List Coloring of Two Matroids Through Reduction to Partition Matroids
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Even Factors, Jump Systems, and Discrete Convexity
Even Factors, Jump Systems, and Discrete Convexity Yusuke KOBAYASHI¤ Kenjiro TAKAZAWAy May, 2007 Abstract A jump system, which is a set of integer lattice points with an exchange property, is an extended concept of a matroid. Some combinatorial structures such as the degree sequences of the matchings in an undirected graph are known to form a jump system. On the other hand, the maximum even factor problem is a generalization of the maximum matching problem into digraphs. When the given digraph has a certain property called odd- cycle-symmetry, this problem is polynomially solvable. The main result of this paper is that the degree sequences of all even factors in a digraph form a jump system if and only if the digraph is odd-cycle-symmetric. Furthermore, as a gen- eralization, we show that the weighted even factors induce M-convex (M-concave) functions on jump systems. These results suggest that even factors are a natural generalization of matchings and the assumption of odd-cycle-symmetry of digraphs is essential. 1 Introduction In the study of combinatorial optimization, extensions of matroids are introduced as abstract con- cepts including many combinatorial objects. A number of optimization problems on matroidal structures can be solved in polynomial time. One of the extensions of matroids is a jump system of Bouchet and Cunningham [2]. A jump system is a set of integer lattice points with an exchange property (to be described in Section 2.1); see also [18, 22]. It is a generalization of a matroid [4], a delta-matroid [1, 3, 8], and a base polyhedron of an integral polymatroid (or a submodular sys- tem) [14]. -
Invitation to Matroid Theory
Invitation to Matroid Theory Gregory Henselman-Petrusek January 21, 2021 Abstract This text is a companion to the short course Invitation to matroid theory taught in the Univer- sity of Oxford Centre for TDA in January 2021. It was first written for algebraic topologists, but should be suitable for all audiences; no background in matroid theory is assumed! 1 How to use this text Matroid theory is a beautiful, powerful, and accessible. Many important subjects can be understood and even researched with just a handful of concepts and definitions from matroid theory. Even so, getting use out of matroids often depends on knowing the right keywords for an internet search, and finding the right keyword depends on two things: (i) a precise knowledge of the core terms/definitions of matroid theory, and (ii) a birds eye view of the network of relationships that interconnect some of the main ideas in the field. These are the focus of this text; it falls far short of a comprehensive overview, but provides a strong start to (i) and enough of (ii) to build an interesting (and fun!) network of ideas including some of the most important concepts in matroid theory. The proofs of most results can be found in standard textbooks (e.g. Oxley, Matroid theory). Exercises are chosen to maximize the fraction intellectual reward time required to solve Therefore, if an exercise takes more than a few minutes to solve, it is fine to look up answers in a textbook! Struggling is not prerequisite to learning, at this level. 2 References The following make no attempt at completeness. -
Masters Thesis: an Approach to the Automatic Synthesis of Controllers with Mixed Qualitative/Quantitative Specifications
An approach to the automatic synthesis of controllers with mixed qualitative/quantitative specifications. Athanasios Tasoglou Master of Science Thesis Delft Center for Systems and Control An approach to the automatic synthesis of controllers with mixed qualitative/quantitative specifications. Master of Science Thesis For the degree of Master of Science in Embedded Systems at Delft University of Technology Athanasios Tasoglou October 11, 2013 Faculty of Electrical Engineering, Mathematics and Computer Science (EWI) · Delft University of Technology *Cover by Orestis Gartaganis Copyright c Delft Center for Systems and Control (DCSC) All rights reserved. Abstract The world of systems and control guides more of our lives than most of us realize. Most of the products we rely on today are actually systems comprised of mechanical, electrical or electronic components. Engineering these complex systems is a challenge, as their ever growing complexity has made the analysis and the design of such systems an ambitious task. This urged the need to explore new methods to mitigate the complexity and to create sim- plified models. The answer to these new challenges? Abstractions. An abstraction of the the continuous dynamics is a symbolic model, where each “symbol” corresponds to an “aggregate” of states in the continuous model. Symbolic models enable the correct-by-design synthesis of controllers and the synthesis of controllers for classes of specifications that traditionally have not been considered in the context of continuous control systems. These include qualitative specifications formalized using temporal logics, such as Linear Temporal Logic (LTL). Be- sides addressing qualitative specifications, we are also interested in synthesizing controllers with quantitative specifications, in order to solve optimal control problems. -
