Applications of fine-grained complexity

Abstract

This thesis is on the topic of the applications of Fine-Grained Complexity (FGC). FGC is concerned with categorizing computational problems by their running time up to low order terms in the exponent. FGC has been a very successful field, explaining the computational hardness of many problems via a network of reductions. Given the explanatory success of FGC in the standard computational model, it would be valuable to apply FGC to new areas. This thesis focuses on studying the core assumptions of FGC and three areas of applications: (1) traditional FGC in the standard model (the worst-case RAM model), (2) average-case FGC (ACFGC), and (3) fine-grained cryptography. If we can strengthen the core of FGC, then we would also strengthen the applications of FGC. This thesis demonstrates that a core hypothesis of FGC (the 3-SUM hypothesis) is equivalent to its small space counterpart. This makes the 3-SUM hypothesis more plausible.

FGC has built a network of reductions between problems that explain the known running time of the problems contained in the network. A core goal of FGC research is to add new problems to this network of reductions. This thesis shows that the sparse All Pairs Shortest Paths problem in n-node m-edge graphs requires (nm)[superscript 1-0(1)] time if the zero-k-clique hypothesis is true. This result gives a novel connection between the hardness of these two problems. A problem of much interest to both traditional complexity and FGC is Boolean Satisfiability (SAT). There is a well-studied average-case variant of SAT called Random k-SAT. In this thesis we study the running time of this problem, seeking to understand its ACFGC. We present an algorithm for Random k-SAT which runs in 2[superscript n](superscript [1-[omega](lg[superscript 2](k))=k)] time, giving the fastest known running time for Random k-SAT. Modern cryptography relies on average-case constructions.

That is, an encryption scheme is shown to be hard to break via reduction from a problem conjectured to be hard on average. Similarly, fine-grained cryptography relies on average-case fine-grained lower bounds and reductions. This is the core connection between fine-grained cryptography and ACFGC. This thesis presents a plausible fine-grained average-case hypothesis which results in a novel public-key cryptosystem. This thesis presents results that strengthen the core hypotheses of FGC, gives more efficient algorithms for problems of interest, and builds fine-grained cryptosystems. The core goal is to apply the tools and techniques of FGC to get novel results in both the worst-case setting as well as the average-case setting.