Minimum-weight Perfect Matching Problem Research Articles

Constrained many-to-many point matching in two dimensions

AbstractIn the minimum-weight many-to-many point matching problem, we are given a set R of red points and a set B of blue points in the plane, of total size N, and we want to pair up each point in R to one or more points in B and vice versa so that the sum of distances between the paired points is minimized. This problem can be solved in $$O(N^3)$$ O ( N 3 ) time by using a reduction to the minimum-weight perfect matching problem, and thus, it is not fast enough to be used for on-line systems where a large number of tunes need to be compared. Motivated by similarity problems in music theory, in this paper we study several constrained minimum-weight many-to-many point matching problems in which the allowed pairings are given by geometric restrictions, i.e., a bichromatic pair can be matched if and only if the corresponding points satisfy a specific condition of closeness. We provide algorithms to solve these constrained versions in O(N) time when the sets R and B are given ordered by abscissa.

Constrained read-once refutations in UTVPI constraint systems: A parallel perspective

AbstractIn this paper, we analyze two types of refutations for Unit Two Variable Per Inequality (UTVPI) constraints. A UTVPI constraint is a linear inequality of the form: $a_{i}\cdot x_{i}+a_{j} \cdot x_{j} \le b_{k}$ , where $a_{i},a_{j}\in \{0,1,-1\}$ and $b_{k} \in \mathbb{Z}$ . A conjunction of such constraints is called a UTVPI constraint system (UCS) and can be represented in matrix form as: ${\bf A \cdot x \le b}$ . UTVPI constraints are used in many domains including operations research and program verification. We focus on two variants of read-once refutation (ROR). An ROR is a refutation in which each constraint is used at most once. A literal-once refutation (LOR), a more restrictive form of ROR, is a refutation in which each literal ( $x_i$ or $-x_i$ ) is used at most once. First, we examine the constraint-required read-once refutation (CROR) problem and the constraint-required literal-once refutation (CLOR) problem. In both of these problems, we are given a set of constraints that must be used in the refutation. RORs and LORs are incomplete since not every system of linear constraints is guaranteed to have such a refutation. This is still true even when we restrict ourselves to UCSs. In this paper, we provide NC reductions between the CROR and CLOR problems in UCSs and the minimum weight perfect matching problem. The reductions used in this paper assume a CREW PRAM model of parallel computation. As a result, the reductions establish that, from the perspective of parallel algorithms, the CROR and CLOR problems in UCSs are equivalent to matching. In particular, if an NC algorithm exists for either of these problems, then there is an NC algorithm for matching.

A logic-based Benders decomposition solution approach for two covering problems that consider the underlying transportation

We investigate two maximal covering location problems with capacity restrictions, minimum workload, and transportation. The problems are inspired by a waste collection problem in which large waste containers are scattered throughout the municipality, and the residents bring their waste to these containers. We take the residents’ preferences into account when allocating them to locations. When a container is full, a vehicle transports an empty container from the disposal facility (depot) to that location and replaces it. We propose a mixed-integer linear programming formulation for the problems in which vehicles can carry one or two containers, and apply a logic-based Benders decomposition approach for the latter. Here, the sub problem is a multi-period minimum weight perfect matching problem. We show that our logic-based Benders decomposition approach outperforms the direct formulation in terms of solution quality and speed. We further show that transportation of two containers at a time reduces the distance to be driven by 29.5% on average, without compromising the covering level. Furthermore, we analyze the effect of imposing a minimum workload as well as the effect of changing the focus between transportation and covering.

On computing discretized Ricci curvatures of graphs: Local algorithms and (localized) fine-grained reductions

Characterizing shapes of high-dimensional objects via Ricci curvatures plays a critical role in many research areas in mathematics and physics. However, even though several discretizations of Ricci curvatures for discrete combinatorial objects such as networks have been proposed and studied by mathematicians, the computational complexity aspects of these discretizations have escaped the attention of theoretical computer scientists to a large extent. In this paper, we study one such discretization, namely the Ollivier-Ricci curvature, from the perspective of efficient computation by fine-grained reductions and local query-based algorithms. Our main contributions are the following.▹We relate our curvature computation problem to minimum weight perfect matching problem on complete bipartite graphs via fine-grained reduction.▹We formalize the computational aspects of the curvature computation problems in suitable frameworks so that they can be studied by researchers in local algorithms.▹We provide the first known lower and upper bounds on queries for query-based algorithms for the curvature computation problems in our local algorithms framework. En route, we also illustrate a localized version of our fine-grained reduction. We believe that our results bring forth an intriguing set of research questions, motivated both in theory and practice, regarding designing efficient algorithms for curvatures of geometrical objects.

