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106
Exact algorithms for NPhard problems: A survey
 Combinatorial Optimization  Eureka, You Shrink!, LNCS
"... Abstract. We discuss fast exponential time solutions for NPcomplete problems. We survey known results and approaches, we provide pointers to the literature, and we discuss several open problems in this area. The list of discussed NPcomplete problems includes the travelling salesman problem, schedu ..."
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Cited by 118 (3 self)
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Abstract. We discuss fast exponential time solutions for NPcomplete problems. We survey known results and approaches, we provide pointers to the literature, and we discuss several open problems in this area. The list of discussed NPcomplete problems includes the travelling salesman problem, scheduling under precedence constraints, satisfiability, knapsack, graph coloring, independent sets in graphs, bandwidth of a graph, and many more. 1
Subexponential Parameterized Algorithms on Graphs of Bounded Genus and HMinorFree Graphs
, 2003
"... We introduce a new framework for designing fixedparameter algorithms with subexponential running time2 . Our results apply to a broad family of graph problems, called bidimensional problems, which includes many domination and covering problems such as vertex cover, feedback vertex set, minimum m ..."
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Cited by 41 (13 self)
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We introduce a new framework for designing fixedparameter algorithms with subexponential running time2 . Our results apply to a broad family of graph problems, called bidimensional problems, which includes many domination and covering problems such as vertex cover, feedback vertex set, minimum maximal matching, dominating set, edge dominating set, cliquetransversal set, and many others restricted to bounded genus graphs. Furthermore, it is fairly straightforward to prove that a problem is bidimensional. In particular, our framework includes as special cases all previously known problems to have such subexponential algorithms. Previously, these algorithms applied to planar graphs, singlecrossingminorfree graphs, and/or map graphs; we extend these results to apply to boundedgenus graphs as well. In a parallel development of combinatorial results, we establish an upper bound on the treewidth (or branchwidth) of a boundedgenus graph that excludes some planar graph H as a minor. This bound depends linearly on the size (H) of the excluded graph H and the genus g(G) of the graph G, and applies and extends the graphminors work of Robertson and Seymour. Building on these results...
On the Complexity of kSAT
, 2001
"... The kSAT problem is to determine if a given kCNF has a satisfying assignment. It is a celebrated open question as to whether it requires exponential time to solve kSAT for k 3. Here exponential time means 2 $n for some $>0. In this paper, assuming that, for k 3, kSAT requires exponential time co ..."
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Cited by 38 (2 self)
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The kSAT problem is to determine if a given kCNF has a satisfying assignment. It is a celebrated open question as to whether it requires exponential time to solve kSAT for k 3. Here exponential time means 2 $n for some $>0. In this paper, assuming that, for k 3, kSAT requires exponential time complexity, we show that the complexity of kSAT increases as k increases. More precisely, for k 3, define s k=inf[$: there exists 2 $n algorithm for solving kSAT]. Define ETH (ExponentialTime Hypothesis) for kSAT as follows: for k 3, s k>0. In this paper, we show that s k is increasing infinitely often assuming ETH for kSAT. Let s be the limit of s k. We will in fact show that s k (1&d k) s for some constant d>0. We prove this result by bringing together the ideas of critical clauses and the Sparsification Lemma to reduce the satisfiability of a kCNF to the satisfiability of a disjunction of 2 =n k$CNFs in fewer variables for some k $ k and arbitrarily small =>0. We also show that such a disjunction can be computed in time 2 =n for arbitrarily small =>0.
Tight lower bounds for certain parameterized NPhard problems
 Information and Computation
, 2004
"... Based on the framework of parameterized complexity theory, we derive tight lower bounds on the computational complexity for a number of wellknown NPhard problems. We start by proving a general result, namely that the parameterized weighted satisfiability problem on deptht circuits cannot be solve ..."
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Cited by 37 (5 self)
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Based on the framework of parameterized complexity theory, we derive tight lower bounds on the computational complexity for a number of wellknown NPhard problems. We start by proving a general result, namely that the parameterized weighted satisfiability problem on deptht circuits cannot be solved in time n o(k) m O(1) , where n is the circuit input length, m is the circuit size, and k is the parameter, unless the (t − 1)st level W [t − 1] of the Whierarchy collapses to FPT. By refining this technique, we prove that a group of parameterized NPhard problems, including weighted sat, hitting set, set cover, and feature set, cannot be solved in time n o(k) m O(1) , where n is the size of the universal set from which the k elements are to be selected and m is the instance size, unless the first level W [1] of the Whierarchy collapses to FPT. We also prove that another group of parameterized problems which includes weighted qsat (for any fixed q ≥ 2), clique, independent set, and dominating set, cannot be solved in time n o(k) unless all search problems in the syntactic class SNP, introduced by Papadimitriou and Yannakakis, are solvable in subexponential time. Note that all these parameterized problems have trivial algorithms of running time either n k m O(1) or O(n k). 1
Subexponential Parameterized Algorithms Collapse the Whierarchy (Extended Abstract)
, 2001
"... It is shown that for essentially all MAX SNPhard optimization problems finding exact solutions in subexponential time is not possible unless W [1] = FPT . In particular, we show that O(2 o(k) p(n)) parameterized algorithms do not exist for Vertex Cover, Max Cut, Max cSat, and a number of pr ..."
