DOCSLIB.ORG

Strong Parameterized Deletion: Bipartite Graphs

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Strong Parameterized Deletion: Bipartite Graphs Strong Parameterized Deletion: Bipartite Graphs Ashutosh Rai1, M. S. Ramanujan2, and Saket Saurabh∗1,3 1 Institute of Mathematical Sciences, India. {ashutosh,saket}@imsc.res.in 2 Vienna Institute of Technology, Austria. [email protected] 3 University of Bergen, Bergen, Norway. Abstract The purpose of this article is two fold: (a) to formally introduce a stronger version of graph deletion problems; and (b) to study this version in the context of bipartite graphs. Given a family of graphs F, a typical instance of parameterized graph deletion problem consists of an undirected graph G and a positive integer k and the objective is to check whether we can delete at most k vertices (or k edges) such that the resulting graph belongs to F. Another version that has been recently studied is the one where the input contains two integers k1 and k2 and the objective is to check whether we can delete at most k1 vertices and k2 edges such that the resulting graph belongs to F. In this paper, we propose and initiate the study of a more general version which we call strong deletion. In this problem, given an undirected graph G and positive integers k1 and k2, the objective is to check whether there exists a vertex subset S of size at most k1 such that each connected component of G−S can be transformed into a graph in F by deleting at most k2 edges. In this paper we study this stronger version of deletion problems for the class of bipartite graphs. In particular, we study Strong Bipartite Deletion, where given an undirected graph G and positive integers k and `, the objective is to check whether there exists a vertex subset S of size at most k such that each connected component of G − S can be made bipartite by deleting at most ` edges. While fixed-parameter tractability when parameterizing by k or ` alone is unlikely, we show that this problem is fixed-parameter tractable (FPT) when parameterized by both k and `. Keywords and phrases fixed parameter tractable, bipartite-editing, recursive understanding 1 Introduction Graph-modification by either deleting vertices or deleting edges or adding edges such that the resulting graph satisfies certain properties or becomes a member of some well-understood graph class is one of the basic problems in graph theory and graph algorithms. However, most of these problems are NP–complete [19,27] and thus they are subjected to intensive study in algorithmic paradigms that are meant for coping with NP-completeness [9,10,22,23]. These paradigms among others include applying restrictions on inputs, approximation algorithms and parameterized complexity. The goal of this paper is to introduce a ‘stronger’ notion of graph deletion in the realm of parameterized complexity and illustrate the difficulties that arise when considering the family of bipartite graphs and provide an approach to obtain fixed-parameter tractability by overcoming these difficulties. A typical instance of parameterized graph deletion is of the following form. Let F be a family of graphs – such as edgeless graphs, forests, cluster graphs, chordal graphs, interval graphs, bipartite graphs, split graphs or planar graphs. The deletion problem corresponding to F is formally stated as follows. ∗ The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement no. 306992 © Ashutosh Rai, M. S. Ramanujan and Saket Saurabh; licensed under Creative Commons License CC-BY 33rd International Symposium on Theoretical Aspects of Computer Science (STACS 2016). Editors: Nicolas Ollinger and Heribert Vollmer; pp. 1–28 Leibniz International Proceedings in Informatics Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany 2 Strong Parameterized Deletion: Bipartite Graphs Figure 1 An example showing the strength of the new parameter for editing problems. F-Vertex (Edge) Deletion Parameters: k Input: An undirected graph G and a non-negative integer k. Question: Does there exist S ⊆ V (G) (or S ⊆ E(G)), such that |S| ≤ k and G − S is in F? In other words, given a graph G, can we delete at most k vertices or k edges such that the resulting graph belongs to F? An algorithm for F-Vertex (Edge) Deletion that runs in time f(k) · |V (G)|O(1) is called fixed-parameter tractable (FPT) algorithm and the problem itself is said to be FPT. The study of parameterized graph deletion problems together with their various restric- tions and generalizations has been an extremely active sub-area over the last few years. In fact, just over the course of the last couple of years there have been results on parameterized algorithms for Chordal Editing [5], Unit