Directed negative-weight percolation.

We consider the negative-weight percolation model on directed graphs. In particular, we study the model on a two-dimensional, weighted, periodic, centered square lattice. Bond weights are taken from a distribution that allows for both positive and negative values. For a given realization of the disorder, a minimally weighted directed loop and path configuration is determined by performing a nontrivial transformation of the original lattice into a minimum-weight perfect matching problem. For this problem, fast polynomial-time algorithms are available, thus we could study large systems with high accuracy. Depending on the fraction of negatively and positively weighted bonds on the lattice, a continuous phase transition is identified and its characterizing critical exponents are estimated by means of a finite-size scaling analysis. We observe a strong change of the universality class with respect to standard directed percolation as well as with respect to undirected negative-weight percolation. Furthermore, the relation to directed polymers in random media is illustrated.

A Polynomial Time Algorithm for Read-Once Certification of Linear Infeasibility in UTVPI Constraints

In this paper, we discuss the design and analysis of polynomial time algorithms for two problems associated with a linearly infeasible system of Unit Two Variable Per Inequality (UTVPI) constraints, viz., (a) the read-once refutation (ROR) problem, and (b) the literal-once refutation (LOR) problem. Recall that a UTVPI constraint is a linear relationship of the form: \(a_{i}\cdot x_{i}+a_{j} \cdot x_{j} \le b_{ij}\), where \(a_{i},a_{j} \in \{0,1,-1\}\). A conjunction of such constraints is called a UTVPI constraint system (UCS) and can be represented in matrix form as: \(\mathbf{A \cdot x \le b}\). These constraints find applications in a host of domains including but not limited to operations research and program verification. For the linear system \(\mathbf{A\cdot x \le b}\), a refutation is a collection of m variables \(\mathbf{y}=[y_{1},y_{2},\ldots , y_{m}]^{T} \in \mathfrak {R}^{m}_{+}\), such that \(\mathbf{y\cdot A =0}\), \(\mathbf{y \cdot b } < 0\). Such a refutation is guaranteed to exist for any infeasible linear program, as per Farkas’ lemma. The refutation is said to be read-once, if each \(y_{i} \in \{0,1\}\). Read-once refutations are incomplete in that their existence is not guaranteed for infeasible linear programs, in general. Indeed they are not complete, even for UCSs. Hence, the question of whether an arbitrary UCS has an ROR is both interesting and non-trivial. In this paper, we reduce this problem to the problem of computing a minimum weight perfect matching (MWPM) in an undirected graph. This transformation results in an algorithm that runs in in time polynomial in the size of the input UCS. Additionally, we design a polynomial time algorithm (also via a reduction to the MWPM problem) for a variant of read-once resolution called literal-once resolution. The advantage of reducing our problems to the MWPM problem is that we can leverage recent advances in algorithm design for the MWPM problem towards solving the ROR and LOR problems in UCSs. Finally, we show that another variant of read-once refutation problem called the non-literal read-once refutation (NLROR) problem is NP-complete in UCSs.

High-precision studies of domain-wall properties in the 2D Gaussian Ising spin glass

In two dimensions, short-range spin glasses order only at zero temperature, where efficient combinatorial optimization techniques can be used to study these systems with high precision. The use of large system sizes and high statistics in disorder averages allows for reliable finite-size extrapolations to the thermodynamic limit. Here, we use a recently introduced mapping of the Ising spin-glass ground-state problem to a minimum-weight perfect matching problem on a sparse auxiliary graph to study square-lattice samples of up to 10 000 × 10 000 spins. We propose a windowing technique that allows to extend this method, that is formally restricted to planar graphs, to the case of systems with fully periodic boundary conditions. These methods enable highly accurate estimates of the spin-stiffness exponent and domain-wall fractal dimension of the 2D Edwards-Anderson spin-glass with Gaussian couplings. Studying the compatibility of domain walls in this system with traces of stochastic Loewner evolution (SLE), we find a strong dependence on boundary conditions and compatibility with SLE only for one out of several setups.

