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Register a new userA novel evolutionary formulation of the maximum independent set problem
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Added by
Valmir Barbosa
Publication date
2003
fields
Informatics Engineering
and research's language is
English
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We introduce a novel evolutionary formulation of the problem of finding a maximum independent set of a graph. The new formulation is based on the relationship that exists between a graphs independence number and its acyclic orientations. It views such orientations as individuals and evolves them with the aid of evolutionary operators that are very heavily based on the structure of the graph and its acyclic orientations. The resulting heuristic has been tested on some of the Second DIMACS Implementation Challenge benchmark graphs, and has been found to be competitive when compared to several of the other heuristics that have also been tested on those graphs.
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Evolutionary algorithms (EA) have been widely accepted as efficient solvers for complex real world optimization problems, including engineering optimization. However, real world optimization problems often involve uncertain environment including noisy and/or dynamic environments, which pose major challenges to EA-based optimization. The presence of noise interferes with the evaluation and the selection process of EA, and thus adversely affects its performance. In addition, as presence of noise poses challenges to the evaluation of the fitness function, it may need to be estimated instead of being evaluated. Several existing approaches attempt to address this problem, such as introduction of diversity (hyper mutation, random immigrants, special operators) or incorporation of memory of the past (diploidy, case based memory). However, these approaches fail to adequately address the problem. In this paper we propose a Distributed Population Switching Evolutionary Algorithm (DPSEA) method that addresses optimization of functions with noisy fitness using a distributed population switching architecture, to simulate a distributed self-adaptive memory of the solution space. Local regression is used in the pseudo-populations to estimate the fitness. Successful applications to benchmark test problems ascertain the proposed methods superior performance in terms of both robustness and accuracy.
We study the Maximum Independent Set of Rectangles (MISR) problem: given a set of $n$ axis-parallel rectangles, find a largest-cardinality subset of the rectangles, such that no two of them overlap. MISR is a basic geometric optimization problem with many applications, that has been studied extensively. Until recently, the best approximation algorithm for it achieved an $O(log log n)$-approximation factor. In a recent breakthrough, Adamaszek and Wiese provided a quasi-polynomial time approximation scheme: a $(1-epsilon)$-approximation algorithm with running time $n^{O(operatorname{poly}(log n)/epsilon)}$. Despite this result, obtaining a PTAS or even a polynomial-time constant-factor approximation remains a challenging open problem. In this paper we make progress towards this goal by providing an algorithm for MISR that achieves a $(1 - epsilon)$-approximation in time $n^{O(operatorname{poly}(loglog{n} / epsilon))}$. We introduce several new technical ideas, that we hope will lead to further progress on this and related problems.
We present improved results for approximating maximum-weight independent set ($MaxIS$) in the CONGEST and LOCAL models of distributed computing. Given an input graph, let $n$ and $Delta$ be the number of nodes and maximum degree, respectively, and let $MIS(n,Delta)$ be the the running time of finding a emph{maximal} independent set ($MIS$) in the CONGEST model. Bar-Yehuda et al. [PODC 2017] showed that there is an algorithm in the CONGEST model that finds a $Delta$-approximation for $MaxIS$ in $O(MIS(n,Delta)log W)$ rounds, where $W$ is the maximum weight of a node in the graph, which can be as high as $poly (n)$. Whether their algorithm is deterministic or randomized depends on the $MIS$ algorithm that is used as a black-box. Our main result in this work is a randomized $(poly(loglog n)/epsilon)$-round algorithm that finds, with high probability, a $(1+epsilon)Delta$-approximation for $MaxIS$ in the CONGEST model. That is, by sacrificing only a tiny fraction of the approximation guarantee, we achieve an emph{exponential} speed-up in the running time over the previous best known result. Due to a lower bound of $Omega(sqrt{log n/log log n})$ that was given by Kuhn, Moscibroda and Wattenhofer [JACM, 2016] on the number of rounds for any (possibly randomized) algorithm that finds a maximal independent set (even in the LOCAL model) this result implies that finding a $(1+epsilon)Delta$-approximation for $MaxIS$ is exponentially easier than $MIS$.
In 1960, Asplund and Grunbaum proved that every intersection graph of axis-parallel rectangles in the plane admits an $O(omega^2)$-coloring, where $omega$ is the maximum size of a clique. We present the first asymptotic improvement over this six-decade-old bound, proving that every such graph is $O(omegalogomega)$-colorable and presenting a polynomial-time algorithm that finds such a coloring. This improvement leads to a polynomial-time $O(loglog n)$-approximation algorithm for the maximum weight independent set problem in axis-parallel rectangles, which improves on the previous approximation ratio of $O(frac{log n}{loglog n})$.
We give a polynomial-time constant-factor approximation algorithm for maximum independent set for (axis-aligned) rectangles in the plane. Using a polynomial-time algorithm, the best approximation factor previously known is $O(loglog n)$. The results are based on a new form of recursive partitioning in the plane, in which faces that are constant-complexity and orthogonally convex are recursively partitioned into a constant number of such faces.
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