Exact Complexity and Satisfiability

Abstract

All NP-complete problems are equivalent as far as polynomial time solvability is concerned. However, their exact complexities (worst-case complexity of algorithms that solve every instance correctly and exactly) differ widely. Starting with Bellman [1], Tarjan and Trojanowski [9], Karp [5], and Monien and Speckenmeyer [7], the design of improved exponential time algorithms for NP-complete problems has been a tremendously fruitful endeavor, and one that has recently been accelerating both in terms of the number of results and the increasing sophistication of algorithmic techniques. There are a vast variety of problems where progress has been made, e.g., k-sat, k-Colorability, Maximum Independent Set, Hamiltonian Path, Chromatic Number, and Circuit Sat for limited classes of circuits. The “current champion” algorithms for these problems deploy a vast variety of algorithmic techniques, e.g., back-tracking search (with its many refinements and novel analysis techniques), divide-and-conquer, dynamic programming, randomized search, algebraic transforms, inclusion-exclusion, color coding, split and list, and algebraic sieving. In many ways, this is analogous to the diversity of approximation ratios and approximation algorithms known for different NP-complete problems. In view of this diversity, it is tempting to focus on the distinctions between problems rather than the interconnections between them. However, over the past two decades, there has been a wave of research showing that such connections do exist. Furthermore, progress on the exact complexity of NP-complete problems is linked to other fundamental questions in computational complexity, such as circuit lower bounds, parameterized complexity, data structures, and the precise complexity of problems within P. We are honored that our work helped to catalyze this wave of research, and humbled by the extent to which later researchers went far beyond our dreams of what might be possible.