On the Expressive Power of Permanents and Perfect Matchings of Matrices of Bounded Pathwidth/Cliquewidth

Some 25 years ago Valiant introduced an algebraic model of computation in order to study the complexity of evaluating families of polynomials. The theory was introduced along with the complexity classes VP and VNP which are analogues of the classical classes P and NP. Families of polynomials that are difficult to evaluate (that is, VNP-complete) include the permanent and hamiltonian polynomials. In a previous paper the authors together with P. Koiran studied the expressive power of permanent and hamiltonian polynomials of matrices of bounded treewidth, as well as the expressive power of perfect matchings of planar graphs. It was established that the permanent and hamiltonian polynomials of matrices of bounded treewidth are equivalent to arithmetic formulas. Also, the sum of weights of perfect matchings of planar graphs was shown to be equivalent to (weakly) skew circuits. In this paper we continue the research in the direction described above, and study the expressive power of permanents, hamiltonians and perfect matchings of matrices that have bounded pathwidth or bounded cliquewidth. In particular, we prove that permanents, hamiltonians and perfect matchings of matrices that have bounded pathwidth express exactly arithmetic formulas. This is an improvement of our previous result for matrices of bounded treewidth. Also, for matrices of bounded weighted cliquewidth we show membership in VP for these polynomials.     

Random Bichromatic Matchings

Given a graph with edges colored Red and Blue, we study the problem of sampling and approximately counting the number of matchings with exactly k Red edges. We solve the problem of estimating the number of perfect matchings with exactly k Red edges for dense graphs. We study a Markov chain on the space of all matchings of a graph that favors matchings with k Red edges. We show that it is rapidly mixing using non-traditional canonical paths that can backtrack. We show that this chain can be used to sample matchings in the 2-dimensional toroidal lattice of any fixed size ℓ with k Red edges, where the horizontal edges are Red and the vertical edges are Blue. An extended abstract appeared in J.R. Correa, A. Hevia and M.A. Kiwi (eds.) Proceedings of the 7th Latin American Theoretical Informatics Symposium, LNCS 3887, pp. 190–201, Springer, 2006. N. Bhatnagar’s and D. Randall’s research was supported in part by NSF grants CCR-0515105 and DMS-0505505. V.V. Vazirani’s research was supported in part by NSF grants 0311541, 0220343 and CCR-0515186. N. Bhatnagar’s and E. Vigoda’s research was supported in part by NSF grant CCR-0455666.     

Almost Exact Matchings

In the exact matching problem we are given a graph G, some of whose edges are colored red, and a positive integer k. The goal is to determine if G has a perfect matching, exactly k edges of which are red. More generally if the matching number of G is m=m(G), the goal is to find a matching with m edges, exactly k edges of which are red, or determine that no such matching exists. This problem is one of the few remaining problems that have efficient randomized algorithms (in fact, this problem is in RNC), but for which no polynomial time deterministic algorithm is known. Our first result shows that, in a sense, this problem is as close to being in P as one can get. We give a polynomial time deterministic algorithm that either correctly decides that no maximum matching has exactly k red edges, or exhibits a matching with m(G)?1 edges having exactly k red edges. Hence, the additive error is one. We also present an efficient algorithm for the exact matching problem in families of graphs for which this problem is known to be tractable. We show how to count the number of exact perfect matchings in K _3,3-minor free graphs (these include all planar graphs as well as many others) in O(n ^3.19) worst case time. Our algorithm can also count the number of perfect matchings in K _3,3-minor free graphs in O(n ^2.19) time.     

