DOCSLIB.ORG

Efficient Algorithms for Graphic Matroid Intersection and Parity

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

\ 194 - 1 \ Automata, Languages and ~rogrammlng 1 , verifying Concurrent Processes Using 12th Colloquium 18 pages.1982, Nafplion, Greece, July 15-19,1985 iomatisingthe ~ogicof Computer Program- 182. s.Proceedings, 1881. Edited by 0. Kozen. ;iw Techniques I- Requirement6 and Logical 3, 1978, Edited by S.B.Yao, S.B. Navathe, .nit. V. 227 pages, 1982. Design Techniques 11: Proceedings, 1979. T.L.Kunii.V, 229-399 pages. 1992. cificalion. Proceedings, 1981. Edited by J. 366.1982. X2ProgiammingLogic.X,292 pages.108 ~age8.1982. Edited by Wilfried Brauer I on Automated Deduciion, Proceedings veland.VII. 389 pages. 1982. sopoulou. G. Per8ch.G. Goos, M. Daismann rchgassner, An Attribute Grammar lor the ida. IX, 511 pages. 1982. guages and programming.Edited by M. Niel- VII, 614 pages, 1982. 3. Hun, â Zimmermann. GAG: A Practical ', 156 pages. 1982. Springer-Verlag Berlin Heidelberg New York Tokyo Efficient Algorithms for Graphic Matroid Intersection and Parity s (Extended Abstract) algori by shorte impro Harold N.Gabow ' Matthias Stallmanu s Department of Computer Science Department of Computer Science O(n 1 University of Colorado North Carolina State University and c Boulder, CO 80309 Raleigh, NC 27695-8206 for s USA USA well- A Abstract matrc An algorithm for matmid intersection, baaed on the phase approach of Dinic for \ network flow and Hopcmft and Karp for matching, is presented. An implementation for the b graphic matmids uses time O(n1P m) if m is Of# fg n), and similar expressions indep otherwise. An implementation to find k edge-disjoint spanning trees on a graph uses eleme time O(VP nlP m) if m is 0(n lg n) and a similar expression otherwise; when m is main 0(fD I@) this improves the previous bound, 0(t? p2). hpmved algorithms for other undir problems are obtained, including maintaining a minimum spanning tree on a planar graph ( subject to changing edge coats, and finding shortest pairs of disjoint paths in a network. two, An algorithm for graphic matmid parity is presented that runs in time O(n m (a6 n). a pal This improves the previous bound of 0(n2 m). if an, The probl = 0, 1. Introduction. Matroid parity generalizes a broad class of combinatorial optimization problems that ! For i includes network flow and matching [La]. Yet it can still be solved efficiently [Lo,SG]. This paper presents more efficient algorithms for several cases of the parity problem. Section 2 develops a phase algorithm for matroid intersection, the "bipartite" parity 3. T problem. This generalizes the phase algorithms of Dinic for network flow, Hopcmft and Karp for matching, and Even and Tarjan for 0-1 network flow [D,HK,ETL It is shown inter that a phase algorithm for matroid intersection reduces to four matroid problems, the recei static-base/dynamic-base cycle/cocycle problems. Some of the results in Section 2 for 1 overlap recent work of Tovey and Trick [TT), and even more recently Cunningham [C]. inter The overlap (indicated precisely in Section 2) is not adequate to achieve the time bounds for I of this paper. equil An efficient implementation of the intersection algorithm for graphic matmids is presented in Sections 3-5. Section 3 gives an overview and timing analysis. The time is cone 0(n112 m) if m is O(n3fl Ig n). This is identical to the Hopcroft-Karp bound for matching, a special case of graphic matmid intersection. The four matroid problems Defu mentioned above make other cases more difficult. The time is O(n m213 1$13 n) if m is ¡1 a fl(n Ig n) and a similar expression otherwise. f Planar graphs can be handled more efficiently because of duality. Section 3 gives a f bound of 0(n3I2 fg n). The approach also gives an algorithm for maintaining a minimum all t spanning tree on a planar graph subject to changing edge costs. The time is 0(lg n) per update, improving the O(Ig2 n) bound of Frederickson IF]. ace Section 3 also givw an implementation of the intersection algorithm to find k Not6 edge-disjoint spanning trees on a graph. It uses time 0(i31? n1D m) if m is Q(n lg n) and that a similar expression otherwise. When m is O(kIl? nap) this improves the bound of folio O(* n2) due to Roskind and Tarjan [RT]. The, ' This work was supported in part by NSF Grant #MCS-8302648. 