Some Open Problems in Fine-Grained Complexity

Abstract

Fine-grained complexity studies problems that are "hard" in the following sense. Consider a computational problem for which existing techniques easily give an algorithm running in a(n) time for inputs of size n, for some a. The algorithm is often brute-force, and despite decades of research, no O(a(n)1-∈) time algorithm for constant " > 0 has been developed. There are many diverse examples of such problems. Here are two: CNF-SAT on n variables and m clauses can be solved via exhaustive search in O(2nmn) time, and no 2(1-∈)npoly(m; n) time algorithm for constant " > 0 is known. The Longest Common Subsequence (LCS) problem on strings of length n has a classical O(n2) time algorithm, and no O(n-∈) time algorithm for " > 0 is known. Let's call these running times the "textbook running times". (Note that this is not well-defined but for many fundamental problems such as SAT or LCS, it is natural. The textbook runtime is the runtime of the algorithm a bright student in an algorithms class would come up with.)