DOCSLIB.ORG

Lecture 14 — February, 1 2014 1 Overview 2 Problems on Matroid

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Lecture 14 — February, 1 2014 1 Overview 2 Problems on Matroid Advanced Graph Algorithms Jan-Apr 2014 Lecture 14 | February, 1 2014 Lecturer: Saket Saurabh Scribe: Sanjukta Roy 1 Overview In the last lecture we learn about matroids and saw some examples of it. Continuing with an example we move to solve some interesting problems of graph theory using matroid intersection and matroid parity problem. 2 Problems on Matroid and Using Them 2.1 Transversal Matroids Recall from the last lecture, transversal matroid M = (E; I) of G has E = A as its ground set, where G is a bipartite graph with the vertex set V (G) being partitioned as A and B and I = fX j X ⊆ A; there is a matching that covers Xg The following shows the incidence matrix of a bipartite graph. columns are indexed by vertices ofA 0 1 rows are x11 x12 x13 ··· x1m indexed byB x x x ··· x C B 21 22 23 2m C verticesB x x x ··· x C T = B 31 32 33 3m C jAj = m and jBj = ` B . .. C of @ . A B x x x ··· x `1 `2 `3 `m `×m ( xi;j if(ai; bj) 2 E(G) ti;j = 0 otherwise xi;j are indeterminants. All of them are independent variables. Next we show that this matrix indeed gives the represention for transversal matroids. 2.1.1 Transversal Matroid is Representable Transversal Matroid is Representable over F[~z] 1 Proof. Forward direction: Suppose some subset X = fa1; :::; aqg is independent. Then there is a matching that saturates X. Let Y = fb1; b2; :::; bqg be the endpoints of this matching and aibi are the matching edges. Consider T [Y; X] a submatrix with rows in Y and columns in X. Consider the determinant of T [Y, X] then q Q we have a term zii which can not be cancelled by any other term. So these columns are linearly i=1 independent. Reverse direction: Suppose some subset X = fa1; ··· ; aqg of columns is independent in T . Then there is a submatrix of T [∗;X] that is nonsingular, say T [Y; X]. Consider the determinant of T [Y; X] q P Q det(T [Y; X]) = sgn(π) ziπ(i) 6= 0 π2S(Y ) i=1 q Q This implies that we have a term ziπ(i) 6= 0 and this gives us that there is a matching that i=1 saturates X in and hence X is independent. In this direction we remove the indeterminants xijs. 2.1.2 Finding the Representation Removing xij To remove the xij we assign xij from values in finite field F of size P uniformly at random. What should be the upper bound on P ? What is the probability that the randomly obtained T is a representation matrix for the transversal matroid. To answer this we use Schwartz-Zippel Lemma. Schwartz-Zippel Lemma Theorem 1. Let p(x1; x2; ··· ; xn) be a non-zero polynomial of degree d over some field F and let S be an N element subset of F. If each xi is independently assigned a value from S with uniform d probability, then p(x1; x2; ··· ; xn) = 0 with probability at most N We think det(T [Y; X]) as polynomial in xij's of degree at most n = jAj. n Probability ( det(T [Y; X]) = 0) < P . There are 2n independent sets in A and thus by union bound, probability that not all of them 2nn are independent in the matroid represented by T is at most P . Thus probability that T is the 2nn representation is at least 1 − P . Take P to be some field with at least 2nn2n elements. Size of this representation with be nO(1) bits. This way we can find the representation in polynomial time using randomisation. Open Problem Can we find the representation of transversal matroids in polynomial time using deterministic algorithm? 2.2 Two Problem Definitions n-Matroid Intersection Given matroids M1 = (E; I1); M2 = (E; I2); ··· ; Mn = (E; In) on the same ground set E, and a number k, does there exist a set X ⊆ E such that jXj > k and X 2 Ii for 1 ≤ i ≤ n? 2 Matroid Pairity Given a matroid M = (E; I), whose elements are partitioned into m pairs. (hence, jEj is even). Objective: Find the maximum cardinality collection of pairs, X ⊆ E such that X 2 I ? Matroid Parity (Linear) Given a matrix whose elements are partitioned into m pairs. z}|{ z}|{ ··· z}|{ 0 1 a11 b11 a12 b12 ··· a1m b1m B a b a b ··· a b C B 21 21 22 22 2m 2m C B a b a b ··· a b C A = B 31 31 32 32 3m 3m C r = rank of the matroid B . .. C @ . A a b a b ··· a b r1 r1 r2 r2 rm rm r×2m Objective: Find the maximum cardinality collection of pairs, such that unoin of columns of these pairs are linearly independent. ? 