21s. Parity Section 4 solves the static-tree cocycle problem used in the graphic intersection algorithm. The solution also speeds an algorithm of Suurballe and Tarjan for finding shortest pairs of disjoint paths in a network [Su TI. The time bound is O(m + n Ig n), improved from 0(m lg^,^ n). Section 6 solves the dynamic-tree cocycle problem. mann Section 6 presents an efficient graphic matroid parity algorithm. The time bound is 1 Com~uterScience 0(n m If n), improving the bound of O(n2 m) of [SG]. In the parity problem the cycle a. State University and cocycle problems involve pairs of edges. This seems to make traditional techniques !7695-8206 for solving cycle and cocycle problems inapplicable. We reduce the problem to a well-studied problem of computational geometry, range searching. Many of the ideas of this paper apply to other matroids, such as job scheduling matroids [GT84] and linear matroids. We will discuss these extensions in a future paper. iroach of Dinic for We close this section with some terminology. The reader is assumed familiar with implementation for the basic notions of matroids (see, e.g., [A, La, W]). It e is an element and B a base (or similar expressions independent set), C(e,Sf denotes the fundamental cycle of e in B. The span of a set of H on a graph uses elements A is denoted q(A). If T C S are seta of elements then S / T denotes the ; when m is matroid restricted to S, with T contracted. A graphic matroid is derived from an ' ilgorithms for other undirected graph; its independent sets are the forests. on a planar graph Consider a matroid M over a set of elements E that is partitioned into blocks of size ~athsin a network. two, called parity pairs. Thus each element e has a mate, denoted 2, such that {e, 2) is time 0(n m If n). a parity pair. A matching M is an independent set consisting of parity pairs, i.e., e â M if and only if s  M. A maximum matching has the greatest number of pairs possible. The (cardinality) parity problem is to find a maximum matching. The matroid intersection problem is the "bipartite" version: M is the direct sum of matroids Mi on elements E', i = 0,1, and every parity pair contains one element from each matroid. T: ;ation problems that Sections 4-6 use these terms for a tree \T\ denotes the number of vertices in T. For edge w of T, v'a subtree is the tree of w containing v. 1 efficiently [Lo,SGL T - jarity problem. ie "bipartite" parity 3. The phane algorithm for matroid htenection. now, ~op~roftand This section derives results that are needed for a phase algorithm for matroid K,ET]. It is shown intersection. It concludes by sketching the algorithm. Some of the results have been itroid problems, the recently obtained using different methods. Tovey and' Trick [TT]derive a phase algorithm esults in Section 2 for polymatroidal network flow. This problem, which generalizes network flow and matroid ly Cunningham [C]. intersection, was introduced by Lawler and Martel [LM], who prove much of the theory eve the time bounds for the phase algorithm. Our Corollary 2.1 is similar to [TT]. Cunningham [C] derives equivalents of our Corollaries 2.1-2.2. graphic matroids is We begin with some facts that apply to the matroid parity problem. The basic lalysis. The time is concept is the augmenting path: >ft-Karp bound for r matroid problems Definition 2.1. An augmenting path for a matching M consists of distinct elements $I3 Iglls n) if m is , B, f M, 0 5 i 5 k, b,, 6. â M, 0 < i 5 k, such that the sets Mf=(M+\)+a,+a,- 6,-8 ,...- ti++(0$i5k),and Section 3 gives a Mi=(M+#,)+a,+a,-b,-8 +~,-~-b,(O<iS.b) staining a minimum ,... all have the same span, sp(M % \)- An diemating path for At is defined similarly, time is 0(lg n) per + + except that the last element may be a, (as in an augmenting path), or a, or b,. lgorithm to ficd k Note that in an augmenting path, Mf is a matching of rank IAfl + 2. In [SG] it is shown ' m is 0(n lg n) and that a nonmaximum matching has an augmenting path. This can be strengthened as ;be bound of follows. Theorem 2.1. In a matroid parity problem, let hi and N be matchings, with IN = IAfl + 2k Then there is a sequence of paths P, ,..., Pi such that Pi is an augmenting path for =M9PI9... 9~.l,q-MCN-M,qnMCM-N,andallpathsPi Suppose i Â¥ are disjoint. by induction ai Proof. For kl, modify N to a matching contained in M U N whose span includes '(e,+,) 5 i t 2, sp(fvf). Then use a swapping argument, as in [SG]. For k>1 use induction. ' Next supy ~~(ff~,~~).by The rest of thii section is restricted to the matroid intersection problem. Recall that in matroid intersection the ground set E is the disjoint union of qronnd sets Ei i = 0.1.
Recommended publications
  • Linear Algebraic Techniques for Spanning Tree Enumeration