Vertex (Edge) Deletion [3], Interval Vertex (Edge) Deletion [4, 6], Planar F Deletion [9, 18], Planar Vertex Dele- tion [15], Block Graph Deletion [17] and Simultaneous Feedback Vertex Set [1]. Several known parameterized algorithms for F-Edge Deletion or the version where the objective is to delete k1 vertices and k2 edges utilize the fact that given a yes-instance (G, k) ∗ or (G, k1, k2) to the problem, there exists a vertex set S of V (G) of size k = k1 + k2 such that G − S∗ belongs to F. Clearly, this is true for any hereditary family of graphs; that is, if G ∈ F then all its induced subgraphs belong to F. Thus, if the corresponding F-Vertex Deletion is FPT then we can apply this algorithm and find a vertex subset S∗ of size at most k such that G − S∗ ∈ F. Having the set S∗ allows one to infer numerous structural properties of the input which can then be utilized in non-trivial ways to solve the original F-Vertex (Edge) Deletion problem. However, the existence of such a set is no longer guaranteed in the proposed stronger version of this problem. Let F be a polynomial-time recognizable family of graphs; that is, given a graph G, in polynomial time we can decide whether G belongs to F. For a fixed integer `, let F + `e denote the class of graphs that can be obtained by adding ` edges to a graph in F [2]. Furthermore, suppose that F-Edge Deletion is FPT with running time O(g(`) · nc). Here, n = |V (G)| and c is a fixed constant. That is, for any fixed integer `, F + `e can be recognized in time O(nc). The Strong F-Deletion problem is defined as follows. Strong F-Deletion Parameters: k1, k2 Input: An undirected graph G and non-negative integers k1 and k2. Question: Does there exist S1 ⊆ V (G) such that |S1| ≤ k1 and every connected component of G − S1 belongs to F + k2e? A. Rai, M. S. Ramanujan and S. Saurabh 3 A close inspection easily shows that Strong F-Deletion is much stronger than F- Deletion. For example, let F be the family of bipartite graphs and consider the graph n−1 G depicted in Figure 1. The graph G has 3 vertex disjoint triangles and the vertex v is adjacent to two vertices from each of the triangles. Clearly, after deleting v every connected component can be made bipartite by deleting exactly one edge. Thus, in the traditional n−1 sense this is a yes-instance of Bipartite Deletion with parameters (1, 3 ); however this is already a yes-instance of Strong Bipartite Deletion with parameters (1, 1). Thus, these problems seem much harder than traditional editing problems as the parameters are much smaller. An alternate viewpoint is that when G has a vertex set S of size at most k1 such that every connected component of G − S is in F + k2e then S can be considered a modulator into the graph class where every connected component of the graph belongs to F + k2e. While modulators to various polynomial-time recognizable graph classes have been studied in a very systematic way [8,9,14], the same is not true of modulators to NP-complete graph classes. Studying the Strong F-Deletion problem on the other hand allows us to do precisely this. In fact, it is not even necessary that the class F is a conventional graph class and instead can be the ‘composition’ of various graph classes. For instance F could be defined as the set of all graphs where each connected component is either chordal or a bipartite graph. The computation of such modulators is a problem with several algorithmic applications. For instance, in a recent work, Ganian et al. [11] used a similar notion in order to design algorithms for the classic Constraint Satisfaction Problem. In this article we study the Strong F-Deletion, when F is the family of bipartite graphs. Henceforth, F denotes the family of bipartite graphs. We call a graph G, t- pseudobipartite, if every connected component of G belongs to F + te. The problem we study is as follows. Strong Bipartite Deletion Parameters: k, ` Input: An undirected graph G and non-negative integers k and `. Question: Does there exist S ⊆ V (G) such that |S| ≤ k and every connected compon- ent of G − S belongs to F + `e? We refer to the set S as the solution for this instance. The primary reason behind the selection of the family of bipartite graphs for our study is that the problems where we are required to delete vertices and/or edges to obtain a bipartite graph are some of the most basic and well studied problems in parameterized complexity and studying these problems has led to the discovery of several new techniques and tools. These problems are called Odd Cycle Transversal and Edge Bipartization in literature and the algorithms with best dependence on the parameter have running time O(2.3146knO(1)) and O(1.977knO(1)), respectively [20, 25].
