Maximizing the reduction ability for near-maximum independent set computation

Abstract

Finding the maximum independent set is a fundamental NP-hard problem in graph theory. Recent studies have paid much attention to designing efficient algorithms that find a maximal independent set of good quality (the more vertices the better). Kernelization is a widely used technique that applies rich reduction rules to determine the vertices that definitely belong to the maximum independent set. When no reduction rules can be applied anymore, greedy strategies including vertex addition or vertex deletion are employed to break the tie. It remains an open problem that how to apply these reduction rules and determine the greedy strategy to optimize the overall performance including both solution quality and time efficiency. Thus we propose a scheduling framework that dynamically determines the reduction rules and greedy strategies rather than applying them in a fixed order. As an important reduction rule, degree-two reduction exhibits powerful pruning ability but suffers from high time complexity O ( nm ), where n and m denote the number of vertices and edges respectively. We propose a novel data structure called representative graph, based on which the worst-case time complexity of degree-two reduction is reduced to O ( m log n ). Moreover, we enrich the naive vertex addition strategy by considering the graph topology and develop efficient methods (active vertex index and lazy update mechanism) to improve the time efficiency. Extensive experiments are conducted on both large real networks and various types of synthetic graphs to confirm the effectiveness, efficiency and robustness of our algorithms.