Two Algorithms for the Matroid Secretary Problem
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Parameterized Algorithms Using Matroids Lecture I: Matroid Basics and Its Use As Data Structure
Parameterized Algorithms using Matroids Lecture I: Matroid Basics and its use as data structure Saket Saurabh The Institute of Mathematical Sciences, India and University of Bergen, Norway, ADFOCS 2013, MPI, August 5{9, 2013 1 Introduction and Kernelization 2 Fixed Parameter Tractable (FPT) Algorithms For decision problems with input size n, and a parameter k, (which typically is the solution size), the goal here is to design an algorithm with (1) running time f (k) nO , where f is a function of k alone. · Problems that have such an algorithm are said to be fixed parameter tractable (FPT). 3 A Few Examples Vertex Cover Input: A graph G = (V ; E) and a positive integer k. Parameter: k Question: Does there exist a subset V 0 V of size at most k such ⊆ that for every edge( u; v) E either u V 0 or v V 0? 2 2 2 Path Input: A graph G = (V ; E) and a positive integer k. Parameter: k Question: Does there exist a path P in G of length at least k? 4 Kernelization: A Method for Everyone Informally: A kernelization algorithm is a polynomial-time transformation that transforms any given parameterized instance to an equivalent instance of the same problem, with size and parameter bounded by a function of the parameter. 5 Kernel: Formally Formally: A kernelization algorithm, or in short, a kernel for a parameterized problem L Σ∗ N is an algorithm that given ⊆ × (x; k) Σ∗ N, outputs in p( x + k) time a pair( x 0; k0) Σ∗ N such that 2 × j j 2 × • (x; k) L (x 0; k0) L , 2 () 2 • x 0 ; k0 f (k), j j ≤ where f is an arbitrary computable function, and p a polynomial. -
Lecture 0: Matroid Basics
Parameterized Algorithms using Matroids Lecture 0: Matroid Basics Saket Saurabh The Institute of Mathematical Sciences, India and University of Bergen, Norway. ADFOCS 2013, MPI, August 04-09, 2013 Kruskal's Greedy Algorithm for MWST Let G = (V; E) be a connected undirected graph and let ≥0 w : E ! R be a weight function on the edges. Kruskal's so-called greedy algorithm is as follows. The algorithm consists of selecting successively edges e1; e2; : : : ; er. If edges e1; e2; : : : ; ek has been selected, then an edge e 2 E is selected so that: 1 e=2f e1; : : : ; ekg and fe; e1; : : : ; ekg is a forest. 2 w(e) is as small as possible among all edges e satisfying (1). We take ek+1 := e. If no e satisfying (1) exists then fe1; : : : ; ekg is a spanning tree. Kruskal's Greedy Algorithm for MWST Let G = (V; E) be a connected undirected graph and let ≥0 w : E ! R be a weight function on the edges. Kruskal's so-called greedy algorithm is as follows. The algorithm consists of selecting successively edges e1; e2; : : : ; er. If edges e1; e2; : : : ; ek has been selected, then an edge e 2 E is selected so that: 1 e=2f e1; : : : ; ekg and fe; e1; : : : ; ekg is a forest. 2 w(e) is as small as possible among all edges e satisfying (1). We take ek+1 := e. If no e satisfying (1) exists then fe1; : : : ; ekg is a spanning tree. It is obviously not true that such a greedy approach would lead to an optimal solution for any combinatorial optimization problem. -
Branch-Depth: Generalizing Tree-Depth of Graphs
Branch-depth: Generalizing tree-depth of graphs ∗1 †‡23 34 Matt DeVos , O-joung Kwon , and Sang-il Oum† 1Department of Mathematics, Simon Fraser University, Burnaby, Canada 2Department of Mathematics, Incheon National University, Incheon, Korea 3Discrete Mathematics Group, Institute for Basic Science (IBS), Daejeon, Korea 4Department of Mathematical Sciences, KAIST, Daejeon, Korea [email protected], [email protected], [email protected] November 5, 2020 Abstract We present a concept called the branch-depth of a connectivity function, that generalizes the tree-depth of graphs. Then we prove two theorems showing that this concept aligns closely with the no- tions of tree-depth and shrub-depth of graphs as follows. For a graph G = (V, E) and a subset A of E we let λG(A) be the number of vertices incident with an edge in A and an edge in E A. For a subset X of V , \ let ρG(X) be the rank of the adjacency matrix between X and V X over the binary field. We prove that a class of graphs has bounded\ tree-depth if and only if the corresponding class of functions λG has arXiv:1903.11988v2 [math.CO] 4 Nov 2020 bounded branch-depth and similarly a class of graphs has bounded shrub-depth if and only if the corresponding class of functions ρG has bounded branch-depth, which we call the rank-depth of graphs. Furthermore we investigate various potential generalizations of tree- depth to matroids and prove that matroids representable over a fixed finite field having no large circuits are well-quasi-ordered by restriction. -
Matroids Graphs Give Isomorphic Matroid Structure?