Load more

Recommended publications
  • Counting Independent Sets in Graphs with Bounded Bipartite Pathwidth∗
    Counting independent sets in graphs with bounded bipartite pathwidth∗ Martin Dyery Catherine Greenhillz School of Computing School of Mathematics and Statistics University of Leeds UNSW Sydney, NSW 2052 Leeds LS2 9JT, UK Australia [email protected] [email protected] Haiko M¨uller∗ School of Computing University of Leeds Leeds LS2 9JT, UK [email protected] 7 August 2019 Abstract We show that a simple Markov chain, the Glauber dynamics, can efficiently sample independent sets almost uniformly at random in polynomial time for graphs in a certain class. The class is determined by boundedness of a new graph parameter called bipartite pathwidth. This result, which we prove for the more general hardcore distribution with fugacity λ, can be viewed as a strong generalisation of Jerrum and Sinclair's work on approximately counting matchings, that is, independent sets in line graphs. The class of graphs with bounded bipartite pathwidth includes claw-free graphs, which generalise line graphs. We consider two further generalisations of claw-free graphs and prove that these classes have bounded bipartite pathwidth. We also show how to extend all our results to polynomially-bounded vertex weights. 1 Introduction There is a well-known bijection between matchings of a graph G and independent sets in the line graph of G. We will show that we can approximate the number of independent sets ∗A preliminary version of this paper appeared as [19]. yResearch supported by EPSRC grant EP/S016562/1 \Sampling in hereditary classes". zResearch supported by Australian Research Council grant DP190100977. 1 in graphs for which all bipartite induced subgraphs are well structured, in a sense that we will define precisely.
    [Show full text]
  • Density Theorems for Bipartite Graphs and Related Ramsey-Type Results
    Density theorems for bipartite graphs and related Ramsey-type results Jacob Fox Benny Sudakov Princeton UCLA and IAS Ramsey’s theorem Definition: r(G) is the minimum N such that every 2-edge-coloring of the complete graph KN contains a monochromatic copy of graph G. Theorem: (Ramsey-Erdos-Szekeres,˝ Erdos)˝ t/2 2t 2 ≤ r(Kt ) ≤ 2 . Question: (Burr-Erd˝os1975) How large is r(G) for a sparse graph G on n vertices? Ramsey numbers for sparse graphs Conjecture: (Burr-Erd˝os1975) For every d there exists a constant cd such that if a graph G has n vertices and maximum degree d, then r(G) ≤ cd n. Theorem: 1 (Chv´atal-R¨odl-Szemer´edi-Trotter 1983) cd exists. 2αd 2 (Eaton 1998) cd ≤ 2 . βd αd log2 d 3 (Graham-R¨odl-Ruci´nski2000) 2 ≤ cd ≤ 2 . Moreover, if G is bipartite, r(G) ≤ 2αd log d n. Density theorem for bipartite graphs Theorem: (F.-Sudakov) Let G be a bipartite graph with n vertices and maximum degree d 2 and let H be a bipartite graph with parts |V1| = |V2| = N and εN edges. If N ≥ 8dε−d n, then H contains G. Corollary: For every bipartite graph G with n vertices and maximum degree d, r(G) ≤ d2d+4n. (D. Conlon independently proved that r(G) ≤ 2(2+o(1))d n.) Proof: Take ε = 1/2 and H to be the graph of the majority color. Ramsey numbers for cubes Definition: d The binary cube Qd has vertex set {0, 1} and x, y are adjacent if x and y differ in exactly one coordinate.
    [Show full text]
  • Constrained Representations of Map Graphs and Half-Squares
    Constrained Representations of Map Graphs and Half-Squares Hoang-Oanh Le Berlin, Germany [email protected] Van Bang Le Universität Rostock, Institut für Informatik, Rostock, Germany [email protected] Abstract The square of a graph H, denoted H2, is obtained from H by adding new edges between two distinct vertices whenever their distance in H is two. The half-squares of a bipartite graph B = (X, Y, EB ) are the subgraphs of B2 induced by the color classes X and Y , B2[X] and B2[Y ]. For a given graph 2 G = (V, EG), if G = B [V ] for some bipartite graph B = (V, W, EB ), then B is a representation of G and W is the set of points in B. If in addition B is planar, then G is also called a map graph and B is a witness of G [Chen, Grigni, Papadimitriou. Map graphs. J. ACM, 49 (2) (2002) 127-138]. While Chen, Grigni, Papadimitriou proved that any map graph G = (V, EG) has a witness with at most 3|V | − 6 points, we show that, given a map graph G and an integer k, deciding if G admits a witness with at most k points is NP-complete. As a by-product, we obtain NP-completeness of edge clique partition on planar graphs; until this present paper, the complexity status of edge clique partition for planar graphs was previously unknown. We also consider half-squares of tree-convex bipartite graphs and prove the following complexity 2 dichotomy: Given a graph G = (V, EG) and an integer k, deciding if G = B [V ] for some tree-convex bipartite graph B = (V, W, EB ) with |W | ≤ k points is NP-complete if G is non-chordal dually chordal and solvable in linear time otherwise.