2.3 Problems Question:1 Phrase bipartite matching problem as intersection of two matroids Let M1 = (E; I1) where I1 = fX ⊂ E j if (i1; j1) 2 X and (i2; j2) 2 X and i1; i2 2 A and j1; j2 2 B and i1 6= i2g that is 8xi 2 X every xi are incident on different vertex on A and M2 = (E, I2) where I2 = fX ⊂ E j if (i1; j1) 2 X and (i2; j2) 2 X and i1; i2 2 B and j1; j2 2 A and i1 6= i2g We want to show that set family M1 and M2 are matroids. Proof. M1 satisfies (I2), since if all the edges in X are incident on different vertices of A then all its subsets are also incident on different vertex of A. Suppose (I3) does not hold. That is there is set X1 and X2 2 I. jX1j < jX2j. For all edges e 0 0 2 X2 nX1. e is incident on a vertex v such that 9 an edge e 2 X1 nX2. e is also incident on v. But since jX2j > jX1j and all edges in X2 are incident on differet vertices then 9 atleast one vertex u such that there is no edge (u; v) 2 X1. Then we can extend X1 using that edge, Hence (I3) holds. Therefore M1 is a matroid. Using the same reason M2 is also a matroid. Therefore M1 \ M2 gives the matching in the bipartite graph. Question:2 Phrase matchng in general graph as matroid parity problem 3 We represent the graph as the following matrix. z }| { z }| { z }| { e11 e12 e21 e22 ··· ejEj ejEj 0 1 2 1 1 0 0 ··· . B . C 8 . 1 if i = u; j = e (u; v) = e 2 E B C > l1 l u B 1 0 ··· C < A = B C a = 1 if i = v; j = e ; (u; v) = e 2 E B C i;j l2 l v B 0 1 ··· C > . B . C :0 otherwise . @ . A jV j · · · jV |×|2Ej that is we take a jV j×j2Ej matrix, we take two columns for each edge. This partitions the columns of the matrix in E pairs. th If two coulms are linearly dependent then they are same. So the i row have only one 1 if ek1 and ek2 are amongst the linearly independent columns and (i; j) = ek 2 E. Hence every vertex in the subgraph formed from these pair of linearly independent columns has degree atmost one, so it corresponds to a matching. Hence the union of columns of these pairs which are linearly independent gives the matching. Question:3 How do you test if a graph has two edge disjoint spanning trees Let G(V,E) be the graph. M1 = (E; I1) be the graphic matroid of G and M2 = (E; I2) be the co-graphic matroid of G. If M1 and M2 has an intersection of size jV j − 1. Call this set of edges S. Then S 2 I1 i.e., S forms spanning trees of G. S 2 I2 i.e., G n S is connected. Therefore 9 another spanning tree of G in E n S. Hence there are atleast two edge disjoint spanning trees. Question:4 3 Matroid intersection Problem is NPC Proof. To prove this we show, directed Hamiltonian path ≤p 3 matroid intersection. Directed Hamiltonian Path Problem: Given directed graph D = (V; A) and s; t 2 V Is there a path from s to t that goes through each vertex exactly once. M1 = (E; I1) is the graphic matroid of underlying undirected graph i.e., E is the same set of edges as A ignoring direction. M2 = (A; I2), I2 = fX 2 E j each v 2 V has at most one incoming edge in X, except s which has no edge entering itg. M3 = (A; I3), I3 = fY 2 E j each v 2 V has at most one outgoing edge in Y , except t which has no edge leaving itg. Now we prove that M2 and M3 are matroids. In question 6 we prove that M2 is a matroid. For M3 heriditary property holds trivially. Suppose A; B 2 I3 and jAj < jBj but @x 2 B n A such that A [ fxg 2 I3. But each edge in B is outgoing edge for exactly one vertex and since no of edges in B is greater than no of edges in A then there is atleast one vertex, v with outdegree 0 in A but has outdegree 4 1 is B. Therefore we can add this edge comming out of v to A.
