with a bipartite representation. Our results generalize Király and Pap's theorem on the maximum-weight stable-marriage problem and rely heavily on their work.

Matching polytopes and Specht modules

We prove that the dimension of the Specht module of a forest $G$ is the same as the normalized volume of the matching polytope of $G$. We also associate to $G$ a symmetric function $s_G$ (analogous to the Schur symmetric function $s_\lambda$ for a partition $\lambda$) and investigate its combinatorial and representation-theoretic properties in relation to the Specht module and Schur module of $G$. We then use this to define notions of standard and semistandard tableaux for forests.

Speeding up IP-based algorithms for constrained quadratic 0–1 optimization

In many practical applications, the task is to optimize a non-linear objective function over the vertices of a well-studied polytope as, e.g., the matching polytope or the travelling salesman polytope (TSP). Prominent examples are the quadratic assignment problem and the quadratic knapsack problem; further applications occur in various areas such as production planning or automatic graph drawing. In order to apply branch-and-cut methods for the exact solution of such problems, the objective function has to be linearized. However, the standard linearization usually leads to very weak relaxations. On the other hand, problem-specific polyhedral studies are often time-consuming. Our goal is the design of general separation routines that can replace detailed polyhedral studies of the resulting polytope and that can be used as a black box. As unconstrained binary quadratic optimization is equivalent to the maximum-cut problem, knowledge about cut polytopes can be used in our setting. Other separation routines are inspired by the local cuts that have been developed by Applegate, Bixby, Chvátal and Cook for faster solution of large-scale traveling salesman instances. Finally, we apply quadratic reformulations of the linear constraints as proposed by Helmberg, Rendl and Weismantel for the quadratic knapsack problem. By extensive experiments, we show that a suitable combination of these methods leads to a drastic speedup in the solution of constrained quadratic 0–1 problems. We also discuss possible generalizations of these methods to arbitrary non-linear objective functions.

König–Egerváry Graphs are Non-Edmonds

Konig–Egervary graphs are those whose maximum matchings are equicardinal to their minimum-order coverings by vertices. Edmonds (J Res Nat Bur Standards Sect B 69B:125–130, 1965) characterized the perfect matching polytope of a graph G = (V, E) as the set of nonnegative vectors $${{\bf{x}}\in\mathbb R^E}$$satisfying two families of constraints: ‘vertex saturation’ and ‘blossom’. Graphs for which the latter constraints are implied by the former are termed non-Edmonds. This note presents two proofs—one combinatorial, one algorithmic—of its title’s assertion. Neither proof relies on the characterization of non-Edmonds graphs due to de Carvalho et al. (J Combin Theory Ser B 92:319–324, 2004).

Gorenstein Polytopes Obtained from Bipartite Graphs

Beck et al. characterized the grid graphs whose perfect matching polytopes are Gorenstein and they also showed that for some parameters, perfect matching polytopes of torus graphs are Gorenstein. In this paper, we complement their result, that is, we characterize the torus graphs whose perfect matching polytopes are Gorenstein. Beck et al. also gave a method to construct an infinite family of Gorenstein polytopes. In this paper, we introduce a new class of polytopes obtained from graphs and we extend their method to construct many more Gorenstein polytopes.

An improved linear bound on the number of perfect matchings in cubic graphs

We show that every cubic bridgeless graph with n vertices has at least 3 n / 4 − 10 perfect matchings. This is the first bound that differs by more than a constant from the maximal dimension of the perfect matching polytope.

Grid Graphs, Gorenstein Polytopes, and Domino Stackings

We examine domino tilings of rectangular boards, which are in natural bijection with perfect matchings of grid graphs. This leads to the study of their associated perfect matching polytopes, and we present some of their properties, in particular, when these polytopes are Gorenstein. We also introduce the notion of domino stackings and present some results and several open questions. Our techniques use results from graph theory, polyhedral geometry, and enumerative combinatorics.

The graph of perfect matching polytope and an extreme problem

For graph G , its perfect matching polytope P o l y ( G ) is the convex hull of incidence vectors of perfect matchings of G . The graph corresponding to the skeleton of P o l y ( G ) is called the perfect matching graph of G , and denoted by P M ( G ) . It is known that P M ( G ) is either a hypercube or hamilton connected [D.J. Naddef, W.R. Pulleyblank, Hamiltonicity and combinatorial polyhedra, J. Combin. Theory Ser. B 31 (1981) 297–312; D.J. Naddef, W.R. Pulleyblank, Hamiltonicity in (0-1)-polytope, J. Combin. Theory Ser. B 37 (1984) 41–52]. In this paper, we give a sharp upper bound of the number of lines for the graphs G whose P M ( G ) is bipartite in terms of sizes of elementary components of G and the order of G , respectively. Moreover, the corresponding extremal graphs are constructed.