    [Show full text]
  • Graph Theory
    1 Graph Theory “Begin at the beginning,” the King said, gravely, “and go on till you come to the end; then stop.” — Lewis Carroll, Alice in Wonderland The Pregolya River passes through a city once known as K¨onigsberg. In the 1700s seven bridges were situated across this river in a manner similar to what you see in Figure 1.1. The city’s residents enjoyed strolling on these bridges, but, as hard as they tried, no residentof the city was ever able to walk a route that crossed each of these bridges exactly once. The Swiss mathematician Leonhard Euler learned of this frustrating phenomenon, and in 1736 he wrote an article [98] about it. His work on the “K¨onigsberg Bridge Problem” is considered by many to be the beginning of the field of graph theory. FIGURE 1.1. The bridges in K¨onigsberg. J.M. Harris et al., Combinatorics and Graph Theory , DOI: 10.1007/978-0-387-79711-3 1, °c Springer Science+Business Media, LLC 2008 2 1. Graph Theory At first, the usefulness of Euler’s ideas and of “graph theory” itself was found only in solving puzzles and in analyzing games and other recreations. In the mid 1800s, however, people began to realize that graphs could be used to model many things that were of interest in society. For instance, the “Four Color Map Conjec- ture,” introduced by DeMorgan in 1852, was a famous problem that was seem- ingly unrelated to graph theory. The conjecture stated that four is the maximum number of colors required to color any map where bordering regions are colored differently.
    [Show full text]
  • On Computing Longest Paths in Small Graph Classes
    On Computing Longest Paths in Small Graph Classes Ryuhei Uehara∗ Yushi Uno† July 28, 2005 Abstract The longest path problem is to find a longest path in a given graph. While the graph classes in which the Hamiltonian path problem can be solved efficiently are widely investigated, few graph classes are known to be solved efficiently for the longest path problem. For a tree, a simple linear time algorithm for the longest path problem is known. We first generalize the algorithm, and show that the longest path problem can be solved efficiently for weighted trees, block graphs, and cacti. We next show that the longest path problem can be solved efficiently on some graph classes that have natural interval representations. Keywords: efficient algorithms, graph classes, longest path problem. 1 Introduction The Hamiltonian path problem is one of the most well known NP-hard problem, and there are numerous applications of the problems [17]. For such an intractable problem, there are two major approaches; approximation algorithms [20, 2, 35] and algorithms with parameterized complexity analyses [15]. In both approaches, we have to change the decision problem to the optimization problem. Therefore the longest path problem is one of the basic problems from the viewpoint of combinatorial optimization. From the practical point of view, it is also very natural approach to try to find a longest path in a given graph, even if it does not have a Hamiltonian path. However, finding a longest path seems to be more difficult than determining whether the given graph has a Hamiltonian path or not.
    [Show full text]
  • Lecture 12 — October 10, 2017 1 Overview 2 Matchings in Bipartite Graphs
    CS 388R: Randomized Algorithms Fall 2017 Lecture 12 | October 10, 2017 Prof. Eric Price Scribe: Shuangquan Feng, Xinrui Hua 1 Overview In this lecture, we will explore 1. Problem of finding matchings in bipartite graphs. 2. Hall's Theorem: sufficient and necessary condition for the existence of perfect matchings in bipartite graphs. 3. An algorithm for finding perfect matching in d-regular bipartite graphs when d = 2k which is based on the idea of Euler tour and has a time complexity of O(nd). 4. A randomized algorithm for finding perfect matchings in all d regular bipartite graphs which has an expected time complexity of O(n log n). 5. Problem of online bipartite matching. 2 Matchings in Bipartite Graphs Definition 1 (Bipartite Graph). A graph G = (V; E) is said to be bipartite if the vertex set V can be partitioned into 2 disjoint sets L and R so that any edge has one vertex in L and the other in R. Definition 2 (Matching). Given an undirected graph G = (V; E), a matching is a subset of edges M ⊆ E that have no endpoints in common. Definition 3 (Maximum Matching). Given an undirected graph G = (V; E), a maximum matching M is a matching of maximum size. Thus for any other matching M 0, we have that jMj ≥ jM 0j. Definition 4 (Perfect Matching). Given an bipartite graph G = (V; E), with the bipartition V = L [ R where jLj = jRj = n, a perfect matching is a maximum matching of size n. There are several well known algorithms for the problem of finding maximum matchings in bipartite graphs.
