80,093 research outputs found
Incomplete MaxSAT Solving by Linear Programming Relaxation and Rounding
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
Here we study the NP-complete -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 -SAT, with as control parameter, can be solved quickly
for small enough values of . 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
-SAT () 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 -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 -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
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 ) nor to approximate it (unless ).
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 . 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
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 in a randomly
generated -dimensional ADK landscape.Comment: 19 pages, 5 figure
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