    Linear Algebraic Techniques for Spanning Tree Enumeration

    LINEAR ALGEBRAIC TECHNIQUES FOR SPANNING TREE ENUMERATION STEVEN KLEE AND MATTHEW T. STAMPS Abstract. Kirchhoff's Matrix-Tree Theorem asserts that the number of spanning trees in a finite graph can be computed from the determinant of any of its reduced Laplacian matrices. In many cases, even for well-studied families of graphs, this can be computationally or algebraically taxing. We show how two well-known results from linear algebra, the Matrix Determinant Lemma and the Schur complement, can be used to elegantly count the spanning trees in several significant families of graphs. 1. Introduction A graph G consists of a finite set of vertices and a set of edges that connect some pairs of vertices. For the purposes of this paper, we will assume that all graphs are simple, meaning they do not contain loops (an edge connecting a vertex to itself) or multiple edges between a given pair of vertices. We will use V (G) and E(G) to denote the vertex set and edge set of G respectively. For example, the graph G with V (G) = f1; 2; 3; 4g and E(G) = ff1; 2g; f2; 3g; f3; 4g; f1; 4g; f1; 3gg is shown in Figure 1. A spanning tree in a graph G is a subgraph T ⊆ G, meaning T is a graph with V (T ) ⊆ V (G) and E(T ) ⊆ E(G), that satisfies three conditions: (1) every vertex in G is a vertex in T , (2) T is connected, meaning it is possible to walk between any two vertices in G using only edges in T , and (3) T does not contain any cycles.
  • Teacher's Guide for Spanning and Weighted Spanning Trees

    Teacher's Guide for Spanning and Weighted Spanning Trees

    Teacher’s Guide for Spanning and weighted spanning trees: a different kind of optimization by sarah-marie belcastro 1TheMath. Let’s talk about spanning trees. No, actually, first let’s talk about graph theory,theareaof mathematics within which the topic of spanning trees lies. 1.1 Graph Theory Background. Informally, a graph is a collection of vertices (that look like dots) and edges (that look like curves), where each edge joins two vertices. Formally, A graph is a pair G =(V,E), where V is a set of dots and E is a set of pairs of vertices. Here are a few examples of graphs, in Figure 1: e b f a Figure 1: Examples of graphs. Note that the word vertex is singular; its plural is vertices. Two vertices that are joined by an edge are called adjacent. For example, the vertices labeled a and b in the leftmost graph of Figure 1 are adjacent. Two edges that meet at a vertex are called incident. For example, the edges labeled e and f in the leftmost graph of Figure 1 are incident. A subgraph is a graph that is contained within another graph. For example, in Figure 1 the second graph is a subgraph of the fourth graph. You can see this at left in Figure 2 where the subgraph in question is emphasized. Figure 2: More examples of graphs. In a connected graph, there is a way to get from any vertex to any other vertex without leaving the graph. The second graph of Figure 2 is not connected.
  • Introduction to Spanning Tree Protocol by George Thomas, Contemporary Controls

