On the Approximability of a Geometric Set Cover Problem
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Independence Number, Vertex (Edge) ୋ Cover Number and the Least Eigenvalue of a Graph
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Linear Algebra and its Applications 433 (2010) 790–795 Contents lists available at ScienceDirect Linear Algebra and its Applications journal homepage: www.elsevier.com/locate/laa The vertex (edge) independence number, vertex (edge) ୋ cover number and the least eigenvalue of a graph ∗ Ying-Ying Tan a, Yi-Zheng Fan b, a Department of Mathematics and Physics, Anhui University of Architecture, Hefei 230601, PR China b School of Mathematical Sciences, Anhui University, Hefei 230039, PR China ARTICLE INFO ABSTRACT Article history: In this paper we characterize the unique graph whose least eigen- Received 4 October 2009 value attains the minimum among all graphs of a fixed order and Accepted 4 April 2010 a given vertex (edge) independence number or vertex (edge) cover Available online 8 May 2010 number, and get some bounds for the vertex (edge) independence Submitted by X. Zhan number, vertex (edge) cover number of a graph in terms of the least eigenvalue of the graph. © 2010 Elsevier Inc. All rights reserved. AMS classification: 05C50 15A18 Keywords: Graph Adjacency matrix Vertex (edge) independence number Vertex (edge) cover number Least eigenvalue 1. Introduction Let G = (V,E) be a simple graph of order n with vertex set V = V(G) ={v1,v2, ...,vn} and edge set E = E(G). The adjacency matrix of G is defined to be a (0, 1)-matrix A(G) =[aij], where aij = 1ifvi is adjacent to vj and aij = 0 otherwise. The eigenvalues of the graph G are referred to the eigenvalues of A(G), which are arranged as λ1(G) λ2(G) ··· λn(G). -
3.1 Matchings and Factors: Matchings and Covers
1 3.1 Matchings and Factors: Matchings and Covers This copyrighted material is taken from Introduction to Graph Theory, 2nd Ed., by Doug West; and is not for further distribution beyond this course. These slides will be stored in a limited-access location on an IIT server and are not for distribution or use beyond Math 454/553. 2 Matchings 3.1.1 Definition A matching in a graph G is a set of non-loop edges with no shared endpoints. The vertices incident to the edges of a matching M are saturated by M (M-saturated); the others are unsaturated (M-unsaturated). A perfect matching in a graph is a matching that saturates every vertex. perfect matching M-unsaturated M-saturated M Contains copyrighted material from Introduction to Graph Theory by Doug West, 2nd Ed. Not for distribution beyond IIT’s Math 454/553. 3 Perfect Matchings in Complete Bipartite Graphs a 1 The perfect matchings in a complete b 2 X,Y-bigraph with |X|=|Y| exactly c 3 correspond to the bijections d 4 f: X -> Y e 5 Therefore Kn,n has n! perfect f 6 matchings. g 7 Kn,n The complete graph Kn has a perfect matching iff… Contains copyrighted material from Introduction to Graph Theory by Doug West, 2nd Ed. Not for distribution beyond IIT’s Math 454/553. 4 Perfect Matchings in Complete Graphs The complete graph Kn has a perfect matching iff n is even. So instead of Kn consider K2n. We count the perfect matchings in K2n by: (1) Selecting a vertex v (e.g., with the highest label) one choice u v (2) Selecting a vertex u to match to v K2n-2 2n-1 choices (3) Selecting a perfect matching on the rest of the vertices. -
Theoretical Computer Science Efficient Approximation of Min Set