Load more

Recommended publications
  • Parity Systems and the Delta-Matroid Intersection Problem
    Parity Systems and the Delta-Matroid Intersection Problem Andr´eBouchet ∗ and Bill Jackson † Submitted: February 16, 1998; Accepted: September 3, 1999. Abstract We consider the problem of determining when two delta-matroids on the same ground-set have a common base. Our approach is to adapt the theory of matchings in 2-polymatroids developed by Lov´asz to a new abstract system, which we call a parity system. Examples of parity systems may be obtained by combining either, two delta- matroids, or two orthogonal 2-polymatroids, on the same ground-sets. We show that many of the results of Lov´aszconcerning ‘double flowers’ and ‘projections’ carry over to parity systems. 1 Introduction: the delta-matroid intersec- tion problem A delta-matroid is a pair (V, ) with a finite set V and a nonempty collection of subsets of V , called theBfeasible sets or bases, satisfying the following axiom:B ∗D´epartement d’informatique, Universit´edu Maine, 72017 Le Mans Cedex, France. [email protected] †Department of Mathematical and Computing Sciences, Goldsmiths’ College, London SE14 6NW, England. [email protected] 1 the electronic journal of combinatorics 7 (2000), #R14 2 1.1 For B1 and B2 in and v1 in B1∆B2, there is v2 in B1∆B2 such that B B1∆ v1, v2 belongs to . { } B Here P ∆Q = (P Q) (Q P ) is the symmetric difference of two subsets P and Q of V . If X\ is a∪ subset\ of V and if we set ∆X = B∆X : B , then we note that (V, ∆X) is a new delta-matroid.B The{ transformation∈ B} (V, ) (V, ∆X) is calledB a twisting.
    [Show full text]
  • CS675: Convex and Combinatorial Optimization Spring 2018 Introduction to Matroid Theory
    CS675: Convex and Combinatorial Optimization Spring 2018 Introduction to Matroid Theory Instructor: Shaddin Dughmi Set system: Pair (X ; I) where X is a finite ground set and I ⊆ 2X are the feasible sets Objective: often “linear”, referred to as modular Analogues of concave and convex: submodular and supermodular (in no particular order!) Today, we will look only at optimizing modular objectives over an extremely prolific family of set systems Related, directly or indirectly, to a large fraction of optimization problems in P Also pops up in submodular/supermodular optimization problems Optimization over Sets Most combinatorial optimization problems can be thought of as choosing the best set from a family of allowable sets Shortest paths Max-weight matching Independent set ... 1/30 Objective: often “linear”, referred to as modular Analogues of concave and convex: submodular and supermodular (in no particular order!) Today, we will look only at optimizing modular objectives over an extremely prolific family of set systems Related, directly or indirectly, to a large fraction of optimization problems in P Also pops up in submodular/supermodular optimization problems Optimization over Sets Most combinatorial optimization problems can be thought of as choosing the best set from a family of allowable sets Shortest paths Max-weight matching Independent set ... Set system: Pair (X ; I) where X is a finite ground set and I ⊆ 2X are the feasible sets 1/30 Analogues of concave and convex: submodular and supermodular (in no particular order!) Today, we will look only at optimizing modular objectives over an extremely prolific family of set systems Related, directly or indirectly, to a large fraction of optimization problems in P Also pops up in submodular/supermodular optimization problems Optimization over Sets Most combinatorial optimization problems can be thought of as choosing the best set from a family of allowable sets Shortest paths Max-weight matching Independent set ..
