80,093 research outputs found

    Incomplete MaxSAT Solving by Linear Programming Relaxation and Rounding

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    NP-hard optimization problems can be found in various real-world settings such as scheduling, planning and data analysis. Coming up with algorithms that can efficiently solve these problems can save various rescources. Instead of developing problem domain specific algorithms we can encode a problem instance as an instance of maximum satisfiability (MaxSAT), which is an optimization extension of Boolean satisfiability (SAT). We can then solve instances resulting from this encoding using MaxSAT specific algorithms. This way we can solve instances in various different problem domains by focusing on developing algorithms to solve MaxSAT instances. Computing an optimal solution and proving optimality of the found solution can be time-consuming in real-world settings. Finding an optimal solution for problems in these settings is often not feasible. Instead we are only interested in finding a good quality solution fast. Incomplete solvers trade guaranteed optimality for better scalability. In this thesis, we study an incomplete solution approach for solving MaxSAT based on linear programming relaxation and rounding. Linear programming (LP) relaxation and rounding has been used for obtaining approximation algorithms on various NP-hard optimization problems. As such we are interested in investigating the effectiveness of this approach on MaxSAT. We describe multiple rounding heuristics that are empirically evaluated on random, crafted and industrial MaxSAT instances from yearly MaxSAT Evaluations. We compare rounding approaches against each other and to state-of-the-art incomplete solvers SATLike and Loandra. The LP relaxation based rounding approaches are not competitive in general against either SATLike or Loandra However, for some problem domains our approach manages to be competitive against SATLike and Loandra

    Phase Transitions of the Typical Algorithmic Complexity of the Random Satisfiability Problem Studied with Linear Programming

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    Here we study the NP-complete KK-SAT problem. Although the worst-case complexity of NP-complete problems is conjectured to be exponential, there exist parametrized random ensembles of problems where solutions can typically be found in polynomial time for suitable ranges of the parameter. In fact, random KK-SAT, with α=M/N\alpha=M/N as control parameter, can be solved quickly for small enough values of α\alpha. It shows a phase transition between a satisfiable phase and an unsatisfiable phase. For branch and bound algorithms, which operate in the space of feasible Boolean configurations, the empirically hardest problems are located only close to this phase transition. Here we study KK-SAT (K=3,4K=3,4) and the related optimization problem MAX-SAT by a linear programming approach, which is widely used for practical problems and allows for polynomial run time. In contrast to branch and bound it operates outside the space of feasible configurations. On the other hand, finding a solution within polynomial time is not guaranteed. We investigated several variants like including artificial objective functions, so called cutting-plane approaches, and a mapping to the NP-complete vertex-cover problem. We observed several easy-hard transitions, from where the problems are typically solvable (in polynomial time) using the given algorithms, respectively, to where they are not solvable in polynomial time. For the related vertex-cover problem on random graphs these easy-hard transitions can be identified with structural properties of the graphs, like percolation transitions. For the present random KK-SAT problem we have investigated numerous structural properties also exhibiting clear transitions, but they appear not be correlated to the here observed easy-hard transitions. This renders the behaviour of random KK-SAT more complex than, e.g., the vertex-cover problem.Comment: 11 pages, 5 figure

    Approximating the least hypervolume contributor: NP-hard in general, but fast in practice

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    The hypervolume indicator is an increasingly popular set measure to compare the quality of two Pareto sets. The basic ingredient of most hypervolume indicator based optimization algorithms is the calculation of the hypervolume contribution of single solutions regarding a Pareto set. We show that exact calculation of the hypervolume contribution is #P-hard while its approximation is NP-hard. The same holds for the calculation of the minimal contribution. We also prove that it is NP-hard to decide whether a solution has the least hypervolume contribution. Even deciding whether the contribution of a solution is at most (1+\eps) times the minimal contribution is NP-hard. This implies that it is neither possible to efficiently find the least contributing solution (unless P=NPP = NP) nor to approximate it (unless NP=BPPNP = BPP). Nevertheless, in the second part of the paper we present a fast approximation algorithm for this problem. We prove that for arbitrarily given \eps,\delta>0 it calculates a solution with contribution at most (1+\eps) times the minimal contribution with probability at least (1−δ)(1-\delta). Though it cannot run in polynomial time for all instances, it performs extremely fast on various benchmark datasets. The algorithm solves very large problem instances which are intractable for exact algorithms (e.g., 10000 solutions in 100 dimensions) within a few seconds.Comment: 22 pages, to appear in Theoretical Computer Scienc

    Fast optimization algorithms and the cosmological constant

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    Denef and Douglas have observed that in certain landscape models the problem of finding small values of the cosmological constant is a large instance of an NP-hard problem. The number of elementary operations (quantum gates) needed to solve this problem by brute force search exceeds the estimated computational capacity of the observable universe. Here we describe a way out of this puzzling circumstance: despite being NP-hard, the problem of finding a small cosmological constant can be attacked by more sophisticated algorithms whose performance vastly exceeds brute force search. In fact, in some parameter regimes the average-case complexity is polynomial. We demonstrate this by explicitly finding a cosmological constant of order 10−12010^{-120} in a randomly generated 10910^9-dimensional ADK landscape.Comment: 19 pages, 5 figure
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