Minimum Color-Degree Perfect B-Matchings
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Lecture 4 1 the Permanent of a Matrix
Grafy a poˇcty - NDMI078 April 2009 Lecture 4 M. Loebl J.-S. Sereni 1 The permanent of a matrix 1.1 Minc's conjecture The set of permutations of f1; : : : ; ng is Sn. Let A = (ai;j)1≤i;j≤n be a square matrix with real non-negative entries. The permanent of the matrix A is n X Y perm(A) := ai,σ(i) : σ2Sn i=1 In 1973, Br`egman[4] proved M´ınc’sconjecture [18]. n×n Pn Theorem 1 (Br`egman,1973). Let A = (ai;j)1≤i;j≤n 2 f0; 1g . Set ri := j=1 ai;j. Then, n Y 1=ri perm(A) ≤ (ri!) : i=1 Further, if ri > 0 for every i 2 f1; 2; : : : ; ng, then there is equality if and only if up to permutations of rows and columns, A is a block-diagonal matrix, each block being a square matrix with all entries equal to 1. Several proofs of this result are known, the original being combinatorial. In 1978, Schrijver [22] found a neat and short proof. A probabilistic description of this proof is presented in the book of Alon and Spencer [3, Chapter 2]. The one we will see in Lecture 5 uses the concept of entropy, and was found by Radhakrishnan [20] in the late nineties. It is a nice illustration of the use of entropy to count combinatorial objects. 1.2 The van der Waerden conjecture A square matrix M = (mij)1≤i;j≤n of non-negative real numbers is doubly stochastic if the sum of the entries of every line is equal to 1, and the same holds for the sum of the entries of each column. -
Lecture 12 – the Permanent and the Determinant
Lecture 12 { The permanent and the determinant Uriel Feige Department of Computer Science and Applied Mathematics The Weizman Institute Rehovot 76100, Israel [email protected] June 23, 2014 1 Introduction Given an order n matrix A, its permanent is X Yn per(A) = aiσ(i) σ i=1 where σ ranges over all permutations on n elements. Its determinant is X Yn σ det(A) = (−1) aiσ(i) σ i=1 where (−1)σ is +1 for even permutations and −1 for odd permutations. A permutation is even if it can be obtained from the identity permutation using an even number of transpo- sitions (where a transposition is a swap of two elements), and odd otherwise. For those more familiar with the inductive definition of the determinant, obtained by developing the determinant by the first row of the matrix, observe that the inductive defini- tion if spelled out leads exactly to the formula above. The same inductive definition applies to the permanent, but without the alternating sign rule. The determinant can be computed in polynomial time by Gaussian elimination, and in time n! by fast matrix multiplication. On the other hand, there is no polynomial time algorithm known for computing the permanent. In fact, Valiant showed that the permanent is complete for the complexity class #P , which makes computing it as difficult as computing the number of solutions of NP-complete problems (such as SAT, Valiant's reduction was from Hamiltonicity). For 0/1 matrices, the matrix A can be thought of as the adjacency matrix of a bipartite graph (we refer to it as a bipartite adjacency matrix { technically, A is an off-diagonal block of the usual adjacency matrix), and then the permanent counts the number of perfect matchings. -
3.1 Matchings and Factors: Matchings and Covers
1 3.1 Matchings and Factors: Matchings and Covers This copyrighted material is taken from Introduction to Graph Theory, 2nd Ed., by Doug West; and is not for further distribution beyond this course. These slides will be stored in a limited-access location on an IIT server and are not for distribution or use beyond Math 454/553. 2 Matchings 3.1.1 Definition A matching in a graph G is a set of non-loop edges with no shared endpoints. The vertices incident to the edges of a matching M are saturated by M (M-saturated); the others are unsaturated (M-unsaturated). A perfect matching in a graph is a matching that saturates every vertex. perfect matching M-unsaturated M-saturated M Contains copyrighted material from Introduction to Graph Theory by Doug West, 2nd Ed. Not for distribution beyond IIT’s Math 454/553. 3 Perfect Matchings in Complete Bipartite Graphs a 1 The perfect matchings in a complete b 2 X,Y-bigraph with |X|=|Y| exactly c 3 correspond to the bijections d 4 f: X -> Y e 5 Therefore Kn,n has n! perfect f 6 matchings. g 7 Kn,n The complete graph Kn has a perfect matching iff… Contains copyrighted material from Introduction to Graph Theory by Doug West, 2nd Ed. Not for distribution beyond IIT’s Math 454/553. 4 Perfect Matchings in Complete Graphs The complete graph Kn has a perfect matching iff n is even. So instead of Kn consider K2n. We count the perfect matchings in K2n by: (1) Selecting a vertex v (e.g., with the highest label) one choice u v (2) Selecting a vertex u to match to v K2n-2 2n-1 choices (3) Selecting a perfect matching on the rest of the vertices. -
Matchgates Revisited