Uniquely Restricted Matchings

A matching in a graph is a set of edges no two of which share a common vertex. In this paper we introduce a new, specialized type of matching which we call uniquely restricted matchings, originally motivated by the problem of determining a lower bound on the rank of a matrix having a specified zero/ non-zero pattern. A uniquely restricted matching is defined to be a matching M whose saturated vertices induce a subgraph which has only one perfect matching, namely M itself. We introduce the two problems of recognizing a uniquely restricted matching and of finding a maximum uniquely restricted matching in a given graph, and present algorithms and complexity results for certain special classes of graphs. We demonstrate that testing whether a given matching M is uniquely restricted can be done in O(|M||E|) time for an arbitrary graph G=(V,E) and in linear time for cacti, interval graphs, bipartite graphs, split graphs and threshold graphs. The maximum uniquely restricted matching problem is shown to be NP-complete for bipartite graphs, split graphs, and hence for chordal graphs and comparability graphs, but can be solved in linear time for threshold graphs, proper interval graphs, cacti and block graphs. Received April 12, 1998; revised June 21, 1999.     

Matchings In Three Catalan Lattices

In this note we consider a series of lattices that are enumerated by the well-known Catalan numbers. For each of these lattices, we exhibit a matching in a constructive way.     

Three-Sided Stable Matchings with Cyclic Preferences

Knuth (Mariages Stables, Les Presses de L’Université de Montréal, 1976) asked whether the stable matching problem can be generalised to three dimensions, e.g., for families containing a man, a woman and a dog. Subsequently, several authors considered the three-sided stable matching problem with cyclic preferences, where men care only about women, women only about dogs, and dogs only about men. In this paper we prove that if the preference lists may be incomplete, then the problem of deciding whether a stable matching exists, given an instance of the three-sided stable matching problem with cyclic preferences, is NP-complete. Considering an alternative stability criterion, strong stability, we show that the problem is NP-complete even for complete lists. These problems can be regarded as special types of stable exchange problems, therefore these results have relevance in some real applications, such as kidney exchange programs.     

Understanding the Generalized Median Stable Matchings

Let I be a stable matching instance with N stable matchings. For each man m, order his (not necessarily distinct) N partners from his most preferred to his least preferred. Denote the ith woman in his sorted list as p _i (m). Let α _i consist of the man-woman pairs where each man m is matched to p _i (m). Teo and Sethuraman proved this surprising result: for i=1 to N, not only is α _i a matching, it is also stable. The α _i ’s are called the generalized median stable matchings of I. Determining if these stable matchings can be computed efficiently is an open problem.     

限制树宽图上的有界聚类

有界聚类问题源于II3M研究院开发的一个分布式流处理系统,即S系统。问题的输入是一个点赋权和边赋权的无向图,并指定若干个称为终端的顶点。称顶点集合的一个子集为一个子类。子类中所有顶点的权和加上该子类边界上所有边的权和称为该子类的费用。有界聚类问题是要得到所有顶点的一个聚类,要求每个子类的费用不超过给定预算召,每个子类至多包含一个终端,并使得所有子类的总费用最小。对于限制树宽图上的有界聚类问题,给出了拟多项式时间精确算法。利用取整的技巧对该算法进行修正,可在多项式时间之内得到(1+ε)-近似解,其中每个子类的费用不超过(1+ε)B,:是任意小的正数。如果进一步要求每个子类恰好包含一个终端,则所给算法可在多项式时间之内得到(1+ε)-近似解,其中每个子类的费用不超过(2+ε)B。     

字符识别中两种改进的模板匹配算法

模板匹配算法是字符识别中最常用的一种算法。本文首先介绍了字符识别与模板匹配算法的基本原理,然后在传统模板匹配方法的基础上提出了两种改进的模板匹配算法。一是多模板匹配法,该算法可以对模糊不清、有倾斜的字符进行了有效识别;二是字符区域分割匹配法,该算法使相似度较大的字符可以得到准确的识别,并且大大减少了计算量,加快了识别速度。本文使基于模板匹配的字符识别达到了更有效,更准确的效果。     

Pareto Optimal Matchings in Many-to-Many Markets with Ties

We consider Pareto optimal matchings (POMs) in a many-to-many market of applicants and courses where applicants have preferences, which may include ties, over individual courses and lexicographic preferences over sets of courses. Since this is the most general setting examined so far in the literature, our work unifies and generalizes several known results. Specifically, we characterize POMs and introduce the Generalized Serial Dictatorship Mechanism with Ties (GSDT) that effectively handles ties via properties of network flows. We show that GSDT can generate all POMs using different priority orderings over the applicants, but it satisfies truthfulness only for certain such orderings. This shortcoming is not specific to our mechanism; we show that any mechanism generating all POMs in our setting is prone to strategic manipulation. This is in contrast to the one-to-one case (with or without ties), for which truthful mechanisms generating all POMs do exist.     