    Introduction to Spanning Tree Protocol by George Thomas, Contemporary Controls

    Volume6•Issue5 SEPTEMBER–OCTOBER 2005 © 2005 Contemporary Control Systems, Inc. Introduction to Spanning Tree Protocol By George Thomas, Contemporary Controls Introduction powered and its memory cleared (Bridge 2 will be added later). In an industrial automation application that relies heavily Station 1 sends a message to on the health of the Ethernet network that attaches all the station 11 followed by Station 2 controllers and computers together, a concern exists about sending a message to Station 11. what would happen if the network fails? Since cable failure is These messages will traverse the the most likely mishap, cable redundancy is suggested by bridge from one LAN to the configuring the network in either a ring or by carrying parallel other. This process is called branches. If one of the segments is lost, then communication “relaying” or “forwarding.” The will continue down a parallel path or around the unbroken database in the bridge will note portion of the ring. The problem with these approaches is the source addresses of Stations that Ethernet supports neither of these topologies without 1 and 2 as arriving on Port A. This special equipment. However, this issue is addressed in an process is called “learning.” When IEEE standard numbered 802.1D that covers bridges, and in Station 11 responds to either this standard the concept of the Spanning Tree Protocol Station 1 or 2, the database will (STP) is introduced. note that Station 11 is on Port B. IEEE 802.1D If Station 1 sends a message to Figure 1. The addition of Station 2, the bridge will do ANSI/IEEE Std 802.1D, 1998 edition addresses the Bridge 2 creates a loop.
  • A Combinatorial Abstraction of the Shortest Path Problem and Its Relationship to Greedoids

    A Combinatorial Abstraction of the Shortest Path Problem and Its Relationship to Greedoids

    A Combinatorial Abstraction of the Shortest Path Problem and its Relationship to Greedoids by E. Andrew Boyd Technical Report 88-7, May 1988 Abstract A natural generalization of the shortest path problem to arbitrary set systems is presented that captures a number of interesting problems, in­ cluding the usual graph-theoretic shortest path problem and the problem of finding a minimum weight set on a matroid. Necessary and sufficient conditions for the solution of this problem by the greedy algorithm are then investigated. In particular, it is noted that it is necessary but not sufficient for the underlying combinatorial structure to be a greedoid, and three ex­ tremely diverse collections of sufficient conditions taken from the greedoid literature are presented. 0.1 Introduction Two fundamental problems in the theory of combinatorial optimization are the shortest path problem and the problem of finding a minimum weight set on a matroid. It has long been recognized that both of these problems are solvable by a greedy algorithm - the shortest path problem by Dijk­ stra's algorithm [Dijkstra 1959] and the matroid problem by "the" greedy algorithm [Edmonds 1971]. Because these two problems are so fundamental and have such similar solution procedures it is natural to ask if they have a common generalization. The answer to this question not only provides insight into what structural properties make the greedy algorithm work but expands the class of combinatorial optimization problems known to be effi­ ciently solvable. The present work is related to the broader question of recognizing gen­ eral conditions under which a greedy algorithm can be used to solve a given combinatorial optimization problem.
  • Arxiv:1403.0920V3 [Math.CO] 1 Mar 2019

    Arxiv:1403.0920V3 [Math.CO] 1 Mar 2019

    Matroids, delta-matroids and embedded graphs Carolyn Chuna, Iain Moffattb, Steven D. Noblec,, Ralf Rueckriemend,1 aMathematics Department, United States Naval Academy, Chauvenet Hall, 572C Holloway Road, Annapolis, Maryland 21402-5002, United States of America bDepartment of Mathematics, Royal Holloway University of London, Egham, Surrey, TW20 0EX, United Kingdom cDepartment of Mathematics, Brunel University, Uxbridge, Middlesex, UB8 3PH, United Kingdom d Aschaffenburger Strasse 23, 10779, Berlin Abstract Matroid theory is often thought of as a generalization of graph theory. In this paper we propose an analogous correspondence between embedded graphs and delta-matroids. We show that delta-matroids arise as the natural extension of graphic matroids to the setting of embedded graphs. We show that various basic ribbon graph operations and concepts have delta-matroid analogues, and illus- trate how the connections between embedded graphs and delta-matroids can be exploited. Also, in direct analogy with the fact that the Tutte polynomial is matroidal, we show that several polynomials of embedded graphs from the liter- ature, including the Las Vergnas, Bollab´as-Riordanand Krushkal polynomials, are in fact delta-matroidal. Keywords: matroid, delta-matroid, ribbon graph, quasi-tree, partial dual, topological graph polynomial 2010 MSC: 05B35, 05C10, 05C31, 05C83 1. Overview Matroid theory is often thought of as a generalization of graph theory. Many results in graph theory turn out to be special cases of results in matroid theory. This is beneficial
  • Sampling Random Spanning Trees Faster Than Matrix Multiplication

