Realizable paths and the NL vs L problem

A celebrated theorem of Savitch [Savitch'70] states that NSPACE(S) is contained in DSPACE(S²). In particular, Savitch gave a deterministic algorithm to solve ST-Connectivity (an NL-complete problem) using O({log}²{n}) space, implying NL (non-deterministic logspace) is contained in DSPACE({log}²{n}). While Savitch's theorem itself has not been improved in the last four decades, several graph connectivity problems are shown to lie between L and NL, providing new insights into the space-bounded complexity classes. All the connectivity problems considered in the literature so far are essentially special cases of ST-Connectivity. In this dissertation, we initiate the study of auxiliary PDAs as graph connectivity problems and define sixteen different "graph realizability problems" and study their relationships. The complexity of these connectivity problems lie between L (logspace) and P (polynomial time). ST-Realizability, the most general graph realizability problem is P-complete. 1DSTREAL(poly), the most specific graph realizability problem is L-complete. As special cases of our graph realizability problems we define two natural problems, Balanced ST-Connectivity and Positive Balanced ST-Connectivity, that lie between L and NL. We study the space complexity of SGSLOGCFL, a graph realizability problem lying between L and LOGCFL. We define generalizations of graph squaring and transitive closure, present efficient parallel algorithms for SGSLOGCFL and use the techniques of Trifonov to show that SGSLOGCFL is contained in DSPACE(lognloglogn). This implies that Balanced ST-Connectivity is contained in DSPACE(lognloglogn). We conclude with several interesting new research directions.PhDCommittee Chair: Richard Lipton; Committee Member: Anna Gal; Committee Member: Maria-Florina Balcan; Committee Member: Merrick Furst; Committee Member: William Coo

Arithmetic Branching Programs with Memory

We extend the well known characterization of VPws as the class of polynomials computed by polynomial size arithmetic branching programs to other complexity classes. In order to do so we add additional memory to the computation of branching programs to make them more expressive. We show that allowing different types of memory in branching programs increases the computational power even for constant width programs. In particular, this leads to very natural and robust characterizations of VP and VNP by branching programs with memory. 1