An Algorithm Framework for the Exact Solution and Improved Approximation of the Maximum Weighted Independent Set Problem

Abstract

The Maximum Weighted Independent Set (MWIS) problem, which considers a graph with weights assigned to nodes and seeks to discover the "heaviest" independent set, that is, a set of nodes with maximum total weight so that no two nodes in the set are connected by an edge. The MWIS problem arises in many application domains, including the resource-constrained scheduling, error-correcting coding, complex system analysis and optimization, and communication networks. Since solving the MWIS problem is the core function for finding the optimum solution of our novel graph-based formulation of the resource-constrained Process Planning and Scheduling (PPS) problem, it is essential to have "good-performance" algorithms to solve the MWIS problem. In this paper, we propose a Novel Hybrid Heuristic Algorithm (NHHA) framework in a divide-and-conquer structure that yields optimum feasible solutions to the MWIS problem. The NHHA framework is optimized to minimize the recurrence. Using the NHHA framework, we also solve the All Maximal Independent Sets Listing (AMISL) problem, which can be seen as the subproblem of the MWIS problem. Moreover, building composed MWIS algorithms that utilizing fast approximation algorithms with the NHHA framework is an effective way to improve the accuracy of approximation MWIS algorithms (e.g., GWMIN and GWMIN2 (Sakai et al., 2003)). Eight algorithms for the MWIS problem, the exact MWIS algorithm, the AMISL algorithm, two approximation algorithms from the literature, and four composed algorithms, are applied and tested for solving the graph-based formulation of the resource-constrained PPS problem to evaluate the scalability, accuracy, and robustness.