Minimum Weight Perfect Matching Research Articles

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.

Optimal selection of essential interconnections for structural controllability in heterogeneous subsystems

This paper deals with structural controllability of a linear time-invariant composite system consisting of several heterogeneous subsystems. The interaction links through which the subsystems interact with other subsystems are referred to as interconnections. We assume the composite system to be structurally controllable if all possible interconnections are present. Our objective is to identify a minimum cardinality set of interconnections required to retain the structural controllability of the composite system. We refer to this problem as the optimal essential interconnection selection problem. We approach the problem in a structured framework, where the zero/nonzero structure of the subsystems is used in the analysis instead of the numerical matrices themselves. This analysis applies to an equivalence class of systems with the same sparsity pattern. Firstly, we propose a polynomial time algorithm to solve the optimal essential interconnection selection problem on a structured composite system when each subsystem is irreducible and no subsystem has a perfect matching in its state bipartite graph. Later, we consider the case where one or more subsystems have perfect matching in their state bipartite graphs. For this case, we first prove a lower bound on the number of minimum number of interconnections needed. Subsequently, we provide a polynomial time algorithm based on a minimum weight perfect matching algorithm and a so-called stub-matching algorithm that achieves this bound. We also discuss about how heterogeneity of the subsystems poses different challenges to the homogeneous counterpart and demonstrate the algorithms using illustrative examples.

A deterministic parallel algorithm for bipartite perfect matching

A fundamental quest in the theory of computing is to understand the power of randomness. It is not known whether every problem with an efficient randomized algorithm also has one that does not use randomness. One of the extensively studied problems under this theme is that of perfect matching. The perfect matching problem has a randomized parallel (NC) algorithm based on the Isolation Lemma of Mulmuley, Vazirani, and Vazirani. It is a long-standing open question whether this algorithm can be derandomized. In this article, we give an almost complete derandomization of the Isolation Lemma for perfect matchings in bipartite graphs. This gives us a deterministic parallel (quasi-NC) algorithm for the bipartite perfect matching problem. Derandomization of the Isolation Lemma means that we deterministically construct a weight assignment so that the minimum weight perfect matching is unique. We present three different ways of doing this construction with a common main idea.

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.

A linear time randomized approximation algorithm for Euclidean matching

We study the problem of computing Euclidean minimum weight matching which investigates the minimum weight perfect matching in a complete geometric graph on a set of 2n points in the plane. We propose a simple randomized approximation algorithm based on the geometrical aspect of the problem with O(n) expected time. The proposed algorithm computes a matching within at most 3 factors of the optimal solution. We also do some experimental tests to evaluate the performance of the proposed algorithm which indicate the efficiency of the proposed algorithm in finding the approximate matching in the practice.

Multi-path Summation for Decoding 2D Topological Codes

Fault tolerance is a prerequisite for scalable quantum computing. Architectures based on 2D topological codes are effective for near-term implementations of fault tolerance. To obtain high performance with these architectures, we require a decoder which can adapt to the wide variety of error models present in experiments. The typical approach to the problem of decoding the surface code is to reduce it to minimum-weight perfect matching in a way that provides a suboptimal threshold error rate, and is specialized to correct a specific error model. Recently, optimal threshold error rates for a variety of error models have been obtained by methods which do not use minimum-weight perfect matching, showing that such thresholds can be achieved in polynomial time. It is an open question whether these results can also be achieved by minimum-weight perfect matching. In this work, we use belief propagation and a novel algorithm for producing edge weights to increase the utility of minimum-weight perfect matching for decoding surface codes. This allows us to correct depolarizing errors using the rotated surface code, obtaining a threshold of 17.76±0.02%. This is larger than the threshold achieved by previous matching-based decoders (14.88±0.02%), though still below the known upper bound of ∼18.9%.

Machine-learning-assisted correction of correlated qubit errors in a topological code

A fault-tolerant quantum computation requires an efficient means to detect and correct errors that accumulate in encoded quantum information. In the context of machine learning, neural networks are a promising new approach to quantum error correction. Here we show that a recurrent neural network can be trained, using only experimentally accessible data, to detect errors in a widely used topological code, the surface code, with a performance above that of the established minimum-weight perfect matching (or blossom) decoder. The performance gain is achieved because the neural network decoder can detect correlations between bit-flip (X) and phase-flip (Z) errors. The machine learning algorithm adapts to the physical system, hence no noise model is needed. The long short-term memory layers of the recurrent neural network maintain their performance over a large number of quantum error correction cycles, making it a practical decoder for forthcoming experimental realizations of the surface code.

