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36
Vertex Cover: Further Observations and Further Improvements
 Journal of Algorithms
, 1999
"... Recently, there have been increasing interests and progresses in lowering the worst case time complexity for wellknown NPhard problems, in particular for the Vertex Cover problem. In this paper, new properties for the Vertex Cover problem are indicated and several simple and new techniques are int ..."
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Cited by 155 (15 self)
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Recently, there have been increasing interests and progresses in lowering the worst case time complexity for wellknown NPhard problems, in particular for the Vertex Cover problem. In this paper, new properties for the Vertex Cover problem are indicated and several simple and new techniques are introduced, which lead to an improved algorithm of time O(kn + 1:271 k k 2 ) for the problem. Our algorithm also induces improvement on previous algorithms for the Independent Set problem on graphs of small degree. 1 Introduction Many optimization problems from industrial applications are NPhard. According to the NPcompleteness theory [10], these problems cannot be solved in polynomial time unless P = NP. However, this fact does not obviate the need for solving these problems for their practical importance. There has been a number of approaches to attacking the NPhardness of optimization problems, including approximation algorithms, heuristic algorithms, and average time analysis. Recent...
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
New methods for 3SAT decision and worstcase analysis
 THEORETICAL COMPUTER SCIENCE
, 1999
"... We prove the worstcase upper bound 1:5045 n for the time complexity of 3SAT decision, where n is the number of variables in the input formula, introducing new methods for the analysis as well as new algorithmic techniques. We add new 2 and 3clauses, called "blocked clauses", generali ..."
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Cited by 69 (14 self)
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We prove the worstcase upper bound 1:5045 n for the time complexity of 3SAT decision, where n is the number of variables in the input formula, introducing new methods for the analysis as well as new algorithmic techniques. We add new 2 and 3clauses, called "blocked clauses", generalizing the extension rule of "Extended Resolution." Our methods for estimating the size of trees lead to a refined measure of formula complexity of 3clausesets and can be applied also to arbitrary trees. Keywords: 3SAT, worstcase upper bounds, analysis of algorithms, Extended Resolution, blocked clauses, generalized autarkness. 1 Introduction In this paper we study the exponential part of time complexity for 3SAT decision and prove the worstcase upper bound 1:5044:: n for n the number of variables in the input formula, using new algorithmic methods as well as new methods for the analysis. These methods also deepen the already existing approaches in a systematically manner. The following results...
Measure and conquer: domination  a case study
 PROCEEDINGS OF THE 32ND INTERNATIONAL COLLOQUIUM ON AUTOMATA, LANGUAGES AND PROGRAMMING (ICALP 2005), SPRINGER LNCS
, 2005
"... DavisPutnamstyle exponentialtime backtracking algorithms are the most common algorithms used for finding exact solutions of NPhard problems. The analysis of such recursive algorithms is based on the bounded search tree technique: a measure of the size of the subproblems is defined; this measure ..."
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Cited by 50 (20 self)
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DavisPutnamstyle exponentialtime backtracking algorithms are the most common algorithms used for finding exact solutions of NPhard problems. The analysis of such recursive algorithms is based on the bounded search tree technique: a measure of the size of the subproblems is defined; this measure is used to lower bound the progress made by the algorithm at each branching step. For the last 30 years the research on exact algorithms has been mainly focused on the design of more and more sophisticated algorithms. However, measures used in the analysis of backtracking algorithms are usually very simple. In this paper we stress that a more careful choice of the measure can lead to significantly better worst case time analysis. As an example, we consider the minimum dominating set problem. The currently fastest algorithm for this problem has running time O(2 0.850n) on nnodes graphs. By measuring the progress of the (same) algorithm in a different way, we refine the time bound to O(2 0.598n). A good choice of the measure can provide such a (surprisingly big) improvement; this suggests that the running time of many other exponentialtime recursive algorithms is largely overestimated because of a “bad” choice of the measure.
Improved Algorithms for 3Coloring, 3EdgeColoring, and Constraint Satisfaction
, 2001
"... We consider worst case time bounds for NPcomplete problems including 3SAT, 3coloring, 3edgecoloring, and 3list coloring. Our algorithms are based on a constraint satisfaction (CSP) formulation of these problems; 3SAT is equivalent to (2, 3)CSP while the other problems above are special cases ..."
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Cited by 45 (3 self)
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We consider worst case time bounds for NPcomplete problems including 3SAT, 3coloring, 3edgecoloring, and 3list coloring. Our algorithms are based on a constraint satisfaction (CSP) formulation of these problems; 3SAT is equivalent to (2, 3)CSP while the other problems above are special cases of (3, 2)CSP. We give a fast algorithm for (3, 2) CSP and use it to improve the time bounds for solving the other problems listed above. Our techniques involve a mixture of DavisPutnamstyle backtracking with more sophisticated matching and network flow based ideas.