Efficient Graph Reachability Query Answering Using Tree Decomposition
Efficient Graph Reachability Query Answering using Tree Decomposition Fang Wei Computer Science Department, University of Freiburg, Germany Abstract. Efficient reachability query answering in large directed graphs has been intensively investigated because of its fundamental importance in many application fields such as XML data processing, ontology rea- soning and bioinformatics. In this paper, we present a novel indexing method based on the concept of tree decomposition. We show analytically that this intuitive approach is both time and space efficient. We demonstrate empirically the efficiency and the effectiveness of our method. 1 Introduction Querying and manipulating large scale graph-like data has attracted much atten- tion in the database community, due to the wide application areas of graph data, such as GIS, XML databases, bioinformatics, social network, and ontologies. The problem of reachability test in a directed graph is among the fundamental operations on the graph data. Given a digraph G = (V; E) and u; v 2 V , a reachability query, denoted as u ! v, ask: is there a path from u to v? One of the fundamental queries on biological networks is for instance, to find all genes whose expressions are directly or indirectly influenced by a given molecule [15]. Given the graph representation of the genes and regulation events, the question can also be reduced to the reachability query in a directed graph. Recently, tree decomposition methodologies have been successfully applied to solving shortest path query answering over undirected graphs [17]. Briefly stated, the vertices in a graph G are decomposed into a tree in which each node contains a set of vertices in G. -
Depth-First Search & Directed Graphs
Depth-first Search and Directed Graphs Story So Far • Breadth-first search • Using breadth-first search for connectivity • Using bread-first search for testing bipartiteness BFS (G, s): Put s in the queue Q While Q is not empty Extract v from Q If v is unmarked Mark v For each edge (v, w): Put w into the queue Q The BFS Tree • Can remember parent nodes (the node at level i that lead us to a given node at level i + 1) BFS-Tree(G, s): Put (∅, s) in the queue Q While Q is not empty Extract (p, v) from Q If v is unmarked Mark v parent(v) = p For each edge (v, w): Put (v, w) into the queue Q Spanning Trees • Definition. A spanning tree of an undirected graph G is a connected acyclic subgraph of G that contains every node of G . • The tree produced by the BFS algorithm (with (( u, parent(u)) as edges) is a spanning tree of the component containing s . • The BFS spanning tree gives the shortest path from s to every other vertex in its component (we will revisit shortest path in a couple of lectures) • BFS trees in general are short and bushy Spanning Trees • Definition. A spanning tree of an undirected graph G is a connected acyclic subgraph of G that contains every node of G . • The tree produced by the BFS algorithm (with (( u, parent(u)) as edges) is a spanning tree of the component containing s . • The BFS spanning tree gives the shortest path from s to every other vertex in its component (we will revisit shortest path in a couple of lectures) • BFS trees in general are short and bushy Generalizing BFS: Whatever-First If we change how we store -
Matroid Theory Release 9.4
Sage 9.4 Reference Manual: Matroid Theory Release 9.4 The Sage Development Team Aug 24, 2021 CONTENTS 1 Basics 1 2 Built-in families and individual matroids 77 3 Concrete implementations 97 4 Abstract matroid classes 149 5 Advanced functionality 161 6 Internals 173 7 Indices and Tables 197 Python Module Index 199 Index 201 i ii CHAPTER ONE BASICS 1.1 Matroid construction 1.1.1 Theory Matroids are combinatorial structures that capture the abstract properties of (linear/algebraic/...) dependence. For- mally, a matroid is a pair M = (E; I) of a finite set E, the groundset, and a collection of subsets I, the independent sets, subject to the following axioms: • I contains the empty set • If X is a set in I, then each subset of X is in I • If two subsets X, Y are in I, and jXj > jY j, then there exists x 2 X − Y such that Y + fxg is in I. See the Wikipedia article on matroids for more theory and examples. Matroids can be obtained from many types of mathematical structures, and Sage supports a number of them. There are two main entry points to Sage’s matroid functionality. The object matroids. contains a number of con- structors for well-known matroids. The function Matroid() allows you to define your own matroids from a variety of sources. We briefly introduce both below; follow the links for more comprehensive documentation. Each matroid object in Sage comes with a number of built-in operations. An overview can be found in the documen- tation of the abstract matroid class. -