    [Show full text]
  • Matroid Intersection
    EECS 495: Combinatorial Optimization Lecture 8 Matroid Intersection Reading: Schrijver, Chapter 41 • colorful spanning forests: for graph G with edges partitioned into color classes fE1;:::;Ekg, colorful spanning forest is Matroid Intersection a forest with edges of different colors. Define M1;M2 on ground set E with: Problem: { I1 = fF ⊂ E : F is acyclicg { I = fF ⊂ E : 8i; jF \ E j ≤ 1g • Given: matroids M1 = (S; I1) and M2 = 2 i (S; I ) on same ground set S 2 Then • Find: max weight (cardinality) common { these are matroids (graphic, parti- independent set J ⊆ I \I 1 2 tion) { common independent sets = color- Applications ful spanning forests Generalizes: Note: In second two examples, (S; I1 \I2) not a matroid, so more general than matroid • max weight independent set of M (take optimization. M = M1 = M2) Claim: Matroid intersection of three ma- troids NP-hard. • matching in bipartite graphs Proof: Reduction from directed Hamilto- For bipartite graph G = ((V1;V2);E), let nian path: given digraph D = (V; E) and Mi = (E; Ii) where vertices s; t, is there a path from s to t that goes through each vertex exactly once. Ii = fJ : 8v 2 Vi; v incident to ≤ one e 2 Jg • M1 graphic matroid of underlying undi- Then rected graph { these are matroids (partition ma- • M2 partition matroid in which F ⊆ E troids) indep if each v has at most one incoming edge in F , except s which has none { common independent sets = match- ings of G • M3 partition :::, except t which has none 1 Intersection is set of vertex-disjoint directed = r1(E) +r2(E)− min(r1(F ) +r2(E nF ) paths with one starting at s and one ending F ⊆E at t, so Hamiltonian path iff max cardinality which is number of vertices minus min intersection has size n − 1.
    [Show full text]
  • Packing of Arborescences with Matroid Constraints Via Matroid Intersection
    Mathematical Programming (2020) 181:85–117 https://doi.org/10.1007/s10107-019-01377-0 FULL LENGTH PAPER Series A Packing of arborescences with matroid constraints via matroid intersection Csaba Király1 · Zoltán Szigeti2 · Shin-ichi Tanigawa3 Received: 29 May 2018 / Accepted: 8 February 2019 / Published online: 2 April 2019 © The Author(s) 2019 Abstract Edmonds’ arborescence packing theorem characterizes directed graphs that have arc- disjoint spanning arborescences in terms of connectivity. Later he also observed a characterization in terms of matroid intersection. Since these fundamental results, intensive research has been done for understanding and extending these results. In this paper we shall extend the second characterization to the setting of reachability-based packing of arborescences. The reachability-based packing problem was introduced by Cs. Király as a common generalization of two different extensions of the spanning arborescence packing problem, one is due to Kamiyama, Katoh, and Takizawa, and the other is due to Durand de Gevigney, Nguyen, and Szigeti. Our new characterization of the arc sets of reachability-based packing in terms of matroid intersection gives an efficient algorithm for the minimum weight reachability-based packing problem, and it also enables us to unify further arborescence packing theorems and Edmonds’ matroid intersection theorem. For the proof, we also show how a new class of matroids can be defined by extending an earlier construction of matroids from intersecting submodular functions to bi-set functions based on an idea of Frank. Mathematics Subject Classification 05C05 · 05C20 · 05C70 · 05B35 · 05C22 B Csaba Király [email protected] Zoltán Szigeti [email protected] Shin-ichi Tanigawa [email protected] 1 Department of Operations Research, ELTE Eötvös Loránd University and the MTA-ELTE Egerváry Research Group on Combinatorial Optimization, Budapest Pázmány Péter sétány 1/C, 1117, Hungary 2 Univ.
