Labeled Search Trees and Amortized Analysis: Improved Upper Bounds for NP-Hard Problems

Abstract

A sequence of exact algorithms to solve the Vertex Cover and Maximum Independent Set problems have been proposed in the literature. All these algorithms appeal to a very conservative analysis that considers the size of the search tree, under a worst-case scenario, to derive an upper bound on the running time of the algorithm. In this paper we propose a different approach to analyze the size of the search tree. We use amortized analysis to show how simple algorithms, if analyzed properly, may perform much better than the upper bounds on their running time derived by considering only a worst-case scenario. This approach allows us to present a simple algorithm of running time O(1.194 k k 2 + n) for the parameterized Vertex Cover problem on degree-3 graphs, and a simple algorithm of running time O(1.1255 n ) for the Maximum Independent Set problem on degree-3 graphs. Both algorithms improve the previous best algorithms for the problems.

Four Shorts Stories on Surprising Algorithmic Uses of Treewidth

“…Together with the fact that a Maximum Independent Set on an n-vertex graph can be solved in time 2 w •n O (1) if a tree decomposition of width w is given, it follows that the problem can be solved in time (1) . The running time obtained as a simple consequence of this pathwidth bound was better than some earlier work at that time [5,20], but since then improved algorithms with more complicated and problem-specific arguments were found for this problem [17,18,67,76]. In a similar way, algorithms for Minimum Dominating Set and Max Cut follow immediately from Theorem 4, which were better than some of the algorithms found by earlier problem specific techniques [42].…”

Maximum weighted independent set: Effective reductions and fast algorithms on sparse graphs

“…Any further improvement may need new observations on the structural properties or new techniques to design and analyze the algorithms. For unweighted Maximum Independent Set on degree-3 graphs, the running time bound was improved for several times [7,29,5,23,28,4,32,15]. Each improvement is small, but each improvement reveals new properties and new analysis.…”