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Theoretical Computer Science 410 (2009) 2184–2195 Contents lists available at ScienceDirect Theoretical Computer Science journal homepage: www.elsevier.com/locate/tcs Efficient approximation of min set cover by moderately exponential algorithms N. Bourgeois, B. Escoffier ∗, V.Th. Paschos LAMSADE, CNRS FRE 3234 and Université Paris-Dauphine, France article info a b s t r a c t Article history: We study the approximation of min set cover combining ideas and results from polynomial Received 13 October 2008 approximation and from exact computation (with non-trivial worst case complexity upper Received in revised form 30 January 2009 bounds) for NP-hard problems. We design approximation algorithms for min set cover Accepted 1 February 2009 achieving ratios that cannot be achieved in polynomial time (unless problems in NP could Communicated by G. Ausiello be solved by slightly super-polynomial algorithms) with worst-case complexity much lower (though super-polynomial) than those of an exact computation. Keywords: 2009 Elsevier B.V. All rights reserved. Exponential algorithms ' Approximation algorithms min set cover 1. Introduction C Given a ground set C of cardinality n and a system S D fS1;:::; Smg ⊂ 2 , min set cover consists of determining a 0 minimum size subsystem S such that [S2S0 S D C. min set cover is a famous NP-hard problem dealt with in the seminal paper [22]. For the last ten years, the issue of exact resolution of NP-hard problems by algorithms having provably non-trivial upper time-complexity bounds has been very actively studied (see, for instance, the surveys [16,26,30]). -
Set Cover Algorithms for Very Large Datasets
Set Cover Algorithms For Very Large Datasets ∗ Graham Cormode Howard Karloff Anthony Wirth AT&T Labs–Research AT&T Labs–Research Department of Computer [email protected] [email protected] Science and Software Engineering The University of Melbourne [email protected] ABSTRACT 1. INTRODUCTION The problem of Set Cover—to find the smallest subcol- The problem of Set Cover arises in a surprisingly broad lection of sets that covers some universe—is at the heart number of places. Although presented as a somewhat ab- of many data and analysis tasks. It arises in a wide range stract problem, it captures many different scenarios that of settings, including operations research, machine learning, arise in the context of data management and knowledge planning, data quality and data mining. Although finding an mining. The basic problem is that we are given a collec- optimal solution is NP-hard, the greedy algorithm is widely tion of sets, each of which is drawn from a common universe used, and typically finds solutions that are close to optimal. of possible items. The goal is to find a subcollection of sets However, a direct implementation of the greedy approach, so that their union includes every item from the universe. which picks the set with the largest number of uncovered Places where Set Cover occurs include: items at each step, does not behave well when the in- • In operations research, the problem of choosing where put is very large and disk resident. The greedy algorithm to locate a number of facilities so that all sites are must make many random accesses to disk, which are unpre- within a certain distance of the closest facility can be dictable and costly in comparison to linear scans. -
Approximation Algorithms
Lecture 21 Approximation Algorithms 21.1 Overview Suppose we are given an NP-complete problem to solve. Even though (assuming P = NP) we 6 can’t hope for a polynomial-time algorithm that always gets the best solution, can we develop polynomial-time algorithms that always produce a “pretty good” solution? In this lecture we consider such approximation algorithms, for several important problems. Specific topics in this lecture include: 2-approximation for vertex cover via greedy matchings. • 2-approximation for vertex cover via LP rounding. • Greedy O(log n) approximation for set-cover. • Approximation algorithms for MAX-SAT. • 21.2 Introduction Suppose we are given a problem for which (perhaps because it is NP-complete) we can’t hope for a fast algorithm that always gets the best solution. Can we hope for a fast algorithm that guarantees to get at least a “pretty good” solution? E.g., can we guarantee to find a solution that’s within 10% of optimal? If not that, then how about within a factor of 2 of optimal? Or, anything non-trivial? As seen in the last two lectures, the class of NP-complete problems are all equivalent in the sense that a polynomial-time algorithm to solve any one of them would imply a polynomial-time algorithm to solve all of them (and, moreover, to solve any problem in NP). However, the difficulty of getting a good approximation to these problems varies quite a bit. In this lecture we will examine several important NP-complete problems and look at to what extent we can guarantee to get approximately optimal solutions, and by what algorithms. -
