Complexity Theoretical Results on Nondeterministic Graph-driven Read-Once Branching Programs

Branching programs are a well-established computation model for boolean functions, especially read-once branching programs (BP1s) have been studied intensively. Recently two restricted nondeterministic (parity) BP1 models, called nondeterministic (parity) graph-driven BP1s and well-structured nondeterministic (parity) graph-driven BP1s, have been investigated. The consistency test for a BP-model M is the test whether a given BP is really a BP of model M. Here it is proved that the consistency test is co-NP-complete for nondeterministic (parity) graph-driven BP1s. Moreover, a lower bound technique for nondeterministic graph-driven BP1s is presented. The method generalizes a technique for the well-structured model and is applied in order to answer in the affirmative the open question whether the model of nondeterministic graph-driven BP1s is a proper restriction of nondeterministic BP1s (with respect to polynomial size)

On symbolic representations of maximum matchings and (un)directed graphs

Abstract. The maximum matching problem is one of the most fundamental algorithmic graph problems and OBDDs are one of the most common dynamic data structures for Boolean functions. Since in some applications graphs become larger and larger, a research branch has emerged which is concerned with the theoretical design and analysis of so-called symbolic algorithms for classical graph problems on OBDD-represented graph instances. Typically problems get harder when their input is represented symbolically, nevertheless not many concrete non-trivial lower bounds are known. Here, it is shown that symbolic OBDD-based algorithms for the maximum matching problem need exponential space (with respect to the OBDD size of the input graph). Furthermore, it is shown that OBDD-representations for undirected graphs can be exponentially larger than OBDD-representations for their directed counterparts and vice versa

Testing Membership in Formal Languages Implicitly Represented by Boolean Functions

Abstract: Combinatorial property testing, initiated formally by Goldreich, Goldwasser, and Ron in [Goldreich et al. (1998)] and inspired by Rubinfeld and Sudan in [Rubinfeld and Sudan 1996], deals with the relaxation of decision problems. Given a property P the aim is to decide whether a given input satisfies the property P or is far from having the property. A property P can be described as a language, i.e., a nonempty family of binary words. The associated property to a family of boolean functions f =(fn) is the set of 1-inputs of f. By an attempt to correlate the notion of testing to other notions of low complexity property testing has been considered in the context of formal languages. Here, a brief summary of results on testing properties defined by formal languages and by languages implicitly represented by small restricted branching programs is provided. Key Words: binary decision diagrams (BDDs), boolean functions, branching programs (BPs), computational complexity, formal languages, property testing, randomness, sublinear algorithms Category: F.1.3, F.2.2, G.3.

nondeterministic graph-driven read-once

A very simple function that requires exponential siz

The Optimal Read-Once Branching Program Complexity for the Direct Storage Access Function

Branching programs are computation models measuring the space of (Turing machine) computations. Read-once branching programs (BP1s) are the most general model where each graph-theoretical path is the computation path for some input. Exponential lower bounds on the size of read-once branching programs are known since a long time. Nevertheless, there are only few functions where the BP1 size is asymptotically known exactly. In this paper, the exact BP1 size of a fundamental function, the direct storage access function, is determined

Parity graph-driven read-once branching programs and an exponential lower bound for integer multiplication

Abstract Branching programs are a well-established computation model for boolean functions, especially read-once branching programs have been studied intensively. Exponential lower bounds for deterministic and nondeterministic read-once branching programs are known for a long time. On the other hand, the problem of proving superpolynomial lower bounds for parity read-once branching programs is still open. In this paper restricted parity read-once branching programs are considered and an exponential lower bound on the size of well-structured parity graph-driven read-once branching programs for integer multiplication is proven. This is the first strongly exponential lower bound on the size of a nonoblivious parity read-once branching program model for an explicitly defined boolean function. In addition, more insight into the structure of integer multiplication is yielded