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Selective Maximum Coverage and Set Packing
Selective Maximum Coverage and Set Packing Felix J. L. Willamowski1∗ and Bj¨ornF. Tauer2y 1 Lehrstuhl f¨urOperations Research RWTH Aachen University [email protected] 2 Lehrstuhl f¨urManagement Science RWTH Aachen University [email protected] Abstract. In this paper we introduce the selective maximum coverage and the selective maximum set packing problem and variants of them. Both problems are strongly related to well studied problems such as maximum coverage, set packing, and (bipartite) hypergraph matching. The two problems are given by a collection of subsets of a ground set and index subsets of the indices of these subsets. Additionally, there are weights either for each element of the ground set or each subset for each index subset. The goal is to find at most one index per index subset such that the total weight of covered elements or of disjoint subsets is maximum. Applications arise in transportation, e.g., dispatching for ride- sharing services. We prove strong intractability results for the problems and provide almost best possible approximation guarantees. 1 Introduction and Preliminaries We introduce the selective maximum coverage problem (smc) generalizing the weighted maximum k coverage problem (wmkc) which is given by a finite ground set X with weights w : X ! Q≥0, a finite collection S of subsets of X, and an 0 integer k 2 Z≥0. The goal is to find a subcollection S ⊆ S containing at most k P subsets and maximizing the total weight of covered elements x2X0 w(x) with 0 X = [S2S0 S. For unit weights, w ≡ 1, the problem is called maximum k coverage problem (mkc). -
Exact Algorithms and APX-Hardness Results for Geometric Packing and Covering Problems∗
Exact Algorithms and APX-Hardness Results for Geometric Packing and Covering Problems∗ Timothy M. Chan† Elyot Grant‡ March 29, 2012 Abstract We study several geometric set cover and set packing problems involv- ing configurations of points and geometric objects in Euclidean space. We show that it is APX-hard to compute a minimum cover of a set of points in the plane by a family of axis-aligned fat rectangles, even when each rectangle is an ǫ-perturbed copy of a single unit square. We extend this result to several other classes of objects including almost-circular ellipses, axis-aligned slabs, downward shadows of line segments, downward shad- ows of graphs of cubic functions, fat semi-infinite wedges, 3-dimensional unit balls, and axis-aligned cubes, as well as some related hitting set prob- lems. We also prove the APX-hardness of a related family of discrete set packing problems. Our hardness results are all proven by encoding a highly structured minimum vertex cover problem which we believe may be of independent interest. In contrast, we give a polynomial-time dynamic programming algo- rithm for geometric set cover where the objects are pseudodisks containing the origin or are downward shadows of pairwise 2-intersecting x-monotone curves. Our algorithm extends to the weighted case where a minimum-cost cover is required. We give similar algorithms for several related hitting set and discrete packing problems. 1 Introduction In a geometric set cover problem, we are given a range space (X, )—a universe X of points in Euclidean space and a pre-specified configurationS of regions or geometric objects such as rectangles or half-planes. -
Reductions and Satisfiability
Reductions and Satisfiability 1 Polynomial-Time Reductions reformulating problems reformulating a problem in polynomial time independent set and vertex cover reducing vertex cover to set cover 2 The Satisfiability Problem satisfying truth assignments SAT and 3-SAT reducing 3-SAT to independent set transitivity of reductions CS 401/MCS 401 Lecture 18 Computer Algorithms I Jan Verschelde, 30 July 2018 Computer Algorithms I (CS 401/MCS 401) Reductions and Satifiability L-18 30 July 2018 1 / 45 Reductions and Satifiability 1 Polynomial-Time Reductions reformulating problems reformulating a problem in polynomial time independent set and vertex cover reducing vertex cover to set cover 2 The Satisfiability Problem satisfying truth assignments SAT and 3-SAT reducing 3-SAT to independent set transitivity of reductions Computer Algorithms I (CS 401/MCS 401) Reductions and Satifiability L-18 30 July 2018 2 / 45 reformulating problems The Ford-Fulkerson algorithm computes maximum flow. By reduction to a flow problem, we could solve the following problems: bipartite matching, circulation with demands, survey design, and airline scheduling. Because the Ford-Fulkerson is an efficient algorithm, all those problems can be solved efficiently as well. Our plan for the remainder of the course is to explore computationally hard problems. Computer Algorithms I (CS 401/MCS 401) Reductions and Satifiability L-18 30 July 2018 3 / 45 imagine a meeting with your boss ... From Computers and intractability. A Guide to the Theory of NP-Completeness by Michael R. Garey and David S. Johnson, Bell Laboratories, 1979. Computer Algorithms I (CS 401/MCS 401) Reductions and Satifiability L-18 30 July 2018 4 / 45 what you want to say is From Computers and intractability. -
