A parameterized runtime analysis of randomized local search and evolutionary algorithm for max-uncut

Pallavi Jain, Lawqueen Kanesh, Jayakrishnan Madathil, Saket Saurabh

Abstract: In the last few years, parameterized complexity has emerged as a new tool to analyze the running time of randomized local search algorithm. However, such analysis are few and far between. In this paper, we do a parameterized running time analysis of a randomized local search algorithm and a (1 + 1) EA for a classical graph partitioning problem, namely, Max l-Uncut, and its balanced counterpart Max Balanced l-Uncut. In Max l-Uncut, given an undirected graph G = (V, E), the objective is to find a partition of V(G) into l parts such that the number of uncut edges - edges within the parts - is maximized. In the last few years, Max l-Uncut and Max Balanced l-Uncut are studied extensively from the approximation point of view. In this paper, we analyze the parameterized running time of a randomized local search algorithm (RLS) for Max Balanced l-Uncut where the parameter is the number of uncut edges. RLS generates a solution of specific fitness in polynomial time for this problem. Furthermore, we design a fixed parameter tractable randomized local search and a (1 + 1) EA for Max l-Uncut and prove that they perform equally well.