We consider the bipartite matching model of customers and servers introduced by Caldentey, Kaplan, and Weiss (Adv. Appl. Probab., 2009). Customers and servers play symmetrical roles. There is a finite set C resp. S, of customer, resp. server, classes. Time is discrete and at each time step, one customer and one server arrive in the system according to a joint probability measure on CxS, independently of the past. Also, at each time step, pairs of matched customer and server, if they exist, depart from the system. Authorized matchings are given by a fixed bipartite graph. A matching policy is chosen, which decides how to match when there are several possibilities. Customers/servers that cannot be matched are stored in a buffer. The evolution of the model can be described by a discrete time Markov chain. We study its stability under various admissible matching policies including: ML (Match the Longest), MS (Match the Shortest), FIFO (match the oldest), priorities. There exist natural necessary conditions for stability (independent of the matching policy) defining the maximal possible stability region. For some bipartite graphs, we prove that the stability region is indeed maximal for any admissible matching policy. For the ML policy, we prove that the stability region is maximal for any bipartite graph. For the MS and priority policies, we exhibit a bipartite graph with a non-maximal stability region.
Let $\Ga$ be the collection of all weighted bipartite graphs each having $σ$ and $m$, as the size of a vertex partition and the total weight, respectively. We give a tight lower bound $\lceil \frac{m-σ}σ \rceil+1$ for the set $\{\textit{Wt}(\textit{mwm}(G))~|~G \in \Ga\}$ which denotes the collection of weights of maximum weight bipartite matchings of all graphs in $\Ga$.
Matching minors are a specialisation of minors fit for the study of graph with perfect matchings. The notion of matching minors has been used to give a structural description of bipartite graphs on which the number of perfect matchings can becomputed efficiently, based on a result of Little, by McCuaig et al. in 1999.In this paper we generalise basic ideas from the graph minor series by Robertson and Seymour to the setting of bipartite graphs with perfect matchings. We introducea version of Erdos-Posa property for matching minors and find a direct link between this property and planarity. From…
The Total Matching Polytope generalizes the Stable Set Polytope and the Matching Polytope. In this paper, we give the perfect formulation for Trees and we derive two new families of valid inequalities, the balanced biclique inequalities which are always facet-defining and the non-balanced lifted biclique inequalities obtained by a lifting procedure, which are facet-defining for bipartite graphs. Finally, we give a complete description for Complete Bipartite Graphs.
Haxell's condition is a natural hypergraph analog of Hall's condition, which is a well-known necessary and sufficient condition for a bipartite graph to admit a perfect matching. That is, when Haxell's condition holds it forces the existence of a perfect matching in the bipartite hypergraph. Unlike in graphs, however, there is no known polynomial time algorithm to find the hypergraph perfect matching that is guaranteed to exist when Haxell's condition is satisfied. We prove the existence of an efficient algorithm to find perfect matchings in bipartite hypergraphs whenever a stronger version of…
Given a graph with edges colored red or blue and an integer $k$, the exact perfect matching problem asks if there exists a perfect matching with exactly $k$ red edges. There exists a randomized polylogarithmic-time parallel algorithm to solve this problem, dating back to the eighties, but no deterministic polynomial-time algorithm is known, even for bipartite graphs. In this paper we show that there is no sub-exponential sized linear program that can describe the convex hull of exact matchings in bipartite graphs. In fact, we prove something stronger, that there is no sub-exponential sized lin…
We show that the bipartite perfect matching problem is in quasi-NC$^2$. That is, it has uniform circuits of quasi-polynomial size $n^{O(\log n)}$, and $O(log^2 n)$ depth. Previously, only an exponential upper bound was known on the size of such circuits with poly-logarithmic depth. We obtain our result by an almost complete derandomization of the famous Isolation Lemma when applied to yield an efficient randomized parallel algorithm for the bipartite perfect matching problem.
We study the classical weighted perfect matchings problem for bipartite graphs or sometimes referred to as the assignment problem, i.e., given a weighted bipartite graph $G = (U\cup V,E)$ with weights $w : E \rightarrow \mathcal{R}$ we are interested to find the maximum matching in $G$ with the minimum/maximum weight. In this work we present a new and arguably simpler analysis of one of the earliest techniques developed for solving the assignment problem, namely the auction algorithm. Using our analysis technique we present tighter and improved bounds on the runtime complexity for finding an a…
We obtain a description of the Bipartite Perfect Matching decision problem as a multilinear polynomial over the Reals. We show that it has full degree and $(1-o_n(1))\cdot 2^{n^2}$ monomials with non-zero coefficients. In contrast, we show that in the dual representation (switching the roles of 0 and 1) the number of monomials is only exponential in $Θ(n \log n)$. Our proof relies heavily on the fact that the lattice of graphs which are "matching-covered" is Eulerian.