    [Show full text]
  • Approximation by Lexicographically Maximal Solutions in Matching And
    Approximation by Lexicographically Maximal Solutions in Matching and Matroid Intersection Problems Krist´of B´erczia Tam´as Kir´alya Yutaro Yamaguchib Yu Yokoic Abstract We study how good a lexicographically maximal solution is in the weighted matching and matroid intersection problems. A solution is lexicographically maximal if it takes as many heaviest elements as possible, and subject to this, it takes as many second heaviest elements as possible, and so on. If the distinct weight values are sufficiently dispersed, e.g., the minimum ratio of two distinct weight values is at least the ground set size, then the lexicographical maximality and the usual weighted optimality are equivalent. We show that the threshold of the ratio for this equivalence to hold is exactly 2. Furthermore, we prove that if the ratio is less than 2, say α, then a lexicographically maximal solution achieves (α/2)-approximation, and this bound is tight. Keywords: Matching, Matroid intersection, Lexicographical maximality, Approximation ratio arXiv:2107.09897v1 [math.CO] 21 Jul 2021 aMTA-ELTE Momentum Matroid Optimization Research Group and MTA-ELTE Egerv´ary Research Group, Department of Operations Research, E¨otv¨os Lor´and University, Budapest, Hungary. Email: {kristof.berczi,tamas.kiraly}@ttk.elte.hu b Department of Informatics, Faculty of Information Science and Electrical Engineering, Kyushu University, Fukuoka, Japan. Email: yutaro [email protected] cPrinciples of Informatics Research Division, National Institute of Informatics, Tokyo, Japan. Email: [email protected] 1 Introduction Matching in bipartite graphs is one of the fundamental topics in combinatorics and optimization. Due to its diverse applications, various optimality criteria of matchings have been proposed based on the number of edges, the total weight of edges, etc.
    [Show full text]
  • The Intersection of a Matroid and a Simplicial Complex 1
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 358, Number 11, November 2006, Pages 4895–4917 S 0002-9947(06)03833-5 Article electronically published on June 19, 2006 THE INTERSECTION OF A MATROID AND A SIMPLICIAL COMPLEX RON AHARONI AND ELI BERGER Abstract. A classical theorem of Edmonds provides a min-max formula re- lating the maximal size of a set in the intersection of two matroids to a “cov- ering” parameter. We generalize this theorem, replacing one of the matroids by a general simplicial complex. One application is a solution of the case r = 3 of a matroidal version of Ryser’s conjecture. Another is an upper bound on the minimal number of sets belonging to the intersection of two matroids, needed to cover their common ground set. This, in turn, is used to derive a weakened version of a conjecture of Rota. Bounds are also found on the dual parameter—the maximal number of disjoint sets, all spanning in each of two given matroids. We study in detail the case in which the complex is the complex of independent sets of a graph, and prove generalizations of known results on “independent systems of representatives” (which are the special case in which the matroid is a partition matroid). In particular, we define a no- tion of k-matroidal colorability of a graph, and prove a fractional version of a conjecture, that every graph G is 2∆(G)-matroidally colorable. The methods used are mostly topological. 1. Introduction The point of departure of this paper is a notion which has been recently developed and studied, that of an independent system of representatives (ISR), which is a generalization of the notion of a system of distinct representatives (SDR).
    [Show full text]
  • On the Graphic Matroid Parity Problem
    On the graphic matroid parity problem Zolt´an Szigeti∗ Abstract A relatively simple proof is presented for the min-max theorem of Lov´asz on the graphic matroid parity problem. 1 Introduction The graph matching problem and the matroid intersection problem are two well-solved problems in Combi- natorial Theory in the sense of min-max theorems [2], [3] and polynomial algorithms [4], [3] for finding an optimal solution. The matroid parity problem, a common generalization of them, turned out to be much more difficult. For the general problem there does not exist polynomial algorithm [6], [8]. Moreover, it contains NP-complete problems. On the other hand, for linear matroids Lov´asz provided a min-max formula in [7] and a polynomial algorithm in [8]. There are several earlier results which can be derived from Lov´asz’ theorem, e.g. Tutte’s result on f-factors [15], a result of Mader on openly disjoint A-paths [11] (see [9]), a result of Nebesky concerning maximum genus of graphs [12] (see [5]), and the problem of Lov´asz on cacti [9]. This latter one is a special case of the graphic matroid parity problem. Our aim is to provide a simple proof for the min-max formula on this problem, i. e. on the graphic matroid parity problem. In an earlier paper [14] of the present author the special case of cacti was considered. We remark that we shall apply the matroid intersection theorem of Edmonds [4]. We refer the reader to [13] for basic concepts on matroids. For a given graph G, the cycle matroid G is defined on the edge set of G in such a way that the independent sets are exactly the edge sets of the forests of G.