An Efficient Reachability Indexing Scheme for Large Directed Graphs
1 Path-Tree: An Efficient Reachability Indexing Scheme for Large Directed Graphs RUOMING JIN, Kent State University NING RUAN, Kent State University YANG XIANG, The Ohio State University HAIXUN WANG, Microsoft Research Asia Reachability query is one of the fundamental queries in graph database. The main idea behind answering reachability queries is to assign vertices with certain labels such that the reachability between any two vertices can be determined by the labeling information. Though several approaches have been proposed for building these reachability labels, it remains open issues on how to handle increasingly large number of vertices in real world graphs, and how to find the best tradeoff among the labeling size, the query answering time, and the construction time. In this paper, we introduce a novel graph structure, referred to as path- tree, to help labeling very large graphs. The path-tree cover is a spanning subgraph of G in a tree shape. We show path-tree can be generalized to chain-tree which theoretically can has smaller labeling cost. On top of path-tree and chain-tree index, we also introduce a new compression scheme which groups vertices with similar labels together to further reduce the labeling size. In addition, we also propose an efficient incremental update algorithm for dynamic index maintenance. Finally, we demonstrate both analytically and empirically the effectiveness and efficiency of our new approaches. Categories and Subject Descriptors: H.2.8 [Database management]: Database Applications—graph index- ing and querying General Terms: Performance Additional Key Words and Phrases: Graph indexing, reachability queries, transitive closure, path-tree cover, maximal directed spanning tree ACM Reference Format: Jin, R., Ruan, N., Xiang, Y., and Wang, H. -
The Surprizing Complexity of Generalized Reachability Games Nathanaël Fijalkow, Florian Horn
The surprizing complexity of generalized reachability games Nathanaël Fijalkow, Florian Horn To cite this version: Nathanaël Fijalkow, Florian Horn. The surprizing complexity of generalized reachability games. 2010. hal-00525762v2 HAL Id: hal-00525762 https://hal.archives-ouvertes.fr/hal-00525762v2 Preprint submitted on 2 Feb 2012 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. The surprising complexity of generalized reachability games Nathanaël Fijalkow1,2 and Florian Horn1 1 LIAFA CNRS & Université Denis Diderot - Paris 7, France {nath,florian.horn}@liafa.jussieu.fr 2 ÉNS Cachan École Normale Supérieure de Cachan, France Abstract. Games on graphs provide a natural and powerful model for reactive systems. In this paper, we consider generalized reachability objectives, defined as conjunctions of reachability objectives. We first prove that deciding the winner in such games is PSPACE-complete, although it is fixed-parameter tractable with the number of reachability objectives as parameter. Moreover, we consider the memory requirements for both players and give matching upper and lower bounds on the size of winning strategies. In order to allow more efficient algorithms, we consider subclasses of generalized reachability games. We show that bounding the size of the reachability sets gives two natural subclasses where deciding the winner can be done efficiently. -
Lecture 16: Directed Graphs
Lecture 16: Directed Graphs BOS ORD JFK SFO DFW LAX MIA Courtesy of Goodrich, Tamassia and Olga Veksler Instructor: Yuzhen Xie Outline Directed Graphs Properties Algorithms for Directed Graphs DFS and BFS Strong Connectivity Transitive Closure DAG and Toppgological Ordering 2 Digraphs A digraph is a ggpraph whose E edges are all directed D Short for “directed graph” C Applications B one-way streets A flights task scheduling 3 Digraph Properties E D A graph G=(V,E) such that Each edge goes in one direction: C Ed(dge (A,B) goes from A to B, Edge (B,A) goes from B to A, B If G is simple, m < n*(n-1). A If we keep in-edges and out-edges in separate adjacency lists, we can perform listing of in-edges and out-edges in time proportional to their size OutgoingEdges(A): (A,C), (A,D), (A,B) IngoingEdges(A): (E,A),(B,A) We say that vertex w is reachable from vertex v if there is a directed path from v to w EiseahablefomAE is reachable from A E is not reachable from D 4 Algorithm DFS(G, v) Directed DFS Input digraph G and a start vertex v of G Output traverses vertices in G reachable from v We can specialize the setLabel(v, VISITED) traversal algorithms (DFS and for all e ∈ G.outgoingEdges(v) BFS) to digraphs by traversing edges only along w ← opposite(v,e) their direction if getLabel(w) = UNEXPLORED DFS(G, w) In the directed DFS algorithm, we have 3 four types of edges E discovery edges 4 bkdback edges D forward edges cross edges A directed DFS starting at a C 2 vertex s visits all the vertices reachable from s B 5 -
An Efficient Strongly Connected Components Algorithm In