We consider, for complete bipartite graphs, the convex hulls of characteristic vectors of all matchings, extended by a binary entry indicating whether the matching contains two specific edges. These polytopes are associated to the quadratic matching problems with a single linearized quadratic term. We provide a complete irredundant inequality description, which settles a conjecture by Klein (Ph.D. thesis, TU Dortmund, 2015). In addition, we also derive facetness and separation results for the polytopes. The completeness proof is based on a geometric relationship to a matching polytope of a non…
We obtain complete characterizations of the Unique Bipartite Perfect Matching function, and of its Boolean dual, using multilinear polynomials over the reals. Building on previous results, we show that, surprisingly, the dual description is sparse and has low $\ell_1$-norm -- only exponential in $Θ(n \log n)$, and this result extends even to other families of matching-related functions. Our approach relies on the Möbius numbers in the matching-covered lattice, and a key ingredient in our proof is Möbius' inversion formula. These polynomial representations yield complexity-theoretic results. Fo…
A dynamic bipartite matching model is given by a bipartite matching graph which determines the possible matchings between the various types of supply and demand items. Both supply and demand items arrive to the system according to a stochastic process. Matched pairs leave the system and the others wait in the queues, which induces a holding cost. We model this problem as a Markov Decision Process and study the discounted cost and the average cost problem. We fully characterize the optimal matching policy for complete matching graphs and for the N -shaped matching graph. In the former case, the…
The approximate degree of a Boolean function is the least degree of a real multilinear polynomial approximating it in the $\ell_\infty$-norm over the Boolean hypercube. We show that the approximate degree of the Bipartite Perfect Matching function, which is the indicator over all bipartite graphs having a perfect matching, is $\widetildeΘ(n^{3/2})$. The upper bound is obtained by fully characterizing the unique multilinear polynomial representing the Boolean dual of the perfect matching function, over the reals. Crucially, we show that this polynomial has very small $\ell_1$-norm -- only expon…
A connected graph G is called matching covered if every edge of G is contained in a perfect matching. Perfect matching width is a width parameter for matching covered graphs based on a branch decomposition. It was introduced by Norine and intended as a tool for the structural study of matching covered graphs, especially in the context of Pfaffian orientations. Norine conjectured that graphs of high perfect matching width would contain a large grid as a matching minor, similar to the result on treewidth by Robertson and Seymour. In this paper we obtain the first results on perfect matching widt…
In a graph, a matching cut is an edge cut that is a matching. Matching Cut is the problem of deciding whether or not a given graph has a matching cut, which is known to be NP-complete even when restricted to bipartite graphs. It has been proved that Matching Cut is polynomially solvable for graphs of diameter two. In this paper, we show that, for any fixed integer $d\ge 3$, Matching Cut is NP-complete in the class of graphs of diameter $d$. This resolves an open problem posed by Borowiecki and Jesse-Józefczyk in [Matching cutsets in graphs of diameter $2$, Theoretical Computer Science 407 (200…
In a recent paper, Beniamini and Nisan gave a closed-form formula for the unique multilinear polynomial for the Boolean function determining whether a given bipartite graph $G \subseteq K_{n,n}$ has a perfect matching, together with an efficient algorithm for computing the coefficients of the monomials of this polynomial. We give the following generalization: Given an arbitrary non-negative weight function $w$ on the edges of $K_{n,n}$, consider its set of minimum weight perfect matchings. We give the real multilinear polynomial for the Boolean function which determines if a graph $G \subseteq…
We consider popular matching problems in both bipartite and non-bipartite graphs with strict preference lists. It is known that every stable matching is a min-size popular matching. A subclass of max-size popular matchings called dominant matchings has been well-studied in bipartite graphs: they always exist and there is a simple linear time algorithm to find one. We show that stable and dominant matchings are the only two tractable subclasses of popular matchings in bipartite graphs; more precisely, we show that it is NP-complete to decide if $G$ admits a popular matching that is neither stab…
We consider Bayesian online selection problem of a matching in bipartite graphs, i.e., online weighted matching problem with edge arrivals where online algorithm knows distributions of weights, that corresponds to the intersection of two matroids in [Kleinberg and Wienberg STOC 12] model. We consider a simple class of non adaptive vertex-additive policies that assign static prices to all vertices in the graph and accept each edge only if its weight exceeds the sum of the prices of the edge's endpoints. We show existence of a vertex-additive policy with the expected payoff of at least one third…