Catalogue of Graph Polynomials
Catalogue of graph polynomials J.A. Makowsky April 6, 2011 Contents 1 Graph polynomials 3 1.1 Comparinggraphpolynomials. ........ 3 1.1.1 Distinctivepower.............................. .... 3 1.1.2 Substitutioninstances . ...... 4 1.1.3 Uniformalgebraicreductions . ....... 4 1.1.4 Substitutioninstances . ...... 4 1.1.5 Substitutioninstances . ...... 4 1.1.6 Substitutioninstances . ...... 4 1.2 Definabilityofgraphpolynomials . ......... 4 1.2.1 Staticdefinitions ............................... ... 4 1.2.2 Dynamicdefinitions .............................. 4 1.2.3 SOL-definablepolynomials ............................ 4 1.2.4 Generalizedchromaticpolynomials . ......... 4 2 A catalogue of graph polynomials 5 2.1 Polynomialsfromthezoo . ...... 5 2.1.1 Chromaticpolynomial . .... 5 2.1.2 Chromatic symmetric function . ...... 6 2.1.3 Adjointpolynomials . .... 6 2.1.4 TheTuttepolynomial . ... 7 2.1.5 Strong Tutte symmetric function . ....... 7 2.1.6 Tutte-Grothendieck invariants . ........ 8 2.1.7 Aweightedgraphpolynomial . ..... 8 2.1.8 Chainpolynomial ............................... 9 2.1.9 Characteristicpolynomial . ....... 10 2.1.10 Matchingpolynomial. ..... 11 2.1.11 Theindependentsetpolynomial . ....... 12 2.1.12 Thecliquepolynomial . ..... 14 2.1.13 Thevertex-coverpolynomial . ...... 14 2.1.14 Theedge-coverpolynomial . ..... 15 2.1.15 TheMartinpolynomial . 16 2.1.16 Interlacepolynomial . ...... 17 2.1.17 Thecoverpolynomial . 19 2.1.18 Gopolynomial ................................. 21 2.1.19 Stabilitypolynomial . ...... 23 1 2.1.20 Strong U-polynomial............................... -
What Is the Set Cover Problem
18.434 Seminar in Theoretical Computer Science 1 of 5 Tamara Stern 2.9.06 Set Cover Problem (Chapter 2.1, 12) What is the set cover problem? Idea: “You must select a minimum number [of any size set] of these sets so that the sets you have picked contain all the elements that are contained in any of the sets in the input (wikipedia).” Additionally, you want to minimize the cost of the sets. Input: Ground elements, or Universe U= { u1, u 2 ,..., un } Subsets SSSU1, 2 ,..., k ⊆ Costs c1, c 2 ,..., ck Goal: Find a set I ⊆ {1,2,...,m} that minimizes ∑ci , such that U SUi = . i∈ I i∈ I (note: in the un-weighted Set Cover Problem, c j = 1 for all j) Why is it useful? It was one of Karp’s NP-complete problems, shown to be so in 1972. Other applications: edge covering, vertex cover Interesting example: IBM finds computer viruses (wikipedia) elements- 5000 known viruses sets- 9000 substrings of 20 or more consecutive bytes from viruses, not found in ‘good’ code A set cover of 180 was found. It suffices to search for these 180 substrings to verify the existence of known computer viruses. Another example: Consider General Motors needs to buy a certain amount of varied supplies and there are suppliers that offer various deals for different combinations of materials (Supplier A: 2 tons of steel + 500 tiles for $x; Supplier B: 1 ton of steel + 2000 tiles for $y; etc.). You could use set covering to find the best way to get all the materials while minimizing cost. -
IEOR 269, Spring 2010 Integer Programming and Combinatorial Optimization