8. NP and Computational Intractability
Chapter 8 NP and Computational Intractability CS 350 Winter 2018 1 Algorithm Design Patterns and Anti-Patterns Algorithm design patterns. Ex. Greedy. O(n log n) interval scheduling. Divide-and-conquer. O(n log n) FFT. Dynamic programming. O(n2) edit distance. Duality. O(n3) bipartite matching. Reductions. Local search. Randomization. Algorithm design anti-patterns. NP-completeness. O(nk) algorithm unlikely. PSPACE-completeness. O(nk) certification algorithm unlikely. Undecidability. No algorithm possible. 2 8.1 Polynomial-Time Reductions Classify Problems According to Computational Requirements Q. Which problems will we be able to solve in practice? A working definition. [von Neumann 1953, Godel 1956, Cobham 1964, Edmonds 1965, Rabin 1966] Those with polynomial-time algorithms. Yes Probably no Shortest path Longest path Matching 3D-matching Min cut Max cut 2-SAT 3-SAT Planar 4-color Planar 3-color Bipartite vertex cover Vertex cover Primality testing Factoring 4 Classify Problems Desiderata. Classify problems according to those that can be solved in polynomial-time and those that cannot. Provably requires exponential-time. Given a Turing machine, does it halt in at most k steps? (the Halting Problem) Given a board position in an n-by-n generalization of chess, can black guarantee a win? Frustrating news. Huge number of fundamental problems have defied classification for decades. This chapter. Show that these fundamental problems are "computationally equivalent" and appear to be different manifestations of one really hard problem. 5 Polynomial-Time Reduction Desiderata'. Suppose we could solve X in polynomial-time. What else could we solve in polynomial time? don't confuse with reduces from Reduction. -
Generalized Sphere-Packing and Sphere-Covering Bounds on the Size
1 Generalized sphere-packing and sphere-covering bounds on the size of codes for combinatorial channels Daniel Cullina, Student Member, IEEE and Negar Kiyavash, Senior Member, IEEE Abstract—Many of the classic problems of coding theory described. Erasure and deletion errors differ from substitution are highly symmetric, which makes it easy to derive sphere- errors in a more fundamental way: the error operation takes packing upper bounds and sphere-covering lower bounds on the an input from one set and produces an output from another. size of codes. We discuss the generalizations of sphere-packing and sphere-covering bounds to arbitrary error models. These In this paper, we will discuss the generalizations of sphere- generalizations become especially important when the sizes of the packing and sphere-covering bounds to arbitrary error models. error spheres are nonuniform. The best possible sphere-packing These generalizations become especially important when the and sphere-covering bounds are solutions to linear programs. sizes of the error spheres are nonuniform. Sphere-packing and We derive a series of bounds from approximations to packing sphere-covering bounds are fundamentally related to linear and covering problems and study the relationships and trade-offs between them. We compare sphere-covering lower bounds with programming and the best possible versions of the bounds other graph theoretic lower bounds such as Turan’s´ theorem. We are solutions to linear programs. In highly symmetric cases, show how to obtain upper bounds by optimizing across a family of including many classical error models, it is often possible to channels that admit the same codes. -
Packing Triangles in Low Degree Graphs and Indifference Graphsଁ Gordana Mani´C∗, Yoshiko Wakabayashi
Discrete Mathematics 308 (2008) 1455–1471 www.elsevier.com/locate/disc Packing triangles in low degree graphs and indifference graphsଁ Gordana Mani´c∗, Yoshiko Wakabayashi Departamento de Ciência da Computação, Universidade de São Paulo, Rua do Matão, 1010—CEP 05508-090–São Paulo, Brazil Received 5 November 2005; received in revised form 28 August 2006; accepted 11 July 2007 Available online 6 September 2007 Abstract We consider the problems of finding the maximum number of vertex-disjoint triangles (VTP) and edge-disjoint triangles (ETP) in a simple graph. Both problems are NP-hard. The algorithm with the best approximation ratio known so far for these problems has ratio 3/2 + ε, a result that follows from a more general algorithm for set packing obtained by Hurkens and Schrijver [On the size of systems of sets every t of which have an SDR, with an application to the worst-case ratio of heuristics for packing problems, SIAM J. Discrete Math. 2(1) (1989) 68–72]. We present improvements on the approximation ratio for restricted cases of VTP and ETP that are known to be APX-hard: we give an approximation algorithm for VTP on graphs with maximum degree 4 with ratio slightly less than 1.2, and for ETP on graphs with maximum degree 5 with ratio 4/3. We also present an exact linear-time algorithm for VTP on the class of indifference graphs. © 2007 Elsevier B.V. All rights reserved. Keywords: Packing triangles; Approximation algorithm; Polynomial algorithm; Low degree graph; Indifference graph 1. Introduction For a given family F of sets, a collection of pairwise disjoint sets of F is called a packing of F.