Scaling Algorithms for Weighted Matching in General Graphs

We present a new scaling algorithm for maximum (or minimum) weight perfect matching on general, edge weighted graphs. Our algorithm runs in O ( m √ n log( nN )) time, O ( m √ n ) per scale, which matches the running time of the best cardinality matching algorithms on sparse graphs [16, 20, 36, 37]. Here, m , n , and N bound the number of edges, vertices, and magnitude, respectively, of any integer edge weight. Our result improves on a 25-year-old algorithm of Gabow and Tarjan, which runs in O ( m √ n log n α ( m , n ) log( nN )) time.

Fault-tolerance thresholds for the surface code with fabrication errors

The construction of topological error correction codes requires the ability to fabricate a lattice of physical qubits embedded on a manifold with a non-trivial topology such that the quantum information is encoded in the global degrees of freedom (i.e. the topology) of the manifold. However, the manufacturing of large-scale topological devices will undoubtedly suffer from fabrication errors---permanent faulty components such as missing physical qubits or failed entangling gates---introducing permanent defects into the topology of the lattice and hence significantly reducing the distance of the code and the quality of the encoded logical qubits. In this work we investigate how fabrication errors affect the performance of topological codes, using the surface code as the testbed. A known approach to mitigate defective lattices involves the use of primitive SWAP gates in a long sequence of syndrome extraction circuits. Instead, we show that in the presence of fabrication errors the syndrome can be determined using the supercheck operator approach and the outcome of the defective gauge stabilizer generators without any additional computational overhead or the use of SWAP gates. We report numerical fault-tolerance thresholds in the presence of both qubit fabrication and gate fabrication errors using a circuit-based noise model and the minimum-weight perfect matching decoder. Our numerical analysis is most applicable to 2D chip-based technologies, but the techniques presented here can be readily extended to other topological architectures. We find that in the presence of 8% qubit fabrication errors, the surface code can still tolerate a computational error rate of up to 0.1%.

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.

Comparing Phylogenetic Trees by Matching Nodes Using the Transfer Distance Between Partitions.

Ability to quantify dissimilarity of different phylogenetic trees describing the relationship between the same group of taxa is required in various types of phylogenetic studies. For example, such metrics are used to assess the quality of phylogeny construction methods, to define optimization criteria in supertree building algorithms, or to find horizontal gene transfer (HGT) events. Among the set of metrics described so far in the literature, the most commonly used seems to be the Robinson-Foulds distance. In this article, we define a new metric for rooted trees-the Matching Pair (MP) distance. The MP metric uses the concept of the minimum-weight perfect matching in a complete bipartite graph constructed from partitions of all pairs of leaves of the compared phylogenetic trees. We analyze the properties of the MP metric and present computational experiments showing its potential applicability in tasks related to finding the HGT events.

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.

Fast fault-tolerant decoder for qubit and qudit surface codes

The surface code is one of the most promising candidates for combating errors in large scale fault-tolerant quantum computation. A fault-tolerant decoder is a vital part of the error correction process---it is the algorithm which computes the operations needed to correct or compensate for the errors according to the measured syndrome, even when the measurement itself is error prone. Previously decoders based on minimum-weight perfect matching have been studied. However, these are not immediately generalizable from qubit to qudit codes. In this work, we develop a fault-tolerant decoder for the surface code, capable of efficient operation for qubits and qudits of any dimension, generalizing the decoder first introduced by Bravyi and Haah [Phys. Rev. Lett. 111, 200501 (2013)]. We study its performance when both the physical qudits and the syndromes measurements are subject to generalized uncorrelated bit-flip noise (and the higher dimensional equivalent). We show that, with appropriate enhancements to the decoder and a high enough qudit dimension, a threshold at an error rate of more than 8% can be achieved.