Log-Space Algorithms for Paths and Matchings in k-Trees

Reachability and shortest path problems are NL-complete for general graphs. They are known to be in L for graphs of tree-width 2 (Jakoby and Tantau in Proceedings of FSTTCS’07: The 27th Annual Conference on Foundations of Software Technology and Theoretical Computer Science, pp. 216–227, 2007). In this paper, we improve these bounds for k-trees, where k is a constant. In particular, the main results of our paper are log-space algorithms for reachability in directed k-trees, and for computation of shortest and longest paths in directed acyclic k-trees. Besides the path problems mentioned above, we also consider the problem of deciding whether a k-tree has a perfect matching (decision version), and if so, finding a perfect matching (search version), and prove that these two problems are L-complete. These problems are known to be in P and in RNC for general graphs, and in SPL for planar bipartite graphs, as shown in Datta et al. (Theory Comput. Syst. 47:737–757, 2010). Our results settle the complexity of these problems for the class of k-trees. The results are also applicable for bounded tree-width graphs, when a tree-decomposition is given as input. The technique central to our algorithms is a careful implementation of the divide-and-conquer approach in log-space, along with some ideas from Jakoby and Tantau (Proceedings of FSTTCS’07: The 27th Annual Conference on Foundations of Software Technology and Theoretical Computer Science, pp. 216–227, 2007) and Limaye et al. (Theory Comput. Syst. 46(3):499–522, 2010).     

VisFM: Visual Analysis of Image Feature Matchings

Feature matching is the most basic and pervasive problem in computer vision and it has become a primary component in big data analytics. Many tools have been developed for extracting and matching features in video streams and image frames. However, one of the most basic tools, that is, a tool for simply visualizing matched features for the comparison and evaluation of computer vision algorithms is not generally available, especially when dealing with a large number of matching lines. We introduce VisFM, an integrated visual analysis system for comprehending and exploring image feature matchings. VisFM presents a matching view with an intuitive line bundling to provide useful insights regarding the quality of matched features. VisFM is capable of showing a summarization of the features and matchings through group view to assist domain experts in observing the feature matching patterns from multiple perspectives. VisFM incorporates a series of interactions for exploring the feature data. We demonstrate the visual efficacy of VisFM by applying it to three scenarios. An informal expert feedback, conducted by our collaborator in computer vision, demonstrates how VisFM can be used for comparing and analysing feature matchings when the goal is to improve an image retrieval algorithm.     

Bounded synthesis

A fundamental challenge in the synthesis of reactive systems is the size of the search space: the number of candidate implementations of a temporal specification is typically superexponential or even, for distributed system architectures, infinite. In this article, we introduce the bounded synthesis approach, which makes it possible to traverse this immense search space in a structured manner. We fix a bound on a system parameter, such as the number of states, and limit the search to those implementations that fall below the bound. By incrementally expanding the search to larger bounds, we maintain completeness, while orienting the search towards the simplest (and often most useful) solutions. The technical backbone of this solution is a novel translation from formulas of linear-time temporal logic to sequences of safety tree automata, which are guaranteed to underapproximate the specification and to eventually become emptiness-equivalent. Bounded synthesis is applicable to the entire range of synthesis problems, from individual processes to synchronous and asynchronous distributed systems, to systems with additional design constraints, such as symmetry. We include experimental results from a SMT-based implementation, which demonstrate that bounded synthesis solves many synthesis problems that were previously considered intractable.     