THEORY OF COMPUTING, Volume 10 (7), 2014, pp. 167–197 www.theoryofcomputing.org RESEARCH SURVEY Matchgates Revisited Jin-Yi Cai∗ Aaron Gorenstein Received May 17, 2013; Revised December 17, 2013; Published August 12, 2014 Abstract: We study a collection of concepts and theorems that laid the foundation of matchgate computation. This includes the signature theory of planar matchgates, and the parallel theory of characters of not necessarily planar matchgates. Our aim is to present a unified and, whenever possible, simplified account of this challenging theory. Our results include: (1) A direct proof that the Matchgate Identities (MGI) are necessary and sufficient conditions for matchgate signatures. This proof is self-contained and does not go through the character theory. (2) A proof that the MGI already imply the Parity Condition. (3) A simplified construction of a crossover gadget. This is used in the proof of sufficiency of the MGI for matchgate signatures. This is also used to give a proof of equivalence between the signature theory and the character theory which permits omittable nodes. (4) A direct construction of matchgates realizing all matchgate-realizable symmetric signatures. ACM Classification: F.1.3, F.2.2, G.2.1, G.2.2 AMS Classification: 03D15, 05C70, 68R10 Key words and phrases: complexity theory, matchgates, Pfaffian orientation 1 Introduction Leslie Valiant introduced matchgates in a seminal paper [24]. In that paper he presented a way to encode computation via the Pfaffian and Pfaffian Sum, and showed that a non-trivial, though restricted, fragment of quantum computation can be simulated in classical polynomial time. Underlying this magic is a way to encode certain quantum states by a classical computation of perfect matchings, and to simulate certain ∗Supported by NSF CCF-0914969 and NSF CCF-1217549. -
Rainbow Matchings and Transversals∗
AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 59(1) (2014), Pages 211–217 Rainbow matchings and transversals∗ Janos´ Barat´ † MTA-ELTE Geometric and Algebraic Combinatorics Research Group 1117 Budapest, P´azm´any P´eter s´et´any 1/C Hungary Ian M. Wanless School of Mathematical Sciences Monash University Clayton, Vic 3800 Australia Abstract We show that there exists a bipartite graph containing n matchings of 2 sizes mi n satisfying i mi = n + n/2−1, such that the matchings have no rainbow matching. This answers a question posed by Aharoni, Charbit and Howard. We also exhibit (n − 1) × n latin rectangles that cannot be decom- posed into transversals, and some related constructions. In the process we answer a question posed by H¨aggkvist and Johansson. Finally, we propose a Hall-type condition for the existence of a rain- bow matching. Two edges in a graph are independent if they do not share an endpoint. A matching is a set of edges that are pairwise independent. If M1,M2,...,Mn are matchings and there exist edges e1 ∈ M1, e2 ∈ M2,...,en ∈ Mn that form a matching then we say that {M1,M2,...,Mn} possesses a rainbow matching.Notethatwedo not require the matchings Mi to be disjoint, so it may even be the case that ej ∈ Mi for some j = i. A k × n matrix R, containing symbols from an alphabet Λ, can be viewed as listing edges in a bipartite graph GR. The two parts of GR correspond respectively to the columns of R, and to Λ. Each cell in R corresponds to an edge in GR that links the column and the symbol in the cell. -
The Geometry of Dimer Models
THE GEOMETRY OF DIMER MODELS DAVID CIMASONI Abstract. This is an expanded version of a three-hour minicourse given at the winterschool Winterbraids IV held in Dijon in February 2014. The aim of these lectures was to present some aspects of the dimer model to a geometri- cally minded audience. We spoke neither of braids nor of knots, but tried to show how several geometrical tools that we know and love (e.g. (co)homology, spin structures, real algebraic curves) can be applied to very natural problems in combinatorics and statistical physics. These lecture notes do not contain any new results, but give a (relatively original) account of the works of Kaste- leyn [14], Cimasoni-Reshetikhin [4] and Kenyon-Okounkov-Sheffield [16]. Contents Foreword 1 1. Introduction 1 2. Dimers and Pfaffians 2 3. Kasteleyn’s theorem 4 4. Homology, quadratic forms and spin structures 7 5. The partition function for general graphs 8 6. Special Harnack curves 11 7. Bipartite graphs on the torus 12 References 15 Foreword These lecture notes were originally not intended to be published, and the lectures were definitely not prepared with this aim in mind. In particular, I would like to arXiv:1409.4631v2 [math-ph] 2 Nov 2015 stress the fact that they do not contain any new results, but only an exposition of well-known results in the field. Also, I do not claim this treatement of the geometry of dimer models to be complete in any way. The reader should rather take these notes as a personal account by the author of some selected chapters where the words geometry and dimer models are not completely irrelevant, chapters chosen and organized in order for the resulting story to be almost self-contained, to have a natural beginning, and a happy ending. -
