Parameterized Complexity and Approximation Algorithms
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1 Introduction and Terminology
A Survey of Solved Problems and Applications on Bandwidth, Edgesum and Pro le of Graphs Yung-Ling Lai National Chiayi Teacher College Chiayi, Taiwan, R.O.C. Kenneth Williams Western Michigan University Abstract This pap er provides a survey of results on the exact bandwidth, edge- sum, and pro le of graphs. A bibliographyof work in these areas is pro- vided. The emphasis is on comp osite graphs. This may be regarded as an up date of the original survey of solved bandwidth problems by Chinn, Chvatalova, Dewdney, and Gibbs[10] in 1982. Also several of the applica- tion areas involving these graph parameters are describ ed. 1 Intro duction and terminology For a graph G, V (G) denotes the set of vertices of G and E (G) denotes the set of edges of G. Let G = (V; E ) be a graph on n vertices. A 1-1 mapping f : V ! f1; 2;:::;ng is called a proper numbering of G. The bandwidth B (G) of aproper f numbering f of G is the number B (G)= maxfjf (u) f (v )j : uv 2 E g; f and the bandwidth B(G) of G is the number B (G)= minfB (G): f is a prop er numb ering of Gg: f A prop er numb ering f is called a bandwidth numbering of G if B (G)=B (G). f For example, Figure 1 shows bandwidth numb erings for the graphs P ;C ;K 4 5 1;4 and K . In general, B (P ) = 1, B (C ) = 2, B (K ) = dn=2e and B (K ) = 2;3 n n 1;n m;n m + dn=2e1form n. -
Fixed-Parameter Tractability Marko Samer and Stefan Szeider
Handbook of Satisfiability 363 Armin Biere, Marijn Heule, Hans van Maaren and Toby Walsch IOS Press, 2008 c 2008 Marko Samer and Stefan Szeider. All rights reserved. Chapter 13 Fixed-Parameter Tractability Marko Samer and Stefan Szeider 13.1. Introduction The propositional satisfiability problem (SAT) is famous for being the first prob- lem shown to be NP-complete—we cannot expect to find a polynomial-time al- gorithm for SAT. However, over the last decade, SAT-solvers have become amaz- ingly successful in solving formulas with thousands of variables that encode prob- lems arising from various application areas. Theoretical performance guarantees, however, are far from explaining this empirically observed efficiency. Actually, theorists believe that the trivial 2n time bound for solving SAT instances with n variables cannot be significantly improved, say to 2o(n) (see the end of Sec- tion 13.2). This enormous discrepancy between theoretical performance guaran- tees and the empirically observed performance of SAT solvers can be explained by the presence of a certain “hidden structure” in instances that come from applica- tions. This hidden structure greatly facilitates the propagation and simplification mechanisms of SAT solvers. Thus, for deriving theoretical performance guaran- tees that are closer to the actual performance of solvers one needs to take this hidden structure of instances into account. The literature contains several sug- gestions for making the vague term of a hidden structure explicit. For example, the hidden structure can be considered as the “tree-likeness” or “Horn-likeness” of the instance (below we will discuss how these notions can be made precise). -
Introduction to Parameterized Complexity
Introduction to Parameterized Complexity Ignasi Sau CNRS, LIRMM, Montpellier, France UFMG Belo Horizonte, February 2018 1/37 Outline of the talk 1 Why parameterized complexity? 2 Basic definitions 3 Kernelization 4 Some techniques 2/37 Next section is... 1 Why parameterized complexity? 2 Basic definitions 3 Kernelization 4 Some techniques 3/37 Karp (1972): list of 21 important NP-complete problems. Nowadays, literally thousands of problems are known to be NP-hard: unlessP = NP, they cannot be solved in polynomial time. But what does it mean for a problem to be NP-hard? No algorithm solves all instances optimally in polynomial time. Some history of complexity: NP-completeness Cook-Levin Theorem (1971): the SAT problem is NP-complete. 