Bounded underapproximations

We show a new and constructive proof of the following language-theoretic result: for every context-free language L, there is a bounded context-free language L′⊆L which has the same Parikh (commutative) image as L. Bounded languages, introduced by Ginsburg and Spanier, are subsets of regular languages of the form w₁^*w₂^*?w_m^*w_{1}^{*}w_{2}^{*}\cdots w_{m}^{*} for some w ₁,…,w _m ∈Σ ^∗. In particular bounded context-free languages have nice structural and decidability properties. Our proof proceeds in two parts. First, we give a new construction that shows that each context free language L has a subset L _N that has the same Parikh image as L and that can be represented as a sequence of substitutions on a linear language. Second, we inductively construct a Parikh-equivalent bounded context-free subset of L _N .     

Bounded time-stamps

Summary Time-stamps are labels which a system adds to its data items. These labels enable the system to keep track of the temporal precedence relations among its data elements. Many distributed protocols and some applications use the natural numbers as time-stamps. The natural numbers however are not useful for bounded protocols. In this paper we develop a theory ofbounded time-stamps. Time-stamp schemes are defined and the complexity of their implementation is analyzed. This indicates a direction for developing a general tool for converting time-stamp based protocols to bounded protocols. Amos Israeli received his B.Sc. in Mathematics and Physics from Hebrew University in 1976, and his M.Sc. and D.Sc. in Computer Science from the Weizmann Institute in 1980 and the Technion in 1985, respectively. Currently he is a senior lecturer at the Tlectrical Engineering Department at the Technion. Prior to this he was a postdoctoral fellow at the Aiken Computation Laboratory at Harvard. His research interests are in Parallel and Distributed Computing and in Robotics. In particular he has worked on the design and analysis of Wait-Free and Self-Stabilizing distributed protocols. Ming Li received his M.S. and Ph.D. in Computer Science from Wayne State University in 1980 and Cornell University 1985, respectively. Currently he is an associate professor at the Computer Science Department at the University of Waterloo. His research interests are in Theory of Computing, Kolmogorov Complexity, and Machine Learning.Supported in part by the Weizmann fellowship and NSF Grant DCR-86-00379Supported in part by ONR Grant N00014-85-k-0445 and Army Research Office Grant DAAL03-86-K-0171 at Harvard University, by NSF Grant kDCR-86-06366 at Ohio State University, and by NSERC Operating Grant OGP0036747. Most of this work was done when the authors were at Aiken Computation Laboratory at Harvard University. The authors also acknowledge the hospitality of the computer science department at York University, Canada     

Sex-Equal Stable Matchings: Complexity and Exact Algorithms

We explore the complexity and exact computation of a variant of the classical stable marriage problem in which we seek matchings that are not only stable, but are also “fair” in a formal sense. In particular, we study the sex-equal stable marriage problem (SESM), in which, roughly speaking, we wish to find a stable matching with the property that the men’s happiness is as close as possible to the women’s happiness. This problem is known to be strongly NP-hard (Kato in Jpn. J. Ind. Appl. Math. 10:1–19, 1993). We specifically consider SESM instances in which the preference lists of the men and/or women are bounded in length by a constant. On the negative side, we show that SESM is NP-hard, even if both the men’s and women’s preference lists are of length at most three, and is not even in the class XP when parameterized by the objective value of the solution. This strengthens the NP-hardness results of Kato (Jpn. J. Ind. Appl. Math. 10:1–19, 1993). On the positive side, we show that our hardness result is “tight” by giving a polynomial-time algorithm for the case in which the preference lists on one side (say the men) are of length at most two, and the lengths of the lists on the other side (the women) are unbounded. Furthermore, we give a low-order exponential-time algorithm for SESM in which the preference lists on one side are of length at most l (and the lengths of the lists on the other side are unbounded). In particular, for every pair of constants $l \in\mathcal{Z}^{+}$ and $\epsilon\in\mathcal{R}^{+}$ there is an algorithm with running time bounded by $O^{\star}(2^{(5 - \sqrt{24})(l-2 + \epsilon)n}) + O^{\star}(2^{\frac {(l-1)}{2\epsilon}})$ . Hence, if ? is chosen to be a sufficiently small constant, the running time is in O ^?(1.0726 ⁿ ), O ^?(1.1504 ⁿ ), O ^?(1.2339 ⁿ ),… for l=3,4,5,…?.