Perfect Matching Quad Layouts for Manifold Meshes

AbstractThis paper introduces a new approach to automatically generate pure quadrilateral patch layouts on manifold meshes. The algorithm is based on a careful construction of a singularity graph of a given input frame field or a given periodic global parameterization. A pure quadrilateral patch layout is then derived as a constrained minimum weight perfect matching of that graph. The resulting layout is optimal relative to a balance between coarseness and geometric feature alignment. We formulate the problem of finding pure quadrilateral patch layouts as a global optimization problem related to a well‐known concept in graph theory. The main advantage of the new method is its simplicity and its computation speed. Patch layouts generated by the present algorithm are high quality and are very competitive compared to current state of the art.

Optimum matchings in weighted bipartite graphs

Given an integer weighted bipartite graph \(\{G,w\}\) we consider the problems of finding all the edges that occur in some minimum weight matching of maximum cardinality and enumerating all the minimum weight perfect matchings. We construct a subgraph \(G_\mathrm{cs}\) of G, which depends on an \(\epsilon \)-optimal solution of the dual linear program associated to the assignment problem on \(\{G,w\}\), that allows us to reduce these problems to their unweighted variants on \(G_\mathrm{cs}\). For instance, starting from scratch we get an algorithm that solves the problem of finding all the edges that occur in some minimum weight perfect matching in time \(O(\sqrt{n}m\log (nW))\), where \(n=|U|\ge |V|\), \(m=|E|\), and \(W=\mathrm{max}\{|w(e)|\, :\, e\in E\}\).

On the transportation problem with market choice

We study a variant of the classical transportation problem in which suppliers with limited capacities have a choice of which demands (markets) to satisfy. We refer to this problem as the transportation problem with market choice (TPMC). While the classical transportation problem is known to be strongly polynomial-time solvable, we show that its market choice counterpart is strongly NP-complete. For the special case when all potential demands are no greater than two, we show that the problem reduces in polynomial time to minimum weight perfect matching in a general graph, and thus can be solved in polynomial time. We give valid inequalities and coefficient update schemes for general mixed-integer sets that are substructures of TPMC. Finally, we give conditions under which these inequalities define facets, and report our preliminary computational experiments with using them in a branch-and-cut algorithm.

Efficient Markov chain Monte Carlo algorithm for the surface code

Minimum-weight perfect matching (MWPM) has been been the primary classical algorithm for error correction in the surface code, since it is of low runtime complexity and achieves relatively low logical error rates [Phys. Rev. Lett. 108, 180501 (2012)]. A Markov chain Monte Carlo (MCMC) algorithm [Phys. Rev. Lett. 109, 160503 (2012)] is able to achieve lower logical error rates and higher thresholds than MWPM, but requires a classical runtime complexity which is super-polynomial in L, the linear size of the code. In this work we present an MCMC algorithm that achieves significantly lower logical error rates than MWPM at the cost of a polynomially increased classical runtime complexity. For error rates p close to the threshold, our algorithm needs a runtime complexity which is increased by O(L^2) relative to MWPM in order to achieve a lower logical error rate. If p is below an L-dependent critical value, no increase in the runtime complexity is necessary any longer. For p->0, the logical error rate achieved by our algorithm is exponentially smaller (in L) than that of MWPM, without requiring an increased runtime complexity. Our algorithm allows for trade-offs between runtime and achieved logical error rates as well as for parallelization, and can be also used to correct in the case of imperfect stabilizer measurements.

Partial 3‐D Correspondence from Shape Extremities

AbstractWe present a 3‐D correspondence method to match the geometric extremities of two shapes which are partially isometric. We consider the most general setting of the isometric partial shape correspondence problem, in which shapes to be matched may have multiple common parts at arbitrary scales as well as parts that are not similar. Our rank‐and‐vote‐and‐combine algorithm identifies and ranks potentially correct matches by exploring the space of all possible partial maps between coarsely sampled extremities. The qualified top‐ranked matchings are then subjected to a more detailed analysis at a denser resolution and assigned with confidence values that accumulate into a vote matrix. A minimum weight perfect matching algorithm is finally iterated to combine the accumulated votes into an optimal (partial) mapping between shape extremities, which can further be extended to a denser map. We test the performance of our method on several data sets and benchmarks in comparison with state of the art.

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.