IEOR 269, Spring 2010 Integer Programming and Combinatorial Optimization Professor Dorit S. Hochbaum Contents 1 Introduction 1 2 Formulation of some ILP 2 2.1 0-1 knapsack problem . 2 2.2 Assignment problem . 2 3 Non-linear Objective functions 4 3.1 Production problem with set-up costs . 4 3.2 Piecewise linear cost function . 5 3.3 Piecewise linear convex cost function . 6 3.4 Disjunctive constraints . 7 4 Some famous combinatorial problems 7 4.1 Max clique problem . 7 4.2 SAT (satisfiability) . 7 4.3 Vertex cover problem . 7 5 General optimization 8 6 Neighborhood 8 6.1 Exact neighborhood . 8 7 Complexity of algorithms 9 7.1 Finding the maximum element . 9 7.2 0-1 knapsack . 9 7.3 Linear systems . 10 7.4 Linear Programming . 11 8 Some interesting IP formulations 12 8.1 The fixed cost plant location problem . 12 8.2 Minimum/maximum spanning tree (MST) . 12 9 The Minimum Spanning Tree (MST) Problem 13 i IEOR269 notes, Prof. Hochbaum, 2010 ii 10 General Matching Problem 14 10.1 Maximum Matching Problem in Bipartite Graphs . 14 10.2 Maximum Matching Problem in Non-Bipartite Graphs . 15 10.3 Constraint Matrix Analysis for Matching Problems . 16 11 Traveling Salesperson Problem (TSP) 17 11.1 IP Formulation for TSP . 17 12 Discussion of LP-Formulation for MST 18 13 Branch-and-Bound 20 13.1 The Branch-and-Bound technique . 20 13.2 Other Branch-and-Bound techniques . 22 14 Basic graph definitions 23 15 Complexity analysis 24 15.1 Measuring quality of an algorithm . -
New Approximation Algorithms for Minimum Weighted Edge Cover
New Approximation Algorithms for Minimum Weighted Edge Cover S M Ferdous∗ Alex Potheny Arif Khanz Abstract The K-Nearest Neighbor graph is used to spar- We describe two new 3=2-approximation algorithms and a sify data sets, which is an important step in graph-based new 2-approximation algorithm for the minimum weight semi-supervised machine learning. Here one has a few edge cover problem in graphs. We show that one of the 3=2-approximation algorithms, the Dual Cover algorithm, labeled items, many unlabeled items, and a measure of computes the lowest weight edge cover relative to previously similarity between pairs of items; we are required to known algorithms as well as the new algorithms reported label the remaining items. A popular approach for clas- here. The Dual Cover algorithm can also be implemented to be faster than the other 3=2-approximation algorithms on sification is to generate a similarity graph between the serial computers. Many of these algorithms can be extended items to represent both the labeled and unlabeled data, to solve the b-Edge Cover problem as well. We show the and then to use a label propagation algorithm to classify relation of these algorithms to the K-Nearest Neighbor graph construction in semi-supervised learning and other the unlabeled items [23]. In this approach one builds applications. a complete graph out of the dataset and then sparsi- fies this graph by computing a K-Nearest Neighbor 1 Introduction graph [22]. This sparsification leads to efficient al- An Edge Cover in a graph is a subgraph such that gorithms, but also helps remove noise which can af- every vertex has at least one edge incident on it in fect label propagation [11]. -
Edge Guarding Plane Graphs
Edge Guarding Plane Graphs Master Thesis of Paul Jungeblut At the Department of Informatics Institute of Theoretical Informatics Reviewers: Prof. Dr. Dorothea Wagner Prof. Dr. Peter Sanders Advisor: Dr. Torsten Ueckerdt Time Period: 26.04.2019 – 25.10.2019 KIT – The Research University in the Helmholtz Association www.kit.edu Statement of Authorship I hereby declare that this document has been composed by myself and describes my own work, unless otherwise acknowledged in the text. I also declare that I have read the Satzung zur Sicherung guter wissenschaftlicher Praxis am Karlsruher Institut für Technologie (KIT). Paul Jungeblut, Karlsruhe, 25.10.2019 iii Abstract Let G = (V; E) be a plane graph. We say that a face f of G is guarded by an edge vw 2 E if at least one vertex from fv; wg is on the boundary of f. For a planar graph class G the function ΓG : N ! N maps n to the minimal number of edges