Battery Swapping Assignment for Electric Vehicles

This paper formulates a multi-period optimal station assignment problem for electric vehicle (EV) battery swapping that takes into account both temporal and spatial couplings. The goal is to reduce the total EV cost and station congestion due to temporary shortage in supply of available batteries. We show that the problem is reducible to the minimum weight perfect bipartite matching problem. This leads to an efficient solution based on the Hungarian algorithm. Numerical results suggest that the proposed solution provides a significant improvement over a greedy heuristic that assigns EVs to nearest stations.

Distributed spatial modulation with dynamic frequency allocation

This paper proposes a distributed implementation of spatial modulation (SM) using cognitive radios. In distributed spatial modulation (DSM), multiple relays form a virtual antenna array and assist a source to transmit its information to a destination. The source broadcasts its signal, which is independently demodulated by all the relays. Each of the relays then divides the received data in two parts: the first part is used to decide which one of the relays will be active, and the other part decides what data it will transmit to the destination. An analytical expression for symbol error probability is derived for DSM in independent and identically distributed (i.i.d.) Rayleigh fading channels. The analytical results are later compared with Monte Carlo simulations. Further, an instantaneous symbol error rate (SER) based selection combining is proposed to incorporate the direct link between the source and destination with existing DSM. Next, DSM implementation is extended to a cognitive network scenario where the source, relays, and destination are all cognitive radios. A dynamic frequency allocation scheme is proposed to improve the performance of DSM in this scenario. The frequency allocation is modeled through a bipartite graph with end-to-end SER as a weight function. The optimal frequency allocation problem is formulated as minimum weight perfect matching problem and is solved using the Hungarian method. Finally, numerical results are provided to illustrate the efficacy of the proposed scheme.

On a cardinality-constrained transportation problem with market choice

It is well-known that the intersection of the matching polytope with a cardinality constraint is integral (Schrijver, 2003) [8]. In this note, we prove a similar result for the polytope corresponding to the transportation problem with market choice (TPMC) (introduced in Damcı-Kurt et al. (2015)) when the demands are in the set {1,2}. This result generalizes the result regarding the matching polytope. The result in this note implies that some special classes of minimum weight perfect matching problem with a cardinality constraint on a subset of edges can be solved in polynomial time.

Algorithmic Applications of Baur-Strassen’s Theorem

Consider a directed or an undirected graph with integral edge weights from the set [-W, W], that does not contain negative weight cycles. In this article, we introduce a general framework for solving problems on such graphs using matrix multiplication. The framework is based on the usage of Baur-Strassen’s theorem and of Strojohann’s determinant algorithm. It allows us to give new and simple solutions to the following problems: Finding Shortest Cycles . We give a simple Õ ( Wnω ) time algorithm for finding shortest cycles in undirected and directed graphs. For directed graphs (and undirected graphs with nonnegative weights), this matches the time bounds obtained in 2011 by Roditty and Williams. On the other hand, no algorithm working in Õ ( Wn ω ) time was previously known for undirected graphs with negative weights. Furthermore, our algorithm for a given directed or undirected graph detects whether it contains a negative weight cycle within the same running time. Computing Diameter and Radius . We give a simple Õ ( Wnω ) time algorithm for computing a diameter and radius of an undirected or directed graphs. To the best of our knowledge, no algorithm with this running time was known for undirected graphs with negative weights. Finding Minimum-Weight Perfect Matchings . We present an Õ ( Wnω ) time algorithm for finding minimum-weight perfect matchings in undirected graphs. This resolves an open problem posted by Sankowski [2009] who presented such an algorithm but only in the case of bipartite graphs. These three problems that are solved in the full generality demonstrate the utility of this framework. Hence, we believe that it can find applications for solving larger spectra of related problems. As an illustrative example, we apply it to the problem of computing a set of vertices that lie on cycles of length at most t , for some given t . We give a simple Õ ( Wnω ) time algorithm for this problem that improves over the Õ ( Wnωt ) time algorithm given by Yuster in 2011. Besides giving this flexible framework, the other main contribution of this article is the development of a novel combinatorial interpretation of the dual solution for the minimum-weight perfect matching problem. Despite the long history of the matching problem, such a combinatorial interpretation was not known previously. This result sheds a new light on the problem, as there exist many structural theorems about unweighted matchings, but almost no results that could cope with the weighted case.