    Sampling Random Spanning Trees Faster Than Matrix Multiplication

    Sampling Random Spanning Trees Faster than Matrix Multiplication David Durfee∗ Rasmus Kyng† John Peebles‡ Anup B. Rao§ Sushant Sachdeva¶ Abstract We present an algorithm that, with high probability, generates a random spanning tree from an edge-weighted undirected graph in Oe(n4/3m1/2 + n2) time 1. The tree is sampled from a distribution where the probability of each tree is proportional to the product of its edge weights. This improves upon the previous best algorithm due to Colbourn et al. that runs in matrix ω multiplication time, O(n ). For the special case of unweighted√ graphs, this improves upon the best previously known running time of O˜(min{nω, m n, m4/3}) for m n5/3 (Colbourn et al. ’96, Kelner-Madry ’09, Madry et al. ’15). The effective resistance metric is essential to our algorithm, as in the work of Madry et al., but we eschew determinant-based and random walk-based techniques used by previous algorithms. Instead, our algorithm is based on Gaussian elimination, and the fact that effective resistance is preserved in the graph resulting from eliminating a subset of vertices (called a Schur complement). As part of our algorithm, we show how to compute -approximate effective resistances for a set S of vertex pairs via approximate Schur complements in Oe(m + (n + |S|)−2) time, without using the Johnson-Lindenstrauss lemma which requires Oe(min{(m + |S|)−2, m + n−4 + |S|−2}) time. We combine this approximation procedure with an error correction procedure for handing edges where our estimate isn’t sufficiently accurate.
  • Matroids You Have Known

    Matroids You Have Known

    26 MATHEMATICS MAGAZINE Matroids You Have Known DAVID L. NEEL Seattle University Seattle, Washington 98122 [email protected] NANCY ANN NEUDAUER Pacific University Forest Grove, Oregon 97116 nancy@pacificu.edu Anyone who has worked with matroids has come away with the conviction that matroids are one of the richest and most useful ideas of our day. —Gian Carlo Rota [10] Why matroids? Have you noticed hidden connections between seemingly unrelated mathematical ideas? Strange that finding roots of polynomials can tell us important things about how to solve certain ordinary differential equations, or that computing a determinant would have anything to do with finding solutions to a linear system of equations. But this is one of the charming features of mathematics—that disparate objects share similar traits. Properties like independence appear in many contexts. Do you find independence everywhere you look? In 1933, three Harvard Junior Fellows unified this recurring theme in mathematics by defining a new mathematical object that they dubbed matroid [4]. Matroids are everywhere, if only we knew how to look. What led those junior-fellows to matroids? The same thing that will lead us: Ma- troids arise from shared behaviors of vector spaces and graphs. We explore this natural motivation for the matroid through two examples and consider how properties of in- dependence surface. We first consider the two matroids arising from these examples, and later introduce three more that are probably less familiar. Delving deeper, we can find matroids in arrangements of hyperplanes, configurations of points, and geometric lattices, if your tastes run in that direction.
  • Parity Systems and the Delta-Matroid Intersection Problem

    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.
  • Oasics-SOSA-2019-14.Pdf (0.4

    Oasics-SOSA-2019-14.Pdf (0.4

    Simple Greedy 2-Approximation Algorithm for the Maximum Genus of a Graph Michal Kotrbčík Department of Computer Science, Comenius University, 842 48 Bratislava, Slovakia [email protected] Martin Škoviera1 Department of Computer Science, Comenius University, 842 48 Bratislava, Slovakia [email protected] Abstract The maximum genus γM (G) of a graph G is the largest genus of an orientable surface into which G has a cellular embedding. Combinatorially, it coincides with the maximum number of disjoint pairs of adjacent edges of G whose removal results in a connected spanning subgraph of G. In this paper we describe a greedy 2-approximation algorithm for maximum genus by proving that removing pairs of adjacent edges from G arbitrarily while retaining connectedness leads to at least γM (G)/2 pairs of edges removed. As a consequence of our approach we also obtain a 2-approximate counterpart of Xuong’s combinatorial characterisation of maximum genus. 2012 ACM Subject Classification Theory of computation → Design and analysis of algorithms → Graph algorithms analysis, Mathematics of computing → Graph algorithms, Mathematics of computing → Graphs and surfaces Keywords and phrases maximum genus, embedding, graph, greedy algorithm Digital Object Identifier 10.4230/OASIcs.SOSA.2019.14 Acknowledgements The authors would like to thank Rastislav Královič and Jana Višňovská for reading preliminary versions of this paper and making useful suggestions. 1 Introduction One of the paradigms in topological graph theory is the study of all surface embeddings of a given graph. The maximum genus γM (G) parameter of a graph G is then the maximum integer g such that G has a cellular embedding in the orientable surface of genus g.A result of Duke [12] implies that a graph G has a cellular embedding in the orientable surface of genus g if and only if γ(G) ≤ g ≤ γM (G) where γ(G) denotes the (minimum) genus of G.
  • Geographic Routing Without Planarization Ben Leong, Barbara Liskov & Robert Morris Require Planarization