4/37 Nowadays, literally thousands of problems are known to be NP-hard: unlessP = NP, they cannot be solved in polynomial time. But what does it mean for a problem to be NP-hard? No algorithm solves all instances optimally in polynomial time. Some history of complexity: NP-completeness Cook-Levin Theorem (1971): the SAT problem is NP-complete. Karp (1972): list of 21 important NP-complete problems. 4/37 But what does it mean for a problem to be NP-hard? No algorithm solves all instances optimally in polynomial time. Some history of complexity: NP-completeness Cook-Levin Theorem (1971): the SAT problem is NP-complete. Karp (1972): list of 21 important NP-complete problems. Nowadays, literally thousands of problems are known to be NP-hard: unlessP = NP, they cannot be solved in polynomial time. 4/37 Some history of complexity: NP-completeness Cook-Levin Theorem (1971): the SAT problem is NP-complete. -
Exploiting C-Closure in Kernelization Algorithms for Graph Problems
Exploiting c-Closure in Kernelization Algorithms for Graph Problems Tomohiro Koana Technische Universität Berlin, Algorithmics and Computational Complexity, Germany [email protected] Christian Komusiewicz Philipps-Universität Marburg, Fachbereich Mathematik und Informatik, Marburg, Germany [email protected] Frank Sommer Philipps-Universität Marburg, Fachbereich Mathematik und Informatik, Marburg, Germany [email protected] Abstract A graph is c-closed if every pair of vertices with at least c common neighbors is adjacent. The c-closure of a graph G is the smallest number such that G is c-closed. Fox et al. [ICALP ’18] defined c-closure and investigated it in the context of clique enumeration. We show that c-closure can be applied in kernelization algorithms for several classic graph problems. We show that Dominating Set admits a kernel of size kO(c), that Induced Matching admits a kernel with O(c7k8) vertices, and that Irredundant Set admits a kernel with O(c5/2k3) vertices. Our kernelization exploits the fact that c-closed graphs have polynomially-bounded Ramsey numbers, as we show. 2012 ACM Subject Classification Theory of computation → Parameterized complexity and exact algorithms; Theory of computation → Graph algorithms analysis Keywords and phrases Fixed-parameter tractability, kernelization, c-closure, Dominating Set, In- duced Matching, Irredundant Set, Ramsey numbers Funding Frank Sommer: Supported by the Deutsche Forschungsgemeinschaft (DFG), project MAGZ, KO 3669/4-1. 1 Introduction Parameterized complexity [9, 14] aims at understanding which properties of input data can be used in the design of efficient algorithms for problems that are hard in general. The properties of input data are encapsulated in the notion of a parameter, a numerical value that can be attributed to each input instance I. -
Lower Bounds Based on the Exponential Time Hypothesis
Lower bounds based on the Exponential Time Hypothesis Daniel Lokshtanov∗ Dániel Marxy Saket Saurabhz Abstract In this article we survey algorithmic lower bound results that have been obtained in the field of exact exponential time algorithms and pa- rameterized complexity under certain assumptions on the running time of algorithms solving CNF-Sat, namely Exponential time hypothesis (ETH) and Strong Exponential time hypothesis (SETH). 1 Introduction The theory of NP-hardness gives us strong evidence that certain fundamen- tal combinatorial problems, such as 3SAT or 3-Coloring, are unlikely to be polynomial-time solvable. However, NP-hardness does not give us any information on what kind of super-polynomial running time is possible for NP-hard problems. For example, according to our current knowledge, the complexity assumption P 6= NP does not rule out the possibility of having an nO(log n) time algorithm for 3SAT or an nO(log log k) time algorithm for k- Clique, but such incredibly efficient super-polynomial algorithms would be still highly surprising. Therefore, in order to obtain qualitative lower bounds that rule out such algorithms, we need a complexity assumption stronger than P 6= NP. Impagliazzo, Paturi, and Zane [38, 37] introduced the Exponential Time Hypothesis (ETH) and the stronger variant, the Strong Exponential Time Hypothesis (SETH). These complexity assumptions state lower bounds on how fast satisfiability problems can be solved. These assumptions can be used as a basis for qualitative lower bounds for other concrete computational problems. ∗Dept. of Comp. Sc. and Engineering, University of California, USA. [email protected] yInstitut für Informatik, Humboldt-Universiät , Berlin, Germany. -
