Structural Parameterizations with Modulator Oblivion

作     者：Jacob, Ashwin Panolan, Fahad Raman, Venkatesh Sahlot, Vibha 

作者机构：HBNI Inst Math Sci Chennai Tamil Nadu India IIT Hyderabad Dept Comp Sci & Engn Hyderabad India Charles Univ Prague Comp Sci Inst CSI Prague 11800 Czech Republic 

出 版 物：《ALGORITHMICA》 (算法)

年 卷 期：2022年第84卷第8期

页      面：2335-2357页

核心收录：

学科分类：08[工学] 0835[工学-软件工程] 0701[理学-数学] 0812[工学-计算机科学与技术（可授工学、理学学位）] 

主　　题：Parameterized complexity Chordal graph Tree decomposition Strong exponential time hypothesis 

摘      要：It is known that problems like VERTEX COVER, FEEDBACK VERTEX SET and ODD CYCLE TRANSVERSAL are polynomial time solvable in the class of chordal graphs. We consider these problems in a graph that has at most k vertices whose deletion results in a chordal graph when parameterized by k. While this investigation fits naturally into the recent trend of what is called structural parameterizations , here we assume that the deletion set is not given. One method to solve them is to compute a k-sized or an approximate (f(k) sized, for a function f) chordal vertex deletion set and then use the structural properties of the graph to design an algorithm. This method leads to at least k(O(k))n(O(1)) running time when we use the known parameterized or approximation algorithms for finding a k-sized chordal deletion set on an n vertex graph. In this work, we design 2(O(k))n(O(1)) time algorithms for these problems. Our algorithms do not compute a chordal vertex deletion set (or even an approximate solution). Instead, we construct a tree decomposition of the given graph in 2(O(k))n(O(1)) time where each bag is a union of four cliques and O(k) vertices. We then apply standard dynamic programming algorithms over this special tree decomposition. This special tree decomposition can be of independent interest. Our algorithms are, what are sometimes called permissive in the sense that given an integer k, they detect whether the graph has no chordal vertex deletion set of size at most k or output the special tree decomposition and solve the problem. We also show lower bounds for the problems we deal with under the strong exponential time hypothesis.