EECS 495: Combinatorial Optimization Lecture 7 Matroid Representation, Matroid Optimization Reading: Schrijver, Chapters 39 and 40 graph and S = E. A set F ⊆ E is indepen- dent if it is acyclic. Food for thought: can two non-isomorphic Matroids graphs give isomorphic matroid structure? Recap Representation Def: A matroid M = (S; I) is a finite ground Def: For a field F , a matroid M is repre- set S together with a collection of indepen- sentable over F if it is isomorphic to a linear dent sets I ⊆ 2S satisfying: matroid with matrix A and linear indepen- dence taken over F . • downward closed: if I 2 I and J ⊆ I, 2 then J 2 I, and Example: Is uniform matroid U4 binary? Need: matrix A with entries in f0; 1g s.t. no • exchange property: if I;J 2 I and jJj > column is the zero vector, no two rows sum jIj, then there exists an element z 2 J nI to zero over GF(2), any three rows sum to s.t. I [ fzg 2 I. GF(2). Def: A basis is a maximal independent set. • if so, can assume A is 2×4 with columns The cardinality of a basis is the rank of the 1/2 being (0; 1) and (1; 0) and remaining matroid. two vectors with entries in 0; 1 neither all k Def: Uniform matroids Un are given by jSj = zero. n, I = fI ⊆ S : jIj ≤ kg. • only three such non-zero vectors, so can't Def: Linear matroids: Let F be a field, A 2 have all pairs indep. -
A Gentle Introduction to Matroid Algorithmics
A Gentle Introduction to Matroid Algorithmics Matthias F. Stallmann April 14, 2016 1 Matroid axioms The term matroid was first used in 1935 by Hassler Whitney [34]. Most of the material in this overview comes from the textbooks of Lawler [25] and Welsh [33]. A matroid is defined by a set of axioms with respect to a ground set S. There are several ways to do this. Among these are: Independent sets. Let I be a collection of subsets defined by • (I1) ; 2 I • (I2) if X 2 I and Y ⊆ X then Y 2 I • (I3) if X; Y 2 I with jXj = jY j + 1 then there exists x 2 X n Y such that Y [ x 2 I A classic example is a graphic matroid. Here S is the set of edges of a graph and X 2 I if X is a forest. (I1) and (I2) are clearly satisfied. (I3) is a simplified version of the key ingredient in the correctness proofs of most minimum spanning tree algorithms. If a set of edges X forms a forest then jV (X)j − jXj = jc(X)j, where V (x) is the set of endpoints of X and c(X) is the set of components induced by X. If jXj = jY j + 1, then jc(X)j = jc(Y )j − 1, meaning X must have at least one edge x that connects two of the components induced by Y . So Y [ x is also a forest. Another classic algorithmic example is the following uniprocessor scheduling problem, let's call it the scheduling matroid. -
Obstructions for Bounded Branch-Depth in Matroids
ADVANCES IN COMBINATORICS, 2021:4, 25 pp. www.advancesincombinatorics.com Obstructions for Bounded Branch-depth in Matroids J. Pascal Gollin* Kevin Hendrey* Dillon Mayhew† Sang-il Oum* Received 1 April 2020; Revised 22 January 2021; Published 24 May 2021 Abstract: DeVos, Kwon, and Oum introduced the concept of branch-depth of matroids as a natural analogue of tree-depth of graphs. They conjectured that a matroid of sufficiently large branch-depth contains the uniform matroid Un;2n or the cycle matroid of a large fan graph as a minor. We prove that matroids with sufficiently large branch-depth either contain the cycle matroid of a large fan graph as a minor or have large branch-width. As a corollary, we prove their conjecture for matroids representable over a fixed finite field and quasi-graphic matroids, where the uniform matroid is not an option. Key words and phrases: matroids, branch-depth, branch-width, twisted matroids, representable ma- troids, quasi-graphic matroids 1 Introduction Motivated by the notion of tree-depth of graphs, DeVos, Kwon, and Oum [3] introduced the branch- 1 depth of a matroid M as follows. Recall that the connectivity function lM of a matroid M is defined as lM(X) = r(X) + r(E(M) n X) − r(E(M)), where r is the rank function of M.A decomposition is a arXiv:2003.13975v2 [math.CO] 20 May 2021 pair (T;s) consisting of a tree T with at least one internal node and a bijection s from E(M) to the set of leaves of T. -
On the Interplay Between Graphs and Matroids