    [Show full text]
  • Vertex Deletion Problems on Chordal Graphs∗†
    Vertex Deletion Problems on Chordal Graphs∗† Yixin Cao1, Yuping Ke2, Yota Otachi3, and Jie You4 1 Department of Computing, Hong Kong Polytechnic University, Hong Kong, China [email protected] 2 Department of Computing, Hong Kong Polytechnic University, Hong Kong, China [email protected] 3 Faculty of Advanced Science and Technology, Kumamoto University, Kumamoto, Japan [email protected] 4 School of Information Science and Engineering, Central South University and Department of Computing, Hong Kong Polytechnic University, Hong Kong, China [email protected] Abstract Containing many classic optimization problems, the family of vertex deletion problems has an important position in algorithm and complexity study. The celebrated result of Lewis and Yan- nakakis gives a complete dichotomy of their complexity. It however has nothing to say about the case when the input graph is also special. This paper initiates a systematic study of vertex deletion problems from one subclass of chordal graphs to another. We give polynomial-time algorithms or proofs of NP-completeness for most of the problems. In particular, we show that the vertex deletion problem from chordal graphs to interval graphs is NP-complete. 1998 ACM Subject Classification F.2.2 Analysis of Algorithms and Problem Complexity, G.2.2 Graph Theory Keywords and phrases vertex deletion problem, maximum subgraph, chordal graph, (unit) in- terval graph, split graph, hereditary property, NP-complete, polynomial-time algorithm Digital Object Identifier 10.4230/LIPIcs.FSTTCS.2017.22 1 Introduction Generally speaking, a vertex deletion problem asks to transform an input graph to a graph in a certain class by deleting a minimum number of vertices.
    [Show full text]
  • Lecture 9-10: Extremal Combinatorics 1 Bipartite Forbidden Subgraphs 2 Graphs Without Any 4-Cycle
    MAT 307: Combinatorics Lecture 9-10: Extremal combinatorics Instructor: Jacob Fox 1 Bipartite forbidden subgraphs We have seen the Erd}os-Stonetheorem which says that given a forbidden subgraph H, the extremal 1 2 number of edges is ex(n; H) = 2 (1¡1=(Â(H)¡1)+o(1))n . Here, o(1) means a term tending to zero as n ! 1. This basically resolves the question for forbidden subgraphs H of chromatic number at least 3, since then the answer is roughly cn2 for some constant c > 0. However, for bipartite forbidden subgraphs, Â(H) = 2, this answer is not satisfactory, because we get ex(n; H) = o(n2), which does not determine the order of ex(n; H). Hence, bipartite graphs form the most interesting class of forbidden subgraphs. 2 Graphs without any 4-cycle Let us start with the ¯rst non-trivial case where H is bipartite, H = C4. I.e., the question is how many edges G can have before a 4-cycle appears. The answer is roughly n3=2. Theorem 1. For any graph G on n vertices, not containing a 4-cycle, 1 p E(G) · (1 + 4n ¡ 3)n: 4 Proof. Let dv denote the degree of v 2 V . Let F denote the set of \labeled forks": F = f(u; v; w):(u; v) 2 E; (u; w) 2 E; v 6= wg: Note that we do not care whether (v; w) is an edge or not. We count the size of F in two possible ways: First, each vertex u contributes du(du ¡ 1) forks, since this is the number of choices for v and w among the neighbors of u.