    [Show full text]
  • Exact and Approximation Algorithms for Weighted Matroid Intersection
    Noname manuscript No. (will be inserted by the editor) Exact and Approximation Algorithms for Weighted Matroid Intersection Chien-Chung Huang · Naonori Kakimura · Naoyuki Kamiyama Received: date / Accepted: date Abstract In this paper, we propose new exact and approximation algorithms for the weighted matroid intersection problem. Our exact algorithm is faster than previous algorithms when the largest weight is relatively small. Our approximation algorithm delivers a (1 − ϵ)-approximate solution with a running time significantly faster than most known exact algorithms. The core of our algorithms is a decomposition technique: we decompose an in- stance of the weighted matroid intersection problem into a set of instances of the unweighted matroid intersection problem. The computational advantage of this ap- proach is that we can make use of fast unweighted matroid intersection algorithms as a black box for designing algorithms. Precisely speaking, we prove that we can solve the weighted matroid intersection problem via solving W instances of the unweighted matroid intersection problem, where W is the largest given weight. Furthermore, we can find a (1 − ϵ)-approximate solution via solving O(ϵ−1 log r) instances of the un- weighted matroid intersection problem, where r is the smallest rank of the given two matroids. Our algorithms are simple and flexible: they can be adapted to special cases of the weighted matroid intersection problem, using unweighted matroid intersection algorithms. In this paper, we will show the following results. A preliminary version of this paper appears in the proceeding of ACM-SIAM Symposium on Discrete Algorithms (SODA), 2016. Chien-Chung Huang CNRS, Ecole´ Normale Superieure,´ Paris, France.
    [Show full text]
  • Matroid Optimization and Algorithms Robert E. Bixby and William H
    Matroid Optimization and Algorithms Robert E. Bixby and William H. Cunningham June, 1990 TR90-15 MATROID OPTIMIZATION AND ALGORITHMS by Robert E. Bixby Rice University and William H. Cunningham Carleton University Contents 1. Introduction 2. Matroid Optimization 3. Applications of Matroid Intersection 4. Submodular Functions and Polymatroids 5. Submodular Flows and other General Models 6. Matroid Connectivity Algorithms 7. Recognition of Representability 8. Matroid Flows and Linear Programming 1 1. INTRODUCTION This chapter considers matroid theory from a constructive and algorithmic viewpoint. A substantial part of the developments in this direction have been motivated by opti­ mization. Matroid theory has led to a unification of fundamental ideas of combinatorial optimization as well as to the solution of significant open problems in the subject. In addition to its influence on this larger subject, matroid optimization is itself a beautiful part of matroid theory. The most basic optimizational property of matroids is that for any subset every max­ imal independent set contained in it is maximum. Alternatively, a trivial algorithm max­ imizes any {O, 1 }-valued weight function over the independent sets. Most of matroid op­ timization consists of attempts to solve successive generalizations of this problem. In one direction it is generalized to the problem of finding a largest common independent set of two matroids: the matroid intersection problem. This problem includes the matching problem for bipartite graphs, and several other combinatorial problems. In Edmonds' solu­ tion of it and the equivalent matroid partition problem, he introduced the notions of good characterization (intimately related to the NP class of problems) and matroid (oracle) algorithm.