Parameterized Algorithms and Kernels for Rainbow Matching
Parameterized Algorithms and Kernels for Rainbow Matching Sushmita Gupta1, Sanjukta Roy2, Saket Saurabh3, and Meirav Zehavi4 1 University of Bergen, Norway [email protected] 2 The Institute of Mathematical Sciences, HBNI, Chennai, India [email protected] 3 University of Bergen, Norway, and The Institute of Mathematical Sciences, HBNI, Chennai, India [email protected] 4 University of Bergen, Norway [email protected] Abstract In this paper, we study the NP-complete colorful variant of the classical Matching problem, namely, the Rainbow Matching problem. Given an edge-colored graph G and a positive integer k, this problem asks whether there exists a matching of size at least k such that all the edges in the matching have distinct colors. We first develop a deterministic algorithm that solves Rainbow √ k ? 1+ 5 Matching on paths in time O ( 2 ) and polynomial space. This algorithm is based on a curious combination of the method of bounded search trees and a “divide-and-conquer-like” approach, where the branching process is guided by the maintenance of an auxiliary bipartite graph where one side captures “divided-and-conquered” pieces of the path. Our second result is a randomized algorithm that solves Rainbow Matching on general graphs in time O?(2k) and polynomial-space. Here, we show how a result by Björklund et al. [JCSS, 2017] can be invoked as a black box, wrapped by a probability-based analysis tailored to our problem. We also complement our two main results by designing kernels for Rainbow Matching on general and bounded-degree graphs. Keywords and phrases Rainbow Matching, Parameterized Algorithm, Bounded Search Trees, Divide-and-Conquer, 3-Set Packing, 3-Dimensional Matching Digital Object Identifier 10.4230/LIPIcs.MFCS.2017.71 1 Introduction The classical notion of matching has been extensively studied for several decades in the area of Combinatorial Optimization [6, 14]. -
Lectures 4 and 6 Lecturer: Michel X
18.438 Advanced Combinatorial Optimization Feb 13 and 25, 2014 Lectures 4 and 6 Lecturer: Michel X. Goemans Scribe: Zhenyu Liao and Michel X. Goemans Today, we will use an algebraic approach to solve the matching problem. Our goal is to derive an algebraic test for deciding if a graph G = (V; E) has a perfect matching. We may assume that the number of vertices is even since this is a necessary condition for having a perfect matching. First, we will define a few basic needed notations. Definition 1 A skew-symmetric matrix A is a square matrix which satisfies AT = −A, i.e. if A = (aij) we have aij = −aji for all i; j. For a graph G = (V; E) with jV j = n and jEj = m, we construct a n × n skew-symmetric matrix A = (aij) with an entry aij = −aji for each edge (i; j) 2 E and aij = 0 if (i; j) is not an edge; the values aij for the edges will be specified later. Recall: Definition 2 The determinant of matrix A is n X Y det(A) = sgn(σ) aiσ(i) σ2Sn i=1 where Sn is the set of all permutations of n elements and the sgn(σ) is defined to be 1 if the number of inversions in σ is even and −1 otherwise. Note that for a skew-symmetric matrix A, det(A) = det(−AT ) = (−1)n det(A). So if n is odd we have det(A) = 0. Consider K4, the complete graph on 4 vertices, and thus 0 1 0 a12 a13 a14 B−a12 0 a23 a24C A = B C : @−a13 −a23 0 a34A −a14 −a24 −a34 0 By computing its determinant one observes that 2 det(A) = (a12a34 − a13a24 + a14a23) : First, it is the square of a polynomial q(a) in the entries of A, and moreover this polynomial has a monomial precisely for each perfect matching of K4. -
Existences of Rainbow Matchings and Rainbow Matching Covers Lo, Allan
CORE Metadata, citation and similar papers at core.ac.uk Provided by University of Birmingham Research Portal University of Birmingham Existences of rainbow matchings and rainbow matching covers Lo, Allan DOI: 10.1016/j.disc.2015.05.015 License: Creative Commons: Attribution-NonCommercial-NoDerivs (CC BY-NC-ND) Document Version Publisher's PDF, also known as Version of record Citation for published version (Harvard): Lo, A 2015, 'Existences of rainbow matchings and rainbow matching covers', Discrete Mathematics, vol. 338, pp. 2119-2124. https://doi.org/10.1016/j.disc.2015.05.015 Link to publication on Research at Birmingham portal Publisher Rights Statement: Eligibility for repository : checked 16/06/2015 General rights Unless a licence is specified above, all rights (including copyright and moral rights) in this document are retained by the authors and/or the copyright holders. The express permission of the copyright holder must be obtained for any use of this material other than for purposes permitted by law. •Users may freely distribute the URL that is used to identify this publication. •Users may download and/or print one copy of the publication from the University of Birmingham research portal for the purpose of private study or non-commercial research. •User may use extracts from the document in line with the concept of ‘fair dealing’ under the Copyright, Designs and Patents Act 1988 (?) •Users may not further distribute the material nor use it for the purposes of commercial gain. Where a licence is displayed above, please note the terms and conditions of the licence govern your use of this document. -