Minimum weight perfect matching of fault-tolerant topological quantum error correction in average O(1) parallel time

Consider a 2-D square array of qubits of extent $L\times L$. We provide a proof that the minimum weight perfect matching problem associated with running a particular class of topological quantum error correction codes on this array can be exactly solved with a 2-D square array of classical computing devices, each of which is nominally associated with a fixed number $N$ of qubits, in constant average time per round of error detection independent of $L$ provided physical error rates are below fixed nonzero values, and other physically reasonable assumptions. This proof is applicable to the fully fault-tolerant case only, not the case of perfect stabilizer measurements.

Loop calculus and bootstrap-belief propagation for perfect matchings on arbitrary graphs

This manuscript discusses computation of the Partition Function (PF) and the Minimum Weight Perfect Matching (MWPM) on arbitrary, non-bipartite graphs. We present two novel problem formulations – one for computing the PF of a Perfect Matching (PM) and one for finding MWPMs – that build upon the inter-related Bethe Free Energy (BFE), Belief Propagation (BP), Loop Calculus (LC), Integer Linear Programming and Linear Programming frameworks. First, we describe an extension of the LC framework to the PM problem. The resulting formulas, coined (fractional) Bootstrap-BP, express the PF of the original model via the BFE of an alternative PM problem. We then study the zero-temperature version of this Bootstrap-BP formula for approximately solving the MWPM problem. We do so by leveraging the Bootstrap-BP formula to construct a sequence of MWPM problems, where each new problem in the sequence is formed by contracting odd-sized cycles (or blossoms) from the previous problem. This Bootstrap-and-Contract procedure converges reliably and generates an empirically tight upper bound for the MWPM. We conclude by discussing the relationship between our iterative procedure and the famous Blossom Algorithm of Edmonds '65 and demonstrate the performance of the Bootstrap-and-Contract approach on a variety of weighted PM problems.

Minimum—weight perfect matching for nonintrinsic distances on the line

Consider a real line equipped with a (not necessarily intrinsic) distance. We deal with the minimum-weight perfect matching problem for a complete graph whose points are located on the line and whose edges have weights equal to distances along the line. This problem is closely related to one-dimensional Monge-Kantorovich trasnport optimization. The main result of the present note is a "bottom-up" recursion relation for weights of partial minimum-weight matchings.

Cross-Layer User Pairing for CSM in IEEE 802.16 Networks

A novel method to solve the pairing problem for Collaborative Spatial Multiplexing (CSM) in IEEE 802.16 networks is proposed in this letter. This method profiles the traffic generated by mobile stations according to their QoS requirements. User pairs with similar traffic profiles and physical layer compatibility are chosen. We show how the pairing problem can be reduced to a minimum weight perfect matching problem which can be polynomially solved using Edmond's algorithm. Simulation results show that the proposed method improves the capacity of the system by at least 20% while reducing the delay for real-time traffic by about 15%.

Configurational statistics of densely and fully packed loops in the negative-weight percolation model

By means of numerical simulations we investigate the configurational properties of densely and fully packed configurations of loops in the negative-weight percolation (NWP) model. In the presented study we consider 2d square, 2d honeycomb, 3d simple cubic and 4d hypercubic lattice graphs, where edge weights are drawn from a Gaussian distribution. For a given realization of the disorder we then compute a configuration of loops, such that the configurational energy, given by the sum of all individual loop weights, is minimized. For this purpose, we employ a mapping of the NWP model to the "minimum-weight perfect matching problem" that can be solved exactly by using sophisticated polynomial-time matching algorithms. We characterize the loops via observables similar to those used in percolation studies and perform finite-size scaling analyses, up to side length L=256 in 2d, L=48 in 3d and L=20 in 4d (for which we study only some observables), in order to estimate geometric exponents that characterize the configurations of densely and fully packed loops. One major result is that the loops behave like uncorrelated random walks from dimension d=3 on, in contrast to the previously studied behavior at the percolation threshold, where random-walk behavior is obtained for d>=6.