12. Exponential Time Hypothesis COMP6741: Parameterized and Exact Computation
12. Exponential Time Hypothesis COMP6741: Parameterized and Exact Computation Serge Gaspers Semester 2, 2017 Contents 1 SAT and k-SAT 1 2 Subexponential time algorithms 2 3 ETH and SETH 2 4 Algorithmic lower bounds based on ETH 2 5 Algorithmic lower bounds based on SETH 3 6 Further Reading 4 1 SAT and k-SAT SAT SAT Input: A propositional formula F in conjunctive normal form (CNF) Parameter: n = jvar(F )j, the number of variables in F Question: Is there an assignment to var(F ) satisfying all clauses of F ? k-SAT Input: A CNF formula F where each clause has length at most k Parameter: n = jvar(F )j, the number of variables in F Question: Is there an assignment to var(F ) satisfying all clauses of F ? Example: (x1 _ x2) ^ (:x2 _ x3 _:x4) ^ (x1 _ x4) ^ (:x1 _:x3 _:x4) Algorithms for SAT • Brute-force: O∗(2n) • ... after > 50 years of SAT solving (SAT association, SAT conference, JSAT journal, annual SAT competitions, ...) • fastest known algorithm for SAT: O∗(2n·(1−1=O(log m=n))), where m is the number of clauses [Calabro, Impagli- azzo, Paturi, 2006] [Dantsin, Hirsch, 2009] • However: no O∗(1:9999n) time algorithm is known 1 • fastest known algorithms for 3-SAT: O∗(1:3303n) deterministic [Makino, Tamaki, Yamamoto, 2013] and O∗(1:3071n) randomized [Hertli, 2014] • Could it be that 3-SAT cannot be solved in 2o(n) time? • Could it be that SAT cannot be solved in O∗((2 − )n) time for any > 0? 2 Subexponential time algorithms NP-hard problems in subexponential time? • Are there any NP-hard problems that can be solved in 2o(n) time? • Yes. -
On Miniaturized Problems in Parameterized Complexity Theory
On miniaturized problems in parameterized complexity theory Yijia Chen Institut fur¨ Informatik, Humboldt-Universitat¨ zu Berlin, Unter den Linden 6, 10099 Berlin, Germany. [email protected] Jorg¨ Flum Abteilung fur¨ Mathematische Logik, Universitat¨ Freiburg, Eckerstr. 1, 79104 Freiburg, Germany. [email protected] Abstract We introduce a general notion of miniaturization of a problem that comprises the different miniaturizations of concrete problems considered so far. We develop parts of the basic theory of miniaturizations. Using the appropriate logical formalism, we show that the miniaturization of a definable problem in W[t] lies in W[t], too. In particular, the miniaturization of the dominating set problem is in W[2]. Furthermore we investigate the relation between f(k) · no(k) time and subexponential time algorithms for the dominating set problem and for the clique problem. 1. Introduction Parameterized complexity theory provides a framework for a refined complexity analysis of algorithmic problems that are intractable in general. Central to the theory is the notion of fixed-parameter tractability, which relaxes the classical notion of tractability, polynomial time computability, by admitting algorithms whose runtime is exponential, but only in terms of some parameter that is usually expected to be small. Let FPT denote the class of all fixed-parameter tractable problems. A well-known example of a problem in FPT is the vertex cover problem, the parameter being the size of the vertex cover we ask for. As a complexity theoretic counterpart, a theory of parameterized intractability has been developed. In classical complex- ity, the notion of NP-completeness is central to a nice and simple theory for intractable problems. -
Exact Complexity and Satisfiability