On the interplay between graphs and matroids James Oxley Abstract “If a theorem about graphs can be expressed in terms of edges and circuits only it probably exemplifies a more general theorem about ma- troids.” This assertion, made by Tutte more than twenty years ago, will be the theme of this paper. In particular, a number of examples will be given of the two-way interaction between graph theory and matroid theory that enriches both subjects. 1 Introduction This paper aims to be accessible to those with no previous experience of matroids; only some basic familiarity with graph theory and linear algebra will be assumed. In particular, the next section introduces matroids by showing how such objects arise from graphs. It then presents a minimal amount of the- ory to make the rest of the paper comprehensible. Throughout, the emphasis is on the links between graphs and matroids. Section 3 begins by showing how 2-connectedness for graphs extends natu- rally to matroids. It then indicates how the number of edges in a 2-connected loopless graph can be bounded in terms of the circumference and the size of a largest bond. The main result of the section extends this graph result to ma- troids. The results in this section provide an excellent example of the two-way interaction between graph theory and matroid theory. In order to increase the accessibility of this paper, the matroid technicalities have been kept to a minimum. Most of those that do arise have been separated from the rest of the paper and appear in two separate sections, 4 and 10, which deal primarily with proofs. -
Matroid Is Representable Over GF (2), but Not Over Any Other field
18.438 Advanced Combinatorial Optimization October 15, 2009 Lecture 9 Lecturer: Michel X. Goemans Scribe: Shashi Mittal* (*This scribe borrows material from scribe notes by Nicole Immorlica in the previous offering of this course.) 1 Circuits Definition 1 Let M = (S; I) be a matroid. Then the circuits of M, denoted by C(M), is the set of all minimally dependent sets of the matroid. Examples: 1. For a graphic matroid, the circuits are all the simple cycles of the graph. k 2. Consider a uniform matroid, Un (k ≤ n), then the circuits are all subsets of S with size exactly k + 1. Proposition 1 The set of circuits C(M) of a matroid M = (S; I) satisfies the following properties: 1. X; Y 2 C(M), X ⊆ Y ) X = Y . 2. X; Y 2 C(M), e 2 X \ Y and X 6= Y ) there exists C 2 C(M) such that C ⊆ X [ Y n feg. (Note that for circuits we implicitly assume that ; 2= C(M), just as we assume that for matroids ; 2 I.) Proof: 1. follows from the definition that a circuit is a minimally dependent set, and therefore a circuit cannot contain another circuit. 2. Let X; Y 2 C(M) where X 6= Y , and e 2 X \ Y . From 1, it follows that X n Y is non-empty; let f 2 X n Y . Assume on the contrary that (X [ Y ) − e is independent. Since X is a circuit, therefore X − f 2 I. Extend X − f to a maximal independent set in X [ Y , call it Z. -
Aspects of Connectivity with Matroid Constraints in Graphs Quentin Fortier
Aspects of connectivity with matroid constraints in graphs Quentin Fortier To cite this version: Quentin Fortier. Aspects of connectivity with matroid constraints in graphs. Modeling and Simulation. Université Grenoble Alpes, 2017. English. NNT : 2017GREAM059. tel-01838231 HAL Id: tel-01838231 https://tel.archives-ouvertes.fr/tel-01838231 Submitted on 13 Jul 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. THÈSE Pour obtenir le grade de DOCTEUR DE L'UNIVERSITÉ GRENOBLE ALPES Spécialité : Mathématiques et Informatique Arrêté ministériel : 25 mai 2016 Présentée par Quentin FORTIER Thèse dirigée par Zoltán SZIGETI, Professeur des Universités, Grenoble INP préparée au sein du Laboratoire G-SCOP dans l'École Doctorale Mathématiques, Sciences et technologies de l'information, Informatique Aspects de la connexité avec contraintes de matroïdes dans les graphes Aspects of connectivity with matroid constraints in graphs Thèse soutenue publiquement le 27 octobre 2017, devant le jury composé de : Monsieur Yann VAXÈS Professeur des Universités, Université Aix-Marseille, Rapporteur Monsieur Denis CORNAZ Maître de conférences, Université Paris-Dauphine, Rapporteur Monsieur Stéphane BESSY Maître de conférences, Université de Montpellier, Examinateur Monsieur Roland GRAPPE Maître de conférences, Université Paris 13, Examinateur Madame Nadia BRAUNER Professeur des Universités, Université Grenoble Alpes, Présidente Monsieur Zoltán SZIGETI Professeur des Universités, Grenoble INP, Directeur de thèse Résumé La notion de connectivité est fondamentale en théorie des graphes. -
What Is a Matroid?