    [Show full text]
  • Faster Matroid Intersection
    2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS) Faster Matroid Intersection Deeparnab Chakrabarty Yin Tat Lee Aaron Sidford Dartmouth College University of Washington Stanford University [email protected] Microsoft Research [email protected] [email protected] Sahil Singla Sam Chiu-wai Wong Princeton University Microsoft Research Institute for Advanced Study [email protected] [email protected] Abstract—In this paper we consider the classic matroid I. Matroids are fundamental objects in combinatorics, and intersection problem: given two matroids M1 =(V,I1) and the abstract definition above generalizes a wide range of M V,I V 2 =( 2) defined over a common ground set , compute concepts ranging from acyclic graphs to linearly independent a set S ∈I1 ∩I2 of largest possible cardinality, denoted by r. We consider this problem both in the setting where each matrices. M V,I M V,I Mi is accessed through an independence oracle, i.e. a routine Given two matroids 1 =( 1) and 2 =( 2) which returns whether or not a set S ∈Ii in Tind time, and over the same ground set, the matroid intersection problem is Mi the setting where each is accessed through a rank oracle, to find a set I ∈I1 ∩I2 with the largest cardinality |I|. This i.e. a routine which returns the size of the largest independent problem generalizes many important combinatorial opti- subset of S in Mi in Trank time. mization problems such as bipartite matching, arborescences In each setting we provide faster exact and approximate in digraphs, and packing spanning trees. Unsurprisingly, algorithms.
    [Show full text]
  • Efficient Theoretic and Practical Algorithms for Linear Matroid
    Journal of Computer and System Sciences 1273 journal of computer and system sciences 53, 129147 (1996) article no. 0054 CORE Efficient Theoretic and Practical AlgorithmsMetadata, for citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector Linear Matroid Intersection Problems Harold N. Gabow* and Ying Xu Department of Computer Science, University of Colorado at Boulder, Boulder, Colorado 80309 Received February 1, 1989 This paper presents efficient algorithms for a number Efficient algorithms for the matroid intersection problem, both car- of matroid intersection problems. The main algorithm is dinality and weighted versions, are presented. The algorithm for for weighted matroid intersection. It is based on cost weighted intersection works by scaling the weights. The cardinality scaling, generalizing recent work in matching [GaT87], algorithm is a special case, but takes advantage of greater structure. Efficiency of the algorithms is illustrated by several implementations on and network flow [GoT]. A special case is of this problem linear matroids. Consider a linear matroid with m elements and rank n. is cardinality matroid intersection, and several improve- Assume all element weights are integers of magnitude at most N. Our ments are given for this case. The efficiency of our fastest algorithms use time O(mn1.77 log(nN)) and O(mn1.62) for intersection algorithms is illustrated by implementa- weighted and unweighted intersection, respectively; this improves the tions on linear matroids. This is the broadest class of previous best bounds, O(mn2.4) and O(mn2 log n), respectively. Corresponding improvements are given for several applications of matroids with efficient representations. (Implementations of matroid intersection to numerical computation and dynamic systems.
    [Show full text]
  • CS675: Convex and Combinatorial Optimization Fall 2019 Introduction to Matroid Theory
    CS675: Convex and Combinatorial Optimization Fall 2019 Introduction to Matroid Theory Instructor: Shaddin Dughmi Set system: Pair (X ; I) where X is a finite ground set and I ⊆ 2X are the feasible sets Objective: often “linear”, referred to as modular Analogues of concave and convex: submodular and supermodular (in no particular order!) Today, we will look only at optimizing modular objectives over an extremely prolific family of set systems Related, directly or indirectly, to a large fraction of optimization problems in P Also pops up in submodular/supermodular optimization problems Optimization over Sets Most combinatorial optimization problems can be thought of as choosing the best set from a family of allowable sets Shortest paths Max-weight matching Independent set ... 1/30 Objective: often “linear”, referred to as modular Analogues of concave and convex: submodular and supermodular (in no particular order!) Today, we will look only at optimizing modular objectives over an extremely prolific family of set systems Related, directly or indirectly, to a large fraction of optimization problems in P Also pops up in submodular/supermodular optimization problems Optimization over Sets Most combinatorial optimization problems can be thought of as choosing the best set from a family of allowable sets Shortest paths Max-weight matching Independent set ... Set system: Pair (X ; I) where X is a finite ground set and I ⊆ 2X are the feasible sets 1/30 Analogues of concave and convex: submodular and supermodular (in no particular order!) Today, we will look only at optimizing modular objectives over an extremely prolific family of set systems Related, directly or indirectly, to a large fraction of optimization problems in P Also pops up in submodular/supermodular optimization problems Optimization over Sets Most combinatorial optimization problems can be thought of as choosing the best set from a family of allowable sets Shortest paths Max-weight matching Independent set ..
    [Show full text]