    Geographic Routing Without Planarization Ben Leong, Barbara Liskov & Robert Morris Require Planarization

    Geographic Routing without Planarization Ben Leong, Barbara Liskov & Robert Morris MIT CSAIL Greedy Distributed Spanning Tree Routing (GDSTR) • New geographic routing algorithm – DOES NOT require planarization – uses spanning tree, not planar graph – low maintenance cost – better routing performance than existing algorithms Overview • Background • Problem • Approach • Simulation Results • Conclusion Geographic Routing • Wireless nodes have x-y coordinates – can use virtual coordinates (Rao et al. 2003) • Nodes know coordinates of immediate neighbors • Packet destinations specified with x-y coordinates • In general, forward packets greedily Geographic Routing Geographic Routing Source Geographic Routing Destination Source Greedy Forwarding Destination Source Greedy Forwarding Destination Source Greedy Forwarding Destination Source Greedy Forwarding Destination Source Geographic Routing: Dealing with Dead Ends Destination Source Whoops. Dead end! Face Routing Destination Source Face Routing Destination Source Face Routing Destination Source Back to Greedy Forwarding Destination Source Back to Greedy Forwarding Destination Source Back to Greedy Forwarding Destination Source Planarization is Costly! • Planarization is hard for real networks – GG and RNG don’t work • Planarization is complicated & costly! – CLDP (Kim et al., 2005) Greedy Distributed Spanning Tree Routing (GDSTR) • Route on a spanning tree • Use convex hulls to “summarize” the area covered by a subtree – convex hulls tells us what points are possibly reachable – reduces the
  • A Tight Extremal Bound on the Lovász Cactus Number in Planar Graphs

    A Tight Extremal Bound on the Lovász Cactus Number in Planar Graphs

    A Tight Extremal Bound on the Lovász Cactus Number in Planar Graphs Parinya Chalermsook Aalto University, Espoo, Finland parinya.chalermsook@aalto.fi Andreas Schmid Max Planck Institute for Informatics, Saarbrücken, Germany [email protected] Sumedha Uniyal Aalto University, Espoo, Finland sumedha.uniyal@aalto.fi Abstract A cactus graph is a graph in which any two cycles are edge-disjoint. We present a constructive proof of the fact that any plane graph G contains a cactus subgraph C where C contains at least 1 a 6 fraction of the triangular faces of G. We also show that this ratio cannot be improved by showing a tight lower bound. Together with an algorithm for linear matroid parity, our bound implies two approximation algorithms for computing “dense planar structures” inside any graph: (i) 1 A 6 approximation algorithm for, given any graph G, finding a planar subgraph with a maximum 1 number of triangular faces; this improves upon the previous 11 -approximation; (ii) An alternate (and 4 arguably more illustrative) proof of the 9 approximation algorithm for finding a planar subgraph with a maximum number of edges. Our bound is obtained by analyzing a natural local search strategy and heavily exploiting the exchange arguments. Therefore, this suggests the power of local search in handling problems of this kind. 2012 ACM Subject Classification Mathematics of computing → Graph theory Keywords and phrases Graph Drawing, Matroid Matching, Maximum Planar Subgraph, Local Search Algorithms Digital Object Identifier 10.4230/LIPIcs.STACS.2019.19 Related Version Full Version: https://arxiv.org/abs/1804.03485. Funding Parinya Chalermsook: Part of this work was done while PC and AS were visiting the Simons Institute for the Theory of Computing.
  • Branch-Depth: Generalizing Tree-Depth of Graphs

    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.