Minimum Cost Edge-Colorings of Trees Can Be Reduced to Matchings

Let C be a set of colors, and let ω(c) be an integer cost assigned to a color c in C. An edge-coloring of a graph G is to color all the edges of G so that any two adjacent edges are colored with different colors in C. The cost ω(ƒ) of an edge-coloring ƒ of G is the sum of costs ω(ƒ(e)) of colors ƒ(e) assigned to all edges e in G. An edge-coloring ƒ of G is optimal if ω(ƒ) is minimum among all edge-colorings of G. In this paper, we show that the problem of finding an optimal edge-coloring of a tree T can be simply reduced in polynomial time to the minimum weight perfect matching problem for a new bipartite graph constructed from T. The reduction immediately yields an efficient simple algorithm to find an optimal edge-coloring of T in time O(n1.5Δlog (nNω)), where n is the number of vertices in T, Δ is the maximum degree of T, and Nω is the maximum absolute cost |ω(c)| of colors c in C. We then show that our result can be extended for multitrees.

A dedicated algorithm for calculating ground states for the triangular random bond Ising model

In the presented article we present an algorithm for the computation of ground state spin configurations for the 2 d random bond Ising model on planar triangular lattice graphs. Therefore, it is explained how the respective ground state problem can be mapped to an auxiliary minimum-weight perfect matching problem, solvable in polynomial time. Consequently, the ground state properties as well as minimum-energy domain wall (MEDW) excitations for very large 2 d systems, e.g. lattice graphs with up to N = 384 × 384 spins, can be analyzed very fast. Here, we investigate the critical behavior of the corresponding T = 0 ferromagnet to spin-glass transition, signaled by a breakdown of the magnetization, using finite-size scaling analyses of the magnetization and MEDW excitation energy and we contrast our numerical results with previous simulations and presumably exact results.

Exact ground states of large two-dimensional planar Ising spin glasses

Studying spin-glass physics through analyzing their ground-state properties has a long history. Although there exist polynomial-time algorithms for the two-dimensional planar case, where the problem of finding ground states is transformed to a minimum-weight perfect matching problem, the reachable system sizes have been limited both by the needed CPU time and by memory requirements. In this work, we present an algorithm for the calculation of exact ground states for two-dimensional Ising spin glasses with free boundary conditions in at least one direction. The algorithmic foundations of the method date back to the work of Kasteleyn from the 1960s for computing the complete partition function of the Ising model. Using Kasteleyn cities, we calculate exact ground states for huge two-dimensional planar Ising spin-glass lattices (up to 3000;{2} spins) within reasonable time. According to our knowledge, these are the largest sizes currently available. Kasteleyn cities were recently also used by Thomas and Middleton in the context of extended ground states on the torus. Moreover, they show that the method can also be used for computing ground states of planar graphs. Furthermore, we point out that the correctness of heuristically computed ground states can easily be verified. Finally, we evaluate the solution quality of heuristic variants of the L. Bieche approach.

A General Approximation Technique for Constrained Forest Problems

We present a general approximation technique for a large class of graph problems. Our technique mostly applies to problems of covering, at minimum cost, the vertices of a graph with trees, cycles, or paths satisfying certain requirements. In particular, many basic combinatorial optimization problems fit in this framework, including the shortest path, minimum-cost spanning tree, minimum-weight perfect matching, traveling salesman, and Steiner tree problems. Our technique produces approximation algorithms that run in $O(n^{2} \log n)$ time and come within a factor of 2 of optimal for most of these problems. For instance, we obtain a 2-approximation algorithm for the minimum-weight perfect matching problem under the triangle inequality. Our running time of $O(n^{2} \log n)$ time compares favorably with the best strongly polynomial exact algorithms running in $O(n^{3})$ time for dense graphs. A similar result is obtained for the 2-matching problem and its variants. We also derive the first approximation algorithms for many NP-complete problems, including the nonfixed point-to-point connection problem, the exact path partitioning problem, and complex location-design problems. Moreover, for the prize-collecting traveling salesman or Steiner tree problems, we obtain 2-approximation algorithms, therefore improving the previously best-known performance guarantees of 2.5 and 3, respectively [Math. Programming, 59 (1993), pp. 413–4201.