A Multistage View on 2-Satisfiability
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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. -
(R18A0507) Design and Analysis of Algorithms.Pdf
DESIGN AND ANALYSIS OF ALGORITHMS [R18A0507] LECTURE NOTES B.TECH III YEAR – I SEM (R18) (2020-2021) DEPARTMENT OF INFORMATION TECHNOLOGY MALLA REDDY COLLEGE OF ENGINEERING & TECHNOLOGY (Autonomous Institution – UGC, Govt. of India) Recognized under 2(f) and 12 (B) of UGC ACT 1956 (Affiliated to JNTUH, Hyderabad, Approved by AICTE - Accredited by NBA & NAAC – „A‟ Grade - ISO 9001:2015 Certified) Maisammaguda, Dhulapally (Post Via. Hakimpet), Secunderabad – 500100, Telangana State, India IIIYearB.Tech.IT–ISem L T/P/D C 3 -/-/- 3 (R18A0507) Design And Analysis of Algorithms Objectives: To analyze performance ofalgorithms. To choose the appropriate data structure and algorithm design method for a specifiedapplication. To understand how the choice of data structures and algorithm design methodsimpacts the performance ofprograms. To solve problems using algorithm design methods such as the greedy method,divide and conquer, dynamic programming, backtracking and branch andbound. To understand the differences between tractable and intractableproblems. To introduce P and NPclasses. UNIT I: Introduction-Algorithm definition, Algorithm Specification, Performance Analysis- Space complexity, Time complexity, Randomized Algorithms. Divide and conquer- General method, applications - Binary search, Merge sort, Quick sort, Strassen‟s Matrix Multiplication. UNIT II: Disjoint set operations, union and find algorithms, AND/OR graphs, Connected Components and Spanning trees, Bi-connected components Backtracking-Generalmethod,applications-The8-queenproblem,sumofsubsetsproblem,graphcoloring, -
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. -
Implications of the Exponential Time Hypothesis
Implications of the Exponential Time Hypothesis Anant Dhayal May 14, 2018 Abstract The Exponential Time Hypothesis (ETH) [14] states that for k ≥ 3, k-SAT doesn't δn have a sub-exponential time algorithm. Let sk = inffδ j 9 2 time algorithm for solving k-SATg, then ETH is equivalent to s3 > 0. Strong ETH (SETH) [13] is a stronger conjecture which states that lim sk = 1. k!1 In this report we survey lower bounds based on ETH. We describe a reduction from k-SAT to NP-complete CSP's [5]. In particular this implies that no NP-complete CSP has a sub-exponential time algorithm. We also survey lower bounds implied by SETH. 1 Introduction CNF-SAT is the canonical NP-complete problem [9, 17]. A literal is a boolean variable or its negation, and a clause is a disjunction of literals. The input of CNF-SAT is a boolean formula in conjunctive normal form (CNF), i.e., conjunction of clauses. The goal is to check if there is an assignment to the variables which satisfies all the clauses. k-SAT is a special case of CNF-SAT where the clause width (the number of literals in a clause) is bounded by the constant k. When k = 2 the problem has a polynomial time algorithm. However for k ≥ 3 it is proved to be NP-complete [9, 17], hence there is no polynomial time algorithm unless P = NP. When k ≥ 4 there is a simple clause width reduction from k-SAT to (k−1)-SAT wherein we replace a k-clause (x1 _x2 _:::_xk) by two clauses (x1 _x2 _:::_xk−2 _y)^(xk−1 _xk _y) by introducing a new variable y. -
A Fast Algorithm for Performance-Driven Module Implementation Selection*
VLSI DESIGN (C) 1999 OPA (Overseas Publishers Association) N.V. 1999, Vol. 10, No. 2, pp. 237-247 Published by license under Reprints available directly from the publisher the Gordon and Breach Science Photocopying permitted by license only Publishers imprint. Printed in Malaysia. A Fast Algorithm for Performance-Driven Module Implementation Selection* EDWARD Y. C. CHENG and SARTAJ SAHNI Department of Computer and Information Science and Engineering, University of Florida, Gainesville, FL 32611-6120 (Received 24 March 1998," In finalform 8 February 1999) We develop an O( p log n) time algorithm to obtain optimal solutions to the p-pin n-net single channel performance-driven implementation selection problem in which each module has at most two possible implementations (2-PDMIS). Although Her, Wang and Wong [1] have also developed an O(p log n) algorithm for this problem, experi- ments indicate that our algorithm is twice as fast on small circuits and up to eleven times as fast on larger circuits. We also develop an O(pnc-l) time algorithm for the c, c > 1, channel version of the 2-PDMIS problem. Keywords and Phrases." Channel density, net span constraint, performance-driven module imple- mentation selection, complexity 1. INTRODUCTION routing problem. Channel routing with rearrange- able pins was studied by Kobayashi and Drozd [3]. In the channel routing problem, we have a routing They proposed a three step algorithm (1) permute channel with modules on the top and bottom of pins so as to maximize the number of aligned pin the channel, the modules have pins and subsets of pairs (a pair of pins on different sides of the chan- pins define nets.