Small Maximal Independent Sets and Faster Exact Graph Coloring
, 2003
"... We show that, for any nvertex graph G and integer parameter k,there I#k,and that all such sets can be listed in time ). These bounds are tight when n/4 n/3. As a consequence, we show how to compute the exact chromatic number of a graph in time O((4/3+3 /4) 2.4150 ..."
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Cited by 44 (2 self)
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We show that, for any nvertex graph G and integer parameter k,there I#k,and that all such sets can be listed in time ). These bounds are tight when n/4 n/3. As a consequence, we show how to compute the exact chromatic number of a graph in time O((4/3+3 /4) 2.4150 , improving a previous algorithm of Lawler (1976).
New Upper Bounds for Maximum Satisfiability
 Journal of Algorithms
, 1999
"... The (unweighted) Maximum Satisfiability problem (MaxSat) is: given a boolean formula in conjunctive normal form, find a truth assignment that satisfies the most number of clauses. This paper describes exact algorithms that provide new upper bounds for MaxSat. We prove that MaxSat can be solved i ..."
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Cited by 37 (2 self)
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The (unweighted) Maximum Satisfiability problem (MaxSat) is: given a boolean formula in conjunctive normal form, find a truth assignment that satisfies the most number of clauses. This paper describes exact algorithms that provide new upper bounds for MaxSat. We prove that MaxSat can be solved in time O(F  1.3803 K ), where F  is the length of a formula F in conjunctive normal form and K is the number of clauses in F . We also prove the time bounds O(F 1.3995 k ), where k is the maximum number of satisfiable clauses, and O(1.1279 F  ) for the same problem. For Max2Sat this implies a bound of O(1.2722 K ). # An extended abstract of this paper was presented at the 26th International Colloquium on Automata, Languages, and Programming (ICALP'99), LNCS 1644, SpringerVerlag, pages 575584, held in Prague, Czech Republic, July 1115, 1999. + Supported by a Feodor Lynen fellowship (1998) of the Alexander von HumboldtStiftung, Bonn, and the Center for Discrete Ma...
Frozen Development in Graph Coloring
 THEORETICAL COMPUTER SCIENCE
, 2000
"... We define the `frozen development' of coloring random graphs. We identify two nodes in a graph as frozen if they are the same color in all legal colorings. This is analogous to studies of the development of a backbone or spine in SAT (the Satisability problem). We first describe in detail the a ..."
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Cited by 36 (5 self)
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We define the `frozen development' of coloring random graphs. We identify two nodes in a graph as frozen if they are the same color in all legal colorings. This is analogous to studies of the development of a backbone or spine in SAT (the Satisability problem). We first describe in detail the algorithmic techniques used to study frozen development. We present strong empirical evidence that freezing in 3coloring is sudden. A single edge typically causes the size of the graph to collapse in size by 28%. We also use the frozen development to calculate unbiased estimates of probability of colorability in random graphs, even where this probability is as low as 10^300. We investigate the links between frozen development and the solution cost of graph coloring. In SAT, a discontinuity in the order parameter has been correlated with the hardness of SAT instances, and our data for coloring is suggestive of an asymptotic discontinuity. The uncolorability threshold is known to give rise to har...
On Limited versus Polynomial Nondeterminism
, 1997
"... In this paper, we show that efficient algorithms for some problems that require limited nondeterminism imply the subexponential simulation of nondeterministic computation by deterministic computation. In particular, if cliques of size O(log n) can be found in polynomial time, then nondeterministic t ..."
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Cited by 24 (0 self)
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In this paper, we show that efficient algorithms for some problems that require limited nondeterminism imply the subexponential simulation of nondeterministic computation by deterministic computation. In particular, if cliques of size O(log n) can be found in polynomial time, then nondeterministic time f(n) is contained in deterministic time 2 O( p f(n)polylogf(n)) .
Worstcase Analysis, 3SAT Decision and Lower Bounds: Approaches for Improved SAT Algorithms
"... . New methods for worstcase analysis and (3)SAT decision are presented. The focus lies on the central ideas leading to the improved bound 1:5045 n for 3SAT decision ([Ku96]; n is the number of variables). The implications for SAT decision in general are discussed and elucidated by a number of h ..."
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Cited by 21 (6 self)
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. New methods for worstcase analysis and (3)SAT decision are presented. The focus lies on the central ideas leading to the improved bound 1:5045 n for 3SAT decision ([Ku96]; n is the number of variables). The implications for SAT decision in general are discussed and elucidated by a number of hypothesis'. In addition an exponential lower bound for a general class of SATalgorithms is given and the only possibilities to remain under this bound are pointed out. In this article the central ideas leading to the improved worstcase upper bound 1:5045 n for 3SAT decision ([Ku96]) are presented. 1) In nine sections the following subjects are treated: 1. "Gauging of branchings": The " function" and the concept of a "distance function" is introduced, our main tools for the analysis of SAT algorithms, and, as we propose, also a basis for (complete) practical algorithms. 2. "Estimating the size of arbitrary trees": The " Lemma" is presented, yielding an upper bound for the number of l...