State Complexity Characterizations of Parameterized Degree-Bounded Graph Connectivity, Sub-Linear Space Computation, and the Linear Space Hypothesis

Abstract

The linear space hypothesis is a practical working hypothesis, which originally states the insolvability of a restricted 2CNF Boolean formula satisfiability problem parameterized by the number of Boolean variables. From this hypothesis, it naturally follows that the degree-3 directed graph connectivity problem (3DSTCON) parameterized by the number of vertices in a given graph cannot belong to PsubLIN, composed of all parameterized decision problems computable by polynomial-time, sub-linear-space deterministic Turing machines. This hypothesis immediately implies L =NL and it was used as a solid foundation to obtain new lower bounds on the computational complexity of various NL search and NL optimization problems. The state complexity of transformation refers to the cost of converting one type of finite automata to another type, where the cost is measured in terms of the increase of the number of inner states of the converted automata from that of the original automata. We relate the linear space hypothesis to the state complexity of transforming restricted 2-way nondeterministic finite automata to computationally equivalent 2-way alternating finite automata having narrow computation graphs. For this purpose, we present state complexity characterizations of 3DSTCON and PsubLIN. We further characterize a nonuniform version of the linear space hypothesis in terms of the state complexity of transformation. PrologueWe provide the background of parameterized decision problems, the linear space hypothesis, and nonuniform state complexity. We then give an overview of major results of this work. Parameterized Problems and the Linear Space HypothesisThe nondeterministic logarithmic-space complexity class NL has been discussed since early days of computational complexity theory. Typical NL decision problems include the 2CNF Boolean formula satisfiability problem (2SAT) as well as the directed s-t connectivity problem 3 (DSTCON) of deciding the existence of a path from a vertex s to another vertex t in a given directed graph G. These problems are known to be NL-complete under log-space many-one reductions. The NL-completeness is so robust that even if we restrict our interest within graphs whose vertices are limited to be of degree at most 3, the corresponding decision problem, 3DSTCON, remains NL-complete. Similarly, although we force 2CNF Boolean formulas in 2SAT to take only variables, each of which appears at most 3 times in the form of literals, the obtained decision problem, 2SAT 3 , is still an NL-complete problem.When we discuss the computational complexity of given problems, we in practice tend to be more concerned with various parameterizations of the problems. We treat the size of specific "input objects" given to a problem as a "practical" size parameter n and use it to measure how much resource is needed for an algorithm to solve this problem. There are, in fact, multiple ways to choose such a size parameter for each given problem. For example, given an instance x = G, s, t to 3DSTCON, where G is a directed 1 Thi...

Parameterizations of Logarithmic-Space Reductions, Stack-State Complexity of Nonuniform Families of Pushdown Automata, and a Road to the LOGCFL$\subseteq$LOGDCFL/poly Question

“…This equivalence makes it possible to translate standard advised complexity classes into nonuniform state complexity classes. This phenomenon has been observed also in other nonuniform state complexity classes [20,21], including classes induced by probabilistic and quantum finite automata. We remark that an important discovery of [20] is the fact that nonuniform state complexity classes are more closely related to parameterized complexity classes, which naturally include standard (non-advised) complexity classes as special cases.…”

“…After a long recess since their initial works, Kapoutsis [10,11] revitalized the study of the subject and started a systematic study on the nonuniform setting of polynomial state complexities of 2dfa's and 2nfa's. Following these works, Kapoutsis [12] and Kapoutsis and Pighizzini [13] later made significant progress, and Yamakami [20,21,22] further expanded the study to a wider subject.…”

“…A Turing machine considered in this work is equipped with a read-only input tape, a rewritable work tape, and (possibly) a write-once 4 output tape. For the basics of Turing machines, the reader refers to [9] as well as [20,21]. Given two alphabets Σ and Γ, a function f : Σ * → Γ * is in FL if there is a deterministic Turing machine (or a DTM, for short) M with a designated write-once output tape such that, given any input x, M halts in polynomial time and produces f (x) on the output tape using only O(log |x|) work space.…”

“…n )} n∈N denote such a family of promise decision problems over Σ. It is important to remark that Σ does not depend on the choice of n (see, e.g., [20,21,22]). All strings in n∈N (L…”

“…Similarly to para-L/poly and para-NL/poly defined in [21,22], we wish to seek proper "parameterizations" of LOGCFL/poly and LOGDCFL/poly, including LOGCFL and LOGDCFL as their special cases. For readability, we will follow the basic terminology used in [18,19,20,21,22]. A parameterized decision problem is a pair (L, m) of a language L and a size parameter m. We are particularly interested in size parameters computable using logarithmic space.…”