needed to guard all faces of any n-vertex graph in G. This thesis contributes new bounds on ΓG for several graph classes, in particular Γ Γ Γ on 4;stacked for stacked triangulations, on for quadrangulations and on sp for series parallel graphs. Specifically we show that • b(2n - 4)=7c 6 Γ4;stacked(n) 6 b2n=7c, b(n - 2)=4c Γ (n) bn=3c • 6 6 and • b(n - 2)=3c 6 Γsp(n) 6 bn=3c. Note that the bounds for stacked triangulations and series parallel graphs are tight (up to a small constant). For quadrangulations we identify the non-trivial subclass 2 Γ (n) bn=4c of -degenerate quadrangulations for which we further prove ;2-deg 6 matching the lower bound. -
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Covering Problems with Hard Capacities Julia Chuzhoy Joseph (Seffi) Naor Computer Science Department Technion, Haifa 32000, Israel E-mail: cjulia,naor ¡ @cs.technion.ac.il. - Abstract elements of . A cover is a multi-set of input sets that * + can cover all the elements, while - contains at most '() We consider the classical vertex cover and set cover copies of each ./ , and each copy covers at most problems with the addition of hard capacity constraints. elements. We assume that the capacity constraints are hard, This means that a set (vertex) can only cover a limited i.e., the number of copies and the capacity of a set cannot number of its elements (adjacent edges) and the number of be exceeded. The capacitated set cover problem is a natu- available copies of each set (vertex) is bounded. This is a ral generalization of a basic and well-studied problem that natural generalization of the classical problems that also captures practical scenarios where resource limitations are captures resource limitations in practical scenarios. present. We obtain the following results. For the unweighted A special case of the capacitated set cover problem that vertex cover problem with hard capacities we give a ¢ - we consider is the capacitated vertex cover problem, de- © £ approximation algorithm which is based on randomized fined as follows. An undirected graph 01¤1+2 is given !36 rounding with alterations. We prove that the weighted ver- and each vertex 3452 is associated with a cost , a * !36 sion is at least as hard as the set cover problem. This is capacity ',!36 , and a multiplicity . -
Fractional Set Cover in the Streaming Model∗
Fractional Set Cover in the Streaming Model∗ Piotr Indyk1, Sepideh Mahabadi2, Ronitt Rubinfeld3, Jonathan Ullman4, Ali Vakilian5, and Anak Yodpinyanee6 1 CSAIL, MIT, Cambridge, MA, USA [email protected] 2 CSAIL, MIT, Cambridge, MA, USA [email protected] 3 CSAIL, MIT and TAU, Cambridge, MA, USA [email protected] 4 CCIS, Northeastern University, Boston, MA, USA [email protected] 5 CSAIL, MIT, Cambridge, MA, USA [email protected] 6 CSAIL, MIT, Cambridge, MA, USA [email protected] Abstract We study the Fractional Set Cover problem in the streaming model. That is, we consider the relaxation of the set cover problem over a universe of n elements and a collection of m sets, where each set can be picked fractionally, with a value in [0, 1]. We present a randomized (1 + ε)- approximation algorithm that makes p passes over the data, and uses Oe(mnO(1/pε) + n) memory space. The algorithm works in both the set arrival and the edge arrival models. To the best of our knowledge, this is the first streaming result for the fractional set cover problem. We obtain our results by employing the multiplicative weights update framework in the streaming settings. 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems Keywords and phrases Streaming Algorithms, Fractional Set Cover, LP relaxation, Multiplica- tive Weight Update Digital Object Identifier 10.4230/LIPIcs.APPROX/RANDOM.2017.12 1 Introduction Set Cover is one of the classical NP-hard problems in combinatorial optimization. In this problem the input consists of a set (universe) of n elements U = {e1, ··· , en} and a collection of m sets F = {S1, ··· ,Sm}.