A Short Course on Matching Theory, ECNU Shanghai, July 2011
A short course on matching theory, ECNU Shanghai, July 2011. Sergey Norin LECTURE 1 Fundamental definitions and theorems. 1.1. Outline of Lecture • Definitions • Hall's theorem • Tutte's Matching theorem 1.2. Basic definitions. A matching in a graph G is a set of edges M such that no two edges share a common end. A vertex v is said to be saturated or matched by a matching M if v is an end of an edge in M. Otherwise, v is unsaturated or unmatched. The matching number ν(G) of G is the maximum number of edges in a matching in G. A matching M is perfect if every vertex of G is saturated. We will be primarily interested in perfect matchings. We will denote by M(G) the set of all perfect matchings of a graph G and by m(G) := jM(G)j the number of perfect matchings. The main goal of this course is to demonstrate classical and new results related to computing or estimating the quantity m(G). Example 1. The graph K4 is the complete graph on 4 vertices. Let V (K4) = f1; 2; 3; 4g. Then f12; 34g; f13; 24g; f14; 23g are all perfect matchings of K4, i.e. all the elements of the set M(G). We have jm(G)j = 3. Every edge of K4 belongs to exactly one perfect matching. 1 2 SERGEY NORIN, MATCHING THEORY Figure 1. A graph with no perfect matching. Computation of m(G) is of interest for the following reasons. If G is a graph representing connections between the atoms in a molecule, then m(G) encodes some stability and thermodynamic properties of the molecule. -
Arxiv:2108.12879V1 [Cs.CC] 29 Aug 2021
Parameterizing the Permanent: Hardness for K8-minor-free graphs Radu Curticapean∗ Mingji Xia† Abstract In the 1960s, statistical physicists discovered a fascinating algorithm for counting perfect matchings in planar graphs. Valiant later showed that the same problem is #P-hard for general graphs. Since then, the algorithm for planar graphs was extended to bounded-genus graphs, to graphs excluding K3;3 or K5, and more generally, to any graph class excluding a fixed minor H that can be drawn in the plane with a single crossing. This stirred up hopes that counting perfect matchings might be polynomial-time solvable for graph classes excluding any fixed minor H. Alas, in this paper, we show #P-hardness for K8-minor-free graphs by a simple and self-contained argument. 1 Introduction A perfect matching in a graph G is an edge-subset M ⊆ E(G) such that every vertex of G has exactly one incident edge in M. Counting perfect matchings is a central and very well-studied problem in counting complexity. It already starred in Valiant’s seminal paper [23] that introduced the complexity class #P, where it was shown that counting perfect matchings is #P-complete. The problem has driven progress in approximate counting and underlies the so-called holographic algorithms [24, 6, 4, 5]. It also occurs outside of counting complexity, e.g., in statistical physics, via the partition function of the dimer model [21, 16, 17]. In algebraic complexity theory, the matrix permanent is a very well-studied algebraic variant of the problem of counting perfect matchings [1]. -
Complexity Results for Rainbow Matchings
Complexity Results for Rainbow Matchings Van Bang Le1 and Florian Pfender2 1 Universität Rostock, Institut für Informatik 18051 Rostock, Germany [email protected] 2 University of Colorado at Denver, Department of Mathematics & Statistics Denver, CO 80202, USA [email protected] Abstract A rainbow matching in an edge-colored graph is a matching whose edges have distinct colors. We address the complexity issue of the following problem, max rainbow matching: Given an edge-colored graph G, how large is the largest rainbow matching in G? We present several sharp contrasts in the complexity of this problem. We show, among others, that max rainbow matching can be approximated by a polynomial algorithm with approxima- tion ratio 2/3 − ε. max rainbow matching is APX-complete, even when restricted to properly edge-colored linear forests without a 5-vertex path, and is solvable in polynomial time for edge-colored forests without a 4-vertex path. max rainbow matching is APX-complete, even when restricted to properly edge-colored trees without an 8-vertex path, and is solvable in polynomial time for edge-colored trees without a 7-vertex path. max rainbow matching is APX-complete, even when restricted to properly edge-colored paths. These results provide a dichotomy theorem for the complexity of the problem on forests and trees in terms of forbidding paths. The latter is somewhat surprising, since, to the best of our knowledge, no (unweighted) graph problem prior to our result is known to be NP-hard for simple paths. We also address the parameterized complexity of the problem.