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Cited by 36 (2 self)
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It is shown that for essentially all MAX SNPhard optimization problems finding exact solutions in subexponential time is not possible unless W [1] = FPT . In particular, we show that O(2 o(k) p(n)) parameterized algorithms do not exist for Vertex Cover, Max Cut, Max cSat, and a number of problems on bounded degree graphs such as Dominating Set and Independent Set, unless W [1] = FPT . Our results are derived via an approach that uses an extended parameterization of optimization problems and associated techniques to relate the parameterized complexity of problems in FPT to the parameterized complexity of extended versions that are W [1]hard.
A measure & conquer approach for the analysis of exact algorithms
, 2007
"... For more than 40 years Branch & Reduce exponentialtime backtracking algorithms have been among the most common tools used for finding exact solutions of NPhard problems. Despite that, the way to analyze such recursive algorithms is still far from producing tight worstcase running time bounds. Mot ..."
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Cited by 30 (7 self)
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For more than 40 years Branch & Reduce exponentialtime backtracking algorithms have been among the most common tools used for finding exact solutions of NPhard problems. Despite that, the way to analyze such recursive algorithms is still far from producing tight worstcase running time bounds. Motivated by this we use an approach, that we call “Measure & Conquer”, as an attempt to step beyond such limitations. The approach is based on the careful design of a nonstandard measure of the subproblem size; this measure is then used to lower bound the progress made by the algorithm at each branching step. The idea is that a smarter measure may capture behaviors of the algorithm that a standard measure might not be able to exploit, and hence lead to a significantly better worstcase time analysis. In order to show the potentialities of Measure & Conquer, we consider two wellstudied NPhard problems: minimum dominating set and maximum independent set. For the first problem, we consider the current best algorithm, and prove (thanks to a better measure) a much tighter running time bound for it. For the second problem, we describe a new, simple algorithm, and show that its running time is competitive with the current best time bounds, achieved with far more complicated algorithms (and standard analysis). Our examples
Subexponential algorithms for Unique Games and related problems
 In 51 st IEEE FOCS
"... We give subexponential time approximation algorithms for the unique games and the small set expansion problems. Specifically, for some absolute constant c, we give: 1. An exp(kn ε)time algorithm that, given as input a kalphabet unique game on n variables that has an assignment satisfying 1 − ε c f ..."
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Cited by 30 (4 self)
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We give subexponential time approximation algorithms for the unique games and the small set expansion problems. Specifically, for some absolute constant c, we give: 1. An exp(kn ε)time algorithm that, given as input a kalphabet unique game on n variables that has an assignment satisfying 1 − ε c fraction of its constraints, outputs an assignment satisfying 1 − ε fraction of the constraints. 2. An exp(n ε /δ)time algorithm that, given as input an nvertex regular graph that has a set S of δn vertices with edge expansion at most ε c, outputs a set S ′ of at most δn vertices with edge expansion at most ε. We also obtain a subexponential algorithm with improved approximation for the MultiCut problem, as well as subexponential algorithms with improved approximations to MaxCut, SparsestCut and Vertex Cover on some interesting subclasses of instances. Khot’s Unique Games Conjecture (UGC) states that it is NPhard to achieve approximation guarantees such as ours for unique games. While our results stop short of refusing the UGC, they do suggest that Unique Games is significantly easier than NPhard problems such as 3SAT,3LIN, Label Cover and more, that are believed not to have a subexponential algorithm achieving a nontrivial approximation ratio. The main component in our algorithms is a new result on graph decomposition that may have other applications. Namely we show that for every δ> 0 and a regular nvertex graph G, by changing at most δ fraction of G’s edges, one can break G into disjoint parts so that the induced graph on each part has at most n ε eigenvalues larger than 1 − η (where ε, η depend polynomially on δ). Our results are based on combining this decomposition with previous algorithms for unique games on graphs with few large eigenvalues (Kolla and Tulsiani 2007, Kolla 2010). 1
Exact algorithms for treewidth and minimum fillin
 In Proceedings of the 31st International Colloquium on Automata, Languages and Programming (ICALP 2004). Lecture Notes in Comput. Sci
, 2004
"... We show that the treewidth and the minimum fillin of an nvertex graph can be computed in time O(1.8899 n). Our results are based on combinatorial proofs that an nvertex graph has O(1.7087 n) minimal separators and O(1.8135 n) potential maximal cliques. We also show that for the class of ATfree g ..."
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Cited by 26 (15 self)
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We show that the treewidth and the minimum fillin of an nvertex graph can be computed in time O(1.8899 n). Our results are based on combinatorial proofs that an nvertex graph has O(1.7087 n) minimal separators and O(1.8135 n) potential maximal cliques. We also show that for the class of ATfree graphs the running time of our algorithms can be reduced to O(1.4142 n).
Inapproximability of combinatorial optimization problems
 Electronic Colloquium on Computational Complexity
, 2004
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