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 computa- tional 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.
As the hot line in NP-hard problems research in recent years, backbone analysis is crucial for phase transition, hardness, and algorithm design. Whereas theoretical analysis of backbone and its applications in algorithm design are still at a begin- ning state yet, this paper took the quadratic assignment problem (QAP) as a case study and proved by theoretical analysis that it is NP-hard to find the backbone, i.e., no algorithm exists to obtain the backbone of a QAP in polynomial time. Results of this paper showed that it is reasonable to acquire approximate backbone by inter- section of local optimal solutions. Furthermore, with the method of constructing biased instances, this paper proposed a new meta-heuristic―biased instance based approximate backbone (BI-AB), whose basic idea is as follows: firstly, con- struct a new biased instance for every QAP instance (the optimal solution of the new instance is also optimal for the original one); secondly, the approximate backbone is obtained by intersection of multiple local optimal solutions computed by some existing algorithm; finally, search for the optimal solutions in the reduced space by fixing the approximate backbone. Work of the paper enhanced the re- search area of theoretical analysis of backbone. The meta-heuristic proposed in this paper provided a new way for general algorithm design of NP-hard problems as well.