Exact Complexity and Satisfiability (Invited Talk) Russell Impagliazzo and Ramamohan Paturi Department of Computer Science and Engineering University of California, San Diego La Jolla, CA 92093-0404, USA {russell,paturi}@cs.ucsd.edu All NP-complete problems are equivalent as far as polynomial time solv- ability is concerned. However, their exact complexities (worst-case complex- ity of algorithms that solve every instance correctly and exactly) differ widely. Starting with Bellman [1], Tarjan and Trojanowski [9], Karp [5], and Monien and Speckenmeyer [7], the design of improved exponential time al- gorithms for NP-complete problems has been a tremendously fruitful en- deavor, and one that has recently been accelerating both in terms of the number of results and the increasing sophistication of algorithmic techniques. There are a vast variety of problems where progress has been made, e.g., k-sat, k-Colorability, Maximum Independent Set, Hamiltonian Path, Chromatic Number,andCircuit Sat for limited classes of circuits. The “current champion” algorithms for these problems deploy a vast variety of al- gorithmic techniques, e.g., back-tracking search (with its many refinements and novel analysis techniques), divide-and-conquer, dynamic programming, random- ized search, algebraic transforms, inclusion-exclusion, color coding, split and list, and algebraic sieving. In many ways, this is analogous to the diversity of approx- imation ratios and approximation algorithms known for different NP-complete problems. In view of this diversity, it is tempting to focus on the distinctions between problems rather than the interconnections between them. However, over the past two decades, there has been a wave of research showing that such con- nections do exist. -
Parameterized Approximation of Dominating Set Problems
Parameterized Approximation of Dominating Set Problems Rodney G. Downey1, Michael R. Fellows2, Catherine McCartin3, and Frances Rosamond2 1 Victoria University, Wellington, New Zealand, [email protected] 2 The University of Newcastle, Callaghan, Australia, {michael.fellows,frances.rosamond}@newcastle.edu.au 3 Massey University, Palmerston North, New Zealand, [email protected] Abstract. A problem open for many years is whether there is an FPT algorithm that given a graph G and parameter k, either: (1) determines that G has no k-Dominating Set, or (2) produces a dominating set of size at most g(k), where g(k) is some fixed function of k. Such an outcome is termed an FPT approximation algorithm. We describe some results that begin to provide some answers. We show that there is no such FPT algorithm for g(k) of the form k + c (where c is a fixed con- stant, termed an additive FPT approximation), unless FPT = W [2]. We answer the analogous problem completely for the related Independent Dominating Set (IDS) problem, showing that IDS does not admit an FPT approximation algorithm, for any g(k), unless FPT = W [2]. 1 Introduction Most of the work in the area of parameterized complexity and algorithmics to date has focused on exact algorithms for decision problems. However, the theory of parameterized complexity, which derives from the contrast between timecost functions of the form f(k)nc (fixed-parameter tractability, FPT,) and those of the form ng(k), where n is the overall input size, and k is a relevant secondary parameter of the situation, can clearly be deployed in a variety of ways in analyzing computational complexity. -
Fixed-Parameter Tractability and Parameterized Complexity, Applied to Problems from Computational Social Choice — Mathematical Programming Glossary Supplement —
Fixed-Parameter Tractability and Parameterized Complexity, Applied to Problems From Computational Social Choice — Mathematical Programming Glossary Supplement — Claudia Lindner and J¨org Rothe Institut f¨ur Informatik Heinrich-Heine-Universit¨at D¨usseldorf 40225 D¨usseldorf, Germany October 17, 2008 Abstract This supplement provides a brief introduction to the field of fixed-parameter tractability and parameterized complexity. Some basic notions are explained and some related results are presented, with a focus on problems arising in the field of computational social choice. 