Search-Space Reduction via Essential Vertices

IF 0.9 3区数学 Q2 MATHEMATICS

SIAM Journal on Discrete Mathematics Pub Date : 2024-08-30 DOI:10.1137/23m1589347

Benjamin Merlin Bumpus, Bart M. P. Jansen, Jari J. H. de Kroon

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引用次数: 0

Abstract

SIAM Journal on Discrete Mathematics, Volume 38, Issue 3, Page 2392-2415, September 2024.
Abstract. We investigate preprocessing for vertex-subset problems on graphs. While the notion of kernelization, originating in parameterized complexity theory, is a formalization of provably effective preprocessing aimed at reducing the total instance size, our focus is on finding a nonempty vertex set that belongs to an optimal solution. This decreases the size of the remaining part of the solution which still has to be found, and therefore shrinks the search space of fixed-parameter tractable algorithms for parameterizations based on the solution size. We introduce the notion of a [math]-essential vertex as one that is contained in all [math]-approximate solutions. For several classic combinatorial problems such as Odd Cycle Transversal and Directed Feedback Vertex Set, we show that under mild conditions a polynomial-time preprocessing algorithm can find a subset of an optimal solution that contains all 2-essential vertices, by exploiting packing/covering duality. This leads to FPT algorithms to solve these problems where the exponential term in the running time depends only on the number of nonessential vertices in the solution.

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通过基本顶点缩小搜索空间

SIAM 离散数学杂志》，第 38 卷第 3 期，第 2392-2415 页，2024 年 9 月。摘要。我们研究图上顶点子集问题的预处理。源于参数化复杂性理论的 "内核化 "概念是对可证明有效的预处理的形式化，旨在减少实例的总大小，而我们的重点是找到属于最优解的非空顶点集。这就减小了仍需找到的解的剩余部分的大小，从而缩小了基于解大小的参数化固定参数可控算法的搜索空间。我们引入了[math]基本顶点的概念，即包含在所有[math]近似解中的顶点。对于奇数循环横向和定向反馈顶点集等几个经典组合问题，我们证明，在温和条件下，多项式时间预处理算法可以利用打包/覆盖对偶性，找到包含所有 2-基本顶点的最优解子集。这就产生了解决这些问题的 FPT 算法，其中运行时间的指数项仅取决于解中非必要顶点的数量。

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来源期刊

SIAM Journal on Discrete Mathematics 数学-应用数学

CiteScore

1.90

自引率

0.00%

发文量

124

审稿时长

4-8 weeks

期刊介绍： SIAM Journal on Discrete Mathematics (SIDMA) publishes research papers of exceptional quality in pure and applied discrete mathematics, broadly interpreted. The journal''s focus is primarily theoretical rather than empirical, but the editors welcome papers that evolve from or have potential application to real-world problems. Submissions must be clearly written and make a significant contribution. Topics include but are not limited to: properties of and extremal problems for discrete structures combinatorial optimization, including approximation algorithms algebraic and enumerative combinatorics coding and information theory additive, analytic combinatorics and number theory combinatorial matrix theory and spectral graph theory design and analysis of algorithms for discrete structures discrete problems in computational complexity discrete and computational geometry discrete methods in computational biology, and bioinformatics probabilistic methods and randomized algorithms.