Algorithms, heuristics and typical case complexity of hard problems

点击翻译按钮获取中文摘要

During the past decade there has been an extensive study of the phase transition phenomena in various combinatorial search problems such as Propositional Satisfiability (SAT), Constraint Satisfaction Problem (CSP), and graph coloring. These problems are computationally hard in the worst-case, but are of great importance in artificial intelligence and other application domains such as planning, scheduling, and communication networks. The growing interest in the study of the phase transitions in combinatorial search is due to the observation that the typical-case hardness of a random instance is closely related to the phase transition of the solution probability under some instance distribution. Far from the threshold, random instances are typically under-constrained (or over-constrained) and are very easy to solve. Around the threshold, however, extremely hard instances tend to appear with high probability. In the proposed research program, we study theoretically and empirically the interplays among the efficiency of heuristics in search algorithms, the typical-case complexity of problem instances randomly-generated from certain probability distributions, and the various problem structures such as problem representation (encoding), constraint consistency, backbones, backdoors, and structural symmetry. The objectives are (1) to gain insight on the impact of problem structures on the performance of different heuristics and to draw guidance regarding the design of efficient heuristics; (2) to understand the fundamental limitations of different classes of search algorithms; and (3) to develop algorithmic solutions to real-world problems. The algorithmic problems to be investigated fall into three categories: (A) Traditional NP-complete problems that are of interests in artificial intelligence; (B) Specific problems in graph theory with special properties such as the dominating clique problem and the tree-width problem; and (C) Problems from application domains such as the phylogeny reconstruction problem  and the multi-constrained path problem in communication networks.