    [Show full text]
  • Hypergraph Packing and Sparse Bipartite Ramsey Numbers
    Hypergraph packing and sparse bipartite Ramsey numbers David Conlon ∗ Abstract We prove that there exists a constant c such that, for any integer ∆, the Ramsey number of a bipartite graph on n vertices with maximum degree ∆ is less than 2c∆n. A probabilistic argument due to Graham, R¨odland Ruci´nskiimplies that this result is essentially sharp, up to the constant c in the exponent. Our proof hinges upon a quantitative form of a hypergraph packing result of R¨odl,Ruci´nskiand Taraz. 1 Introduction For a graph G, the Ramsey number r(G) is defined to be the smallest natural number n such that, in any two-colouring of the edges of Kn, there exists a monochromatic copy of G. That these numbers exist was proven by Ramsey [19] and rediscovered independently by Erd}osand Szekeres [10]. Whereas the original focus was on finding the Ramsey numbers of complete graphs, in which case it is known that t t p2 r(Kt) 4 ; ≤ ≤ the field has broadened considerably over the years. One of the most famous results in the area to date is the theorem, due to Chvat´al,R¨odl,Szemer´ediand Trotter [7], that if a graph G, on n vertices, has maximum degree ∆, then r(G) c(∆)n; ≤ where c(∆) is just some appropriate constant depending only on ∆. Their proof makes use of the regularity lemma and because of this the bound it gives on the constant c(∆) is (and is necessarily [11]) very bad. The situation was improved somewhat by Eaton [8], who proved, by using a variant of the regularity lemma, that the function c(∆) may be taken to be of the form 22c∆ .
    [Show full text]
  • THE CRITICAL GROUP of a LINE GRAPH: the BIPARTITE CASE Contents 1. Introduction 2 2. Preliminaries 2 2.1. the Graph Laplacian 2
    THE CRITICAL GROUP OF A LINE GRAPH: THE BIPARTITE CASE JOHN MACHACEK Abstract. The critical group K(G) of a graph G is a finite abelian group whose order is the number of spanning forests of the graph. Here we investigate the relationship between the critical group of a regular bipartite graph G and its line graph line G. The relationship between the two is known completely for regular nonbipartite graphs. We compute the critical group of a graph closely related to the complete bipartite graph and the critical group of its line graph. We also discuss general theory for the critical group of regular bipartite graphs. We close with various examples demonstrating what we have observed through experimentation. The problem of classifying the the relationship between K(G) and K(line G) for regular bipartite graphs remains open. Contents 1. Introduction 2 2. Preliminaries 2 2.1. The graph Laplacian 2 2.2. Theory of lattices 3 2.3. The line graph and edge subdivision graph 3 2.4. Circulant graphs 5 2.5. Smith normal form and matrices 6 3. Matrix reductions 6 4. Some specific regular bipartite graphs 9 4.1. The almost complete bipartite graph 9 4.2. Bipartite circulant graphs 11 5. A few general results 11 5.1. The quotient group 11 5.2. Perfect matchings 12 6. Looking forward 13 6.1. Odd primes 13 6.2. The prime 2 14 6.3. Example exact sequences 15 References 17 Date: December 14, 2011. A special thanks to Dr. Victor Reiner for his guidance and suggestions in completing this work.
    [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]
  • Introducing Subclasses of Basic Chordal Graphs
    Introducing subclasses of basic chordal graphs Pablo De Caria 1 Marisa Gutierrez 2 CONICET/ Universidad Nacional de La Plata, Argentina Abstract Basic chordal graphs arose when comparing clique trees of chordal graphs and compatible trees of dually chordal graphs. They were defined as those chordal graphs whose clique trees are exactly the compatible trees of its clique graph. In this work, we consider some subclasses of basic chordal graphs, like hereditary basic chordal graphs, basic DV and basic RDV graphs, we characterize them and we find some other properties they have, mostly involving clique graphs. Keywords: Chordal graph, dually chordal graph, clique tree, compatible tree. 1 Introduction 1.1 Definitions For a graph G, V (G) is the set of its vertices and E(G) is the set of its edges. A clique is a maximal set of pairwise adjacent vertices. The family of cliques of G is denoted by C(G). For v ∈ V (G), Cv will denote the family of cliques containing v. All graphs considered will be assumed to be connected. A set S ⊆ V (G) is a uv-separator if vertices u and v are in different connected components of G−S. It is minimal if no proper subset of S has the 1 Email: [email protected] aaURL: www.mate.unlp.edu.ar/~pdecaria 2 Email: [email protected] same property. S is a minimal vertex separator if there exist two nonadjacent vertices u and v such that S is a minimal uv-separator. S(G) will denote the family of all minimal vertex separators of G.
    [Show full text]