1 Fixed-Parameter Tractability and Parameterized Complexity The study of fixed-parameter tractability and parameterized complexity has emerged as a new field within computational complexity theory in the late 1980s and the early 1990s. Since the early pioneering work of Downey, Fellows, and other researchers this area has established plenty of results, notions, and methods, and it provides a useful framework for dealing in practice with problems considered intractable by classical complexity theory. This supplement gives only a rough sketch of some basic notions and techniques within parameterized complexity theory; for a deeper and more comprehensive treatise, the textbooks by Downey and Fellows [DF99], Flum and Grohe [FG06], and Niedermeier [Nie06] and the survey by Buss and Islam [BI08] are highly recommendable. Let P and NP, respectively, denote the classical (worst-case) complexity classes deterministic polynomial time and nondeterministic polynomial time. Given any two decision problems, A and B, ¡ p we say A polynomial-time many-one reduces to B (denoted A m B) if there is a polynomial-time £ ¤ ¢ computable function f such that, for each input x, x ¢ A if and only if f x B. -
On Bounding the Bandwidth of Graphs with Symmetry
On bounding the bandwidth of graphs with symmetry E.R. van Dam∗ R. Sotirovy Abstract We derive a new lower bound for the bandwidth of a graph that is based on a new lower bound for the minimum cut problem. Our new semidefinite program- ming relaxation of the minimum cut problem is obtained by strengthening the known semidefinite programming relaxation for the quadratic assignment problem (or for the graph partition problem) by fixing two vertices in the graph; one on each side of the cut. This fixing results in several smaller subproblems that need to be solved to obtain the new bound. In order to efficiently solve these subproblems we exploit symmetry in the data; that is, both symmetry in the min-cut problem and symmetry in the graphs. To obtain upper bounds for the bandwidth of graphs with symmetry, we de- velop a heuristic approach based on the well-known reverse Cuthill-McKee algorithm, and that improves significantly its performance on the tested graphs. Our approaches result in the best known lower and upper bounds for the bandwidth of all graphs un- der consideration, i.e., Hamming graphs, 3-dimensional generalized Hamming graphs, Johnson graphs, and Kneser graphs, with up to 216 vertices. Keywords: bandwidth, minimum cut, semidefinite programming, Hamming graphs, John- son graphs, Kneser graphs 1 Introduction For (undirected) graphs, the bandwidth problem (BP) is the problem of labeling the vertices of a given graph with distinct integers such that the maximum difference between the labels of adjacent vertices is minimal. Determining the bandwidth is NP-hard (see [35]) and it remains NP-hard even if it is restricted to trees with maximum degree three (see [17]) or to caterpillars with hair length three (see [34]). -
Exact and Approximate Bandwidth
Exact and Approximate Bandwidth Marek Cygan and Marcin Pilipczuk Dept. of Mathematics, Computer Science and Mechanics, University of Warsaw, Poland {cygan,malcin}@mimuw.edu.pl Abstract. In this paper we gather several improvements in the field of exact and approximate exponential-time algorithms for the BANDWIDTH problem. For graphs with treewidth t we present a O(nO(t)2n) exact algorithm. Moreover for the same class of graphs we introduce a subexponential constant-approximation scheme – for anyα>0 there exists a (1 + α)-approximation algorithm running in O(exp(c(t + n/α)logn)) time where c is a universal constant. These re- sults seem interesting since Unger has proved that BANDWIDTH does not belong to AP X even when the input graph is a tree (assuming P = NP). So some- what surprisingly, despite Unger’s result it turns out that not only a subexponen- tial constant approximation is possible but also a subexponential approximation scheme exists. Furthermore, for any positive integer r,wepresenta(4r − 1)- approximation algorithm that solves BANDWIDTH for an arbitrary input graph ∗ n in O (2 r ) time and polynomial space.1 Finally we improve the currently best known exact algorithm for arbitrary graphs with a O(4.473n ) time and space algorithm. In the algorithms for the small treewidth we develop a technique based on the Fast Fourier Transform, parallel to the Fast Subset Convolution techniques introduced by Bj¨orklund et al. This technique can be also used as a simple method of finding a chromatic number of all subgraphs of a given graph in O∗(2n) time and space, what matches the best known results.