WHAT IS A MATROID? JAMES OXLEY Abstract. Matroids were introduced by Whitney in 1935 to try to capture abstractly the essence of dependence. Whitney’s definition em- braces a surprising diversity of combinatorial structures. Moreover, ma- troids arise naturally in combinatorial optimization since they are pre- cisely the structures for which the greedy algorithm works. This survey paper introduces matroid theory, presents some of the main theorems in the subject, and identifies some of the major problems of current research interest. 1. Introduction A paper with this title appeared in Cubo 5 (2003), 179–218. This pa- per is a revision of that paper. It contains some new material, including some exercises, along with updates of some results from the original paper. It was prepared for presentation at the Workshop on Combinatorics and its Applications run by the New Zealand Institute of Mathematics and its Applications in Auckland in July, 2004. This survey of matroid theory will assume only that the reader is familiar with the basic concepts of linear algebra. Some knowledge of graph theory and field theory would also be helpful but is not essential since the concepts needed will be reviewed when they are introduced. The name “matroid” suggests a structure related to a matrix and, indeed, matroids were intro- duced by Whitney [61] in 1935 to provide a unifying abstract treatment of dependence in linear algebra and graph theory. Since then, it has been rec- ognized that matroids arise naturally in combinatorial optimization and can be used as a framework for approaching a diverse variety of combinatorial problems. -
Matroid Theory.) Given a Matroid M, There Is a Naturally Constructed Dual Matroid M ∗ on the Same Underlying Set
TROPICAL SCHEME THEORY 3. Matroids Matroids take \It's useful to have multiple perspectives on this thing" to a ridicu- lous extent. Let E be a finite set of size n. Sometimes we will identify it with [n] = f1; : : : ; ng, sometimes not. Definition 3.1. A matroid on E is a collection I ⊂ 2E of independent sets satisfying (I1) ; 2 I, (I2) If S 2 I and S0 ⊂ S then S0 2 I, and (I3) If S; T 2 I and jSj > jT j then there exists x 2 S such that T [ fxg 2 I. If S2 = I we say that S is a dependent set. (I3) is called the exchange/replacement axiom. Example 3.2. (1) A finite collection of vectors fve j e 2 Eg in a vector space determines a matroid, where S 2 I if and only if fve j e 2 Sg is linearly independent. (2) The matroid of a subspace V of a coordinate vector space KE is defined by S 2 I if and only if the coordinate functions fxe j e 2 Sg are linearly independent when restricted to V . Note that this an instance of (1), as ∗ fxejV j e 2 Eg is a vector arrangement in V . (3) A finite collection of linear hyperplanes fHeje 2 Eg ⊂ V in a finite-dimensional \ vector space determines a matroid, where S 2 I if and only if codim He = e2S jSj. (We allow the degenerate hyperplane He = V .) Again, this is an instance ∗ of (1): If He is the vanishing locus of a functional xe 2 V , then the matroid of the arrangement of hyperplanes He ⊂ V is identical to the matroid of the ∗ arrangement of vectors xe 2 V . -
Finding Branch-Decompositions and Rank-Decompositions
Finding Branch-decompositions and Rank-decompositions Petr Hlinˇen´y ∗† Faculty of Informatics Masaryk University Botanick´a 68a, 602 00 Brno, Czech Rep. Sang-il Oum ‡§ Department of Mathematical Sciences KAIST, Daejeon, 305-701 Republic of Korea. February 5, 2008 Abstract We present a new algorithm that can output the rank-decomposition of width at most k of a graph if such exists. For that we use an algorithm that, for an input matroid represented over a fixed finite field, outputs its branch-decomposition of width at most k if such exists. This algorithm works also for partitioned matroids. Both these algorithms are fixed-parameter tractable, that is, they run in time O(n3)wheren is the number of vertices / elements of the input, for each constant value of k and any fixed finite field. The previous best algorithm for construction of a branch-decomposition or a rank-decomposition of optimal width due to Oum and Seymour [Testing branch-width. J.Combin.TheorySer.B,97(3) (2007) 385–393] is not fixed-parameter tractable. Keywords: rank-width, clique-width, branch-width, fixed parameter tractable algo- rithm, graph, matroid. ∗[email protected] †Supported by Czech research grant GACRˇ 201/08/0308 and by Research intent MSM0021622419 of the Czech Ministry of Education. ‡[email protected] §This research was done while the second author was at Georgia Institute of Technology and University of Waterloo. Partially supported by NSF grant DMS 0354742 while the second author was at Georgia Institute of Technology. 1 1 Introduction Many graph problems are known to be NP-hard in general; however, for practical application we still need to solve them.