A major step in the graph minors theory of Robertson and Seymour is the transition from the Grid Theorem which, in some sense uniquely, describes areas of large treewidth within a graph, to a notion of local flatness of these areas in form of the existence of a large flat wall within any huge grid of an H-minor free graph. In this paper, we prove a matching theoretic analogue of the Flat Wall Theorem for bipartite graphs excluding a fixed matching minor. Our result builds on a a tight relationship between structural digraph theory and matching theory and allows us to deduce a Flat Wall Theorem…
Two-sided popular matchings in bipartite graphs are a well-known generalization of stable matchings in the marriage setting, and they are especially relevant when preference lists are incomplete. In this case, the cardinality of a stable matching can be as small as half the size of a maximum matching. Popular matchings allow for assignments of larger size while still guaranteeing a certain fairness condition. In fact, stable matchings are popular matchings of minimum size, and a maximum size popular matching can be as large as twice the size of a(ny) stable matching in a given instance. The st…
A bipartite graph B is called a brace if it is connected and every matching of size at most two in B is contained in some perfect matching of B and a cycle C in B is called conformal if B-V(C) has a perfect matching. We show that there do not exist two disjoint alternating paths that form a cross over a conformal cycle C in a brace B if and only if one can reduce B, by an application of a matching theoretic analogue of small clique sums, to a planar brace H in which C bounds a face. We then utilise this result and provide a polynomial time algorithm which solves the 2-linkage problem for alter…
We consider the following question: given an $(X,Y)$-bigraph $G$ and a set $S \subset X$, does $G$ contain two disjoint matchings $M_1$ and $M_2$ such that $M_1$ saturates $X$ and $M_2$ saturates $S$? When $|S|\geq |X|-1$, this question is solvable by finding an appropriate factor of the graph. In contrast, we show that when $S$ is allowed to be an arbitrary subset of $X$, the problem is NP-hard.
We propose a framework for optimal $t$-matchings excluding the prescribed $t$-factors in bipartite graphs. The proposed framework is a generalization of the nonbipartite matching problem and includes several problems, such as the triangle-free $2$-matching, square-free $2$-matching, even factor, and arborescence problems. In this paper, we demonstrate a unified understanding of these problems by commonly extending previous important results. We solve our problem under a reasonable assumption, which is sufficiently broad to include the specific problems listed above. We first present a min-max …
We propose a simple yet very predictive form, based on a Poisson's equation, for the functional dependence of the cost from the density of points in the Euclidean bipartite matching problem. This leads, for quadratic costs, to the analytic prediction of the large $N$ limit of the average cost in dimension $d=1,2$ and of the subleading correction in higher dimension. A non-trivial scaling exponent, $γ_d=\frac{d-2}{d}$, which differs from the monopartite's one, is found for the subleading correction. We argue that the same scaling holds true for a generic cost exponent in dimension $d>2$.
We present a new approach, called a lazy matching, to the problem of on-line matching on bipartite graphs. Imagine that one side of a graph is given and the vertices of the other side are arriving on-line. Originally, incoming vertex is either irrevocably matched to an another element or stays forever unmatched. A lazy algorithm is allowed to match a new vertex to a group of elements (possibly empty) and afterwords, forced against next vertices, may give up parts of the group. The restriction is that all the time each element is in at most one group. We present an optimal lazy algorithm (deter…
We give a sharp lower bound on the number of matchings of a given size in a bipartite graph. When specialized to regular bipartite graphs, our results imply Friedland's Lower Matching Conjecture and Schrijver's theorem proven by Gurvits and Csikvari. Indeed, our work extends the recent work of Csikvari done for regular and bi-regular bipartite graphs. Moreover, our lower bounds are order optimal as they are attained for a sequence of $2$-lifts of the original graph as well as for random $n$-lifts of the original graph when $n$ tends to infinity. We then extend our results to permanents and sub…
Barnette's Conjecture claims that all cubic, 3-connected, planar, bipartite graphs are Hamiltonian. We give a translation of this conjecture into the matching-theoretic setting. This allows us to relax the requirement of planarity to give the equivalent conjecture that all cubic, 3-connected, Pfaffian, bipartite graphs are Hamiltonian. A graph, other than the path of length three, is a brace if it is bipartite and any two disjoint edges are part of a perfect matching. Our perspective allows us to observe that Barnette's Conjecture can be reduced to cubic, planar braces. We show a similar reduc…
Relaxation and rounding approaches became a standard and extremely versatile tool for constrained submodular function maximization. One of the most common rounding techniques in this context are contention resolution schemes. Such schemes round a fractional point by first rounding each coordinate independently, and then dropping some elements to reach a feasible set. Also the second step, where elements are dropped, is typically randomized. This leads to an additional source of randomization within the procedure, which can complicate the analysis. We suggest a different, polyhedral viewpoint t…