An Efficient Strongly Connected Components Algorithm in the Fault Tolerant Model∗† Surender Baswana1, Keerti Choudhary2, and Liam Roditty3 1 Department of Computer Science and Engineering, IIT Kanpur, Kanpur, India [email protected] 2 Department of Computer Science and Engineering, IIT Kanpur, Kanpur, India [email protected] 3 Department of Computer Science, Bar Ilan University, Ramat Gan, Israel [email protected] Abstract In this paper we study the problem of maintaining the strongly connected components of a graph in the presence of failures. In particular, we show that given a directed graph G = (V, E) with n = |V | and m = |E|, and an integer value k ≥ 1, there is an algorithm that computes in O(2kn log2 n) time for any set F of size at most k the strongly connected components of the graph G \ F . The running time of our algorithm is almost optimal since the time for outputting the SCCs of G \ F is at least Ω(n). The algorithm uses a data structure that is computed in a preprocessing phase in polynomial time and is of size O(2kn2). Our result is obtained using a new observation on the relation between strongly connected components (SCCs) and reachability. More specifically, one of the main building blocks in our result is a restricted variant of the problem in which we only compute strongly connected com- ponents that intersect a certain path. Restricting our attention to a path allows us to implicitly compute reachability between the path vertices and the rest of the graph in time that depends logarithmically rather than linearly in the size of the path. -
Matroid Matching: the Power of Local Search
Matroid Matching: the Power of Local Search [Extended Abstract] Jon Lee Maxim Sviridenko Jan Vondrák IBM Research IBM Research IBM Research Yorktown Heights, NY Yorktown Heights, NY San Jose, CA [email protected] [email protected] [email protected] ABSTRACT can be easily transformed into an NP-completeness proof We consider the classical matroid matching problem. Un- for a concrete class of matroids (see [46]). An important weighted matroid matching for linear matroids was solved result of Lov´asz is that (unweighted) matroid matching can by Lov´asz, and the problem is known to be intractable for be solved in polynomial time for linear matroids (see [35]). general matroids. We present a PTAS for unweighted ma- There have been several attempts to generalize Lov´asz' re- troid matching for general matroids. In contrast, we show sult to the weighted case. Polynomial-time algorithms are that natural LP relaxations have an Ω(n) integrality gap known for some special cases (see [49]), but for general linear and moreover, Ω(n) rounds of the Sherali-Adams hierarchy matroids there is only a pseudopolynomial-time randomized are necessary to bring the gap down to a constant. exact algorithm (see [8, 40]). More generally, for any fixed k ≥ 2 and > 0, we obtain a In this paper, we revisit the matroid matching problem (k=2 + )-approximation for matroid matching in k-uniform for general matroids. Our main result is that while LP- hypergraphs, also known as the matroid k-parity problem. based approaches including the Sherali-Adams hierarchy fail As a consequence, we obtain a (k=2 + )-approximation for to provide any meaningful approximation, a simple local- the problem of finding the maximum-cardinality set in the search algorithm gives a PTAS (in the unweighted case). -
Small Space Stream Summary for Matroid Center Sagar Kale EPFL, Lausanne, Switzerland Sagar.Kale@Epfl.Ch
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Dagstuhl Research Online Publication Server Small Space Stream Summary for Matroid Center Sagar Kale EPFL, Lausanne, Switzerland sagar.kale@epfl.ch Abstract In the matroid center problem, which generalizes the k-center problem, we need to pick a set of centers that is an independent set of a matroid with rank r. We study this problem in streaming, where elements of the ground set arrive in the stream. We first show that any randomized one-pass streaming algorithm that computes a better than ∆-approximation for partition-matroid center must use Ω(r2) bits of space, where ∆ is the aspect ratio of the metric and can be arbitrarily large. This shows a quadratic separation between matroid center and k-center, for which the Doubling algorithm [7] gives an 8-approximation using O(k)-space and one pass. To complement this, we give a one-pass algorithm for matroid center that stores at most O(r2 log(1/ε)/ε) points (viz., stream summary) among which a (7 + ε)-approximate solution exists, which can be found by brute force, or a (17 + ε)-approximation can be found with an efficient algorithm. If we are allowed a second pass, we can compute a (3 + ε)-approximation efficiently. We also consider the problem of matroid center with z outliers and give a one-pass algorithm that outputs a set of O((r2 + rz) log(1/ε)/ε) points that contains a (15 + ε)-approximate solution. Our techniques extend to knapsack center and knapsack center with z outliers in a straightforward way, and we get algorithms that use space linear in the size of a largest feasible set (as opposed to quadratic space for matroid center).