Unique optimal solution instance and computational complexity of backbone in the graph bi-partitioning problem

Abstract

As an important tool for heuristic design of NP-hard problems, backbone analysis has become a hot spot in theoretical computer science in recent years. Due to the difficulty in the research on computational complexity of the backbone, many researchers analyzed the backbone by statistic ways. Aiming to increase the backbone size which is usually very small by the existing methods, the unique optimal solution instance construction (UOSIC) is proposed for the graph bi-partitioning problem (GBP). Also, we prove by using the UOSIC that it is NP-hard to obtain the backbone, i.e. no algorithm exists to obtain the backbone of a GBP in polynomial time under the assumption that P ≠ NP. Our work expands the research area of computational complexity of the backbone. And the UOSIC provides a new way for heuristic design of NP-hard problems.graph bi-partitioning problem, NP-hard, backbone analysis, unique optimal solution instance, computational complexity Backbone analysis [1] has been a hot spot in the research of NP-hard problems [2] in recent years. The backbone, the shared common part of all optimal solutions for an NP-hard problem instance, is important to the evaluation of the hardness and phase transition of NP-hard problems. Monasson et al. [3] investigated the effect of the backbone on the hardness and phase transition of the satisfiability problem (SAT) in 1998. Zhang [4] analyzed the effect of the backbone size on the asymmetric traveling salesman problem (ATSP). Moreover, backbone analysis has been currently an important way to design heuristics for NP-hard problems. Schneider [5] and Zou et al. [6] proposed multilevel reduction algorithms for the TSP by using the intersection of local optimal solutions as the approximate backbone, respectively. Zhang et al. [7] presented a backbone guided LK algorithm for the TSP. Zhang [8] , Dubois et al. [9] , and Valnir et al. [10] gave backbone guided local search algorithms for the SAT, respectively. Zou et al. [11] developed the approximate backbone-guided fant (ABFANT) for the quadratic assignment problem (QAP). However, backbone analysis is still in the beginning state using experimental statistic methods. Also, there are few theoretical analysis results concerning the backbone. As to our knowledge, the only theoretical result is that Kilby et al. [12] proved that it is NP-hard to obtain the backbone of the TSP.The graph bi-partitioning problem (GBP) [2] is one of the classical NP-hard problems, with extensive applications in parallel computing, very large scale integration (VLSI), transportation scheduling, and data mining. According to the computational complexity theory [13] , there exists no exact algorithm to solve NP-hard problems in polynomial time unless P=NP. For large instances, many heuristics offering very good near optimal solutions in a reasonable time have been developed [14][15][16][17][18] . Yet there