Nondeterministic Read-once Branching Programs Research Articles

On Tseitin Formulas, Read-Once Branching Programs and Treewidth

We show that any nondeterministic read-once branching program that decides a satisfiable Tseitin formula based on an n × n grid graph has size at least 2Ω(n). Then using the Excluded Grid Theorem by Robertson and Seymour we show that for an arbitrary graph G(V, E) any nondeterministic read-once branching program that computes a satisfiable Tseitin formula based on G has size at least $2^{\Omega (\text {tw}(G)^{\delta })}$ for all δ < 1/36, where tw(G) is the treewidth of G (for planar graphs and some other classes of graphs the statement holds for δ = 1). We apply the mentioned results to the analysis of the complexity of derivations in the proof system OBDD(∧,reordering) and show that any OBDD(∧,reordering)-refutation of an unsatisfiable Tseitin formula based on a graph G has size at least $2^{\Omega (\text {tw}(G)^{\delta })}$ . We also show an upper bound O(|E|2pw(G)) on the size of OBDD representations of a satisfiable Tseitin formula based on G and an upper bound $O(|E||V| 2^{\text {pw}(G)}+|\text {TS}_{G,c}|^{2})$ on the size of OBDD(∧)-refutation of an unsatisifable Tseitin formula TSG, c, where pw(G) is the pathwidth of G.

Foreword: Special Issue on IPEC 2014

We are pleased to present this special issue of Algorithmica, which contains the extended journal versions of selected papers previously presented at the 9th International Symposium on Parameterized and Exact Computation, IPEC 2014, September 10–12, Wroclaw, Poland. The symposium is an established annual meeting of the multivariate and exact algorithms communities. This issue consists of nine papers, reviewed thoroughly according to the usual, high standard of the journal. In The Parameterized Complexity of Geometric Graph Isomorphism, V. Arvind and Gaurav Rattan present improved FPT algorithm for Geometric Graph Isomorphism problem, where one is to decide whether there is a distance preserving bijection between two sets of points in k-dimensional euclidian space. Igor Razgon in On the read-once property of branching programs and CNFs of bounded treewidthproves a space lower bound for non-deterministic read-oncebranching programs on functions expressible as CNFswith treewidth at most k of their primal graphs. In Finding Shortest Paths between Graph Colourings, Matthew Johnson, Dieter Kratsch, Stefan Kratsch, Viresh Patel, and Daniel Paulusma give a complete picture of the parameterized complexity of the k-colouring reconfiguration problem, where the goal is to modify one proper colouring into another one, by changing the colour of one vertex at a time. Given a system of linear equations Ax = b over the binary field one can ask whether there is a solution of weight at most t , exactly t or at least t . In Solving Linear

On the Read-Once Property of Branching Programs and CNFs of Bounded Treewidth

In this paper we prove a space lower bound of $$n^{\varOmega (k)}$$ for non-deterministic (syntactic) read-once branching programs (nrobps) on functions expressible as cnfs with treewidth at most k of their primal graphs. This lower bound rules out the possibility of fixed-parameter space complexity of nrobps parameterized by k. We use lower bound for nrobps to obtain a quasi-polynomial separation between Free Binary Decision Diagrams and Decision Decomposable Negation Normal Forms, essentially matching the existing upper bound introduced by Beame et al. (Proceedings of the twenty-ninth conference on uncertainty in artificial intelligence, Bellevue, 2013) and thus proving the tightness of the latter.

On the Incompressibility of Monotone DNFs

We prove optimal lower bounds for multilinear circuits and for monotone circuits with bounded depth. These lower bounds state that, in order to compute certain functions, these circuits need exactly as many OR gates as the respective DNFs. The proofs exploit a property of the functions that is based solely on prime implicant structure. Due to this feature, the lower bounds proved also hold for approximations of the considered functions that are similar to slice functions. Known lower bound arguments cannot handle these kinds of approximations. In order to show limitations of our approach, we prove that cliques of size n - 1 can be detected in a graph with n vertices by monotone formulas with O(log n) OR gates. Our lower bound for multilinear circuits improves a lower bound due to Borodin, Razborov and Smolensky for nondeterministic read-once branching programs computing the clique function.

A Lower Bound Technique for Nondeterministic Graph-Driven Read-Once-Branching Programs and Its Applications

We present a new lower bound technique for a restricted branching program model, namely for nondeterministic graph-driven read-once branching programs (g.d.-BP1s). The technique is derived by drawing a connection between ω-nondeterministic g.d.-BP1s and ω-nondeterministic communication complexity (for the nondeterministic acceptance modes ω∈{⋁,⋀,⊕}). We apply the technique in order to prove an exponential lower bound for integer multiplication for ω-nondeterministic well-structured g.d.-BP1s. (For ω=⊕ an exponential lower bound was already obtained in [5] by using a different technique.) Further, we use the lower bound technique to prove for an explicitly defined function which can be represented by polynomial size ω-nondeterministic BP1s that it has exponential complexity in the ω-nondeterministic well-structured g.d.-BP1 model for ω∈{⋁,⊕}. This answers an open question from Brosenne et al., whether the nondeterministic BP1 model is in fact more powerful than the well-structured graph-driven variant.

On multi-partition communication complexity

We study k-partition communication protocols, an extension of the standard two-party best-partition model to k input partitions. The main results are as follows. 1. A strong explicit hierarchy on the degree of non-obliviousness is established by proving that, using k + 1 partitions instead of k may decrease the communication complexity from Θ ( n ) to Θ (log k ). 2. Certain linear codes are hard for k -partition protocols even when k may be exponentially large (in the input size). On the other hand, one can show that all characteristic functions of linear codes are easy for randomized OBDDs. 3. It is proved that there are subfunctions of the triangle-freeness function and the function ⊕ Clique 3, n that are hard for multi-partition protocols. As an application, strongly exponential lower bounds on the size of nondeterministic read-once branching programs for these functions are obtained, solving an open problem of Razborov [Proceedings of eighth FCT NCS 529, Springer, 1991, pp. 47–60].

Approximation of boolean functions by combinatorial rectangles

This paper deals with the number of monochromatic combinatorial rectangles required to approximate a boolean function on a constant fraction of all inputs, where each rectangle may use its own partition of the input variables. The main result of the paper is that the number of rectangles required for the approximation of boolean functions in this model is very sensitive to the allowed error. There is an explicitly defined sequence of boolean functions fn on n variables such that fn has rectangle approximations with a constant number of rectangles and one-sided error 13+o(1) or two-sided error 14+o(1), but, on the other hand, fn requires exponentially many rectangles if the error bounds are decreased by an arbitrarily small constant.As applications of this result, the following separation results for read-once branching programs are obtained. The functions from the main result require only linear size for nondeterministic read-once branching programs and randomized read-once branching programs with two-sided error 13+o(1), while randomized read-once branching programs with constant two-sided error smaller than 13 and unambiguous nondeterministic read-once branching programs require exponential size.

Restricted Nondeterministic Read-Once Branching Programs and an Exponential Lower Bound for Integer Multiplication

Branching programs are a well established computation model for Boolean functions, especially read-once branching programs have been studied intensively. In this paper the expressive power of nondeterministic read-once branching programs, more precisely the class of functions representable in polynomial size, is investigated. For that reason two restricted models of nondeterministic read-once branching programs are defined and a lower bound method is presented. Furthermore, the first exponential lower bound for integer multiplication on the size of a nondeterministic nonoblivious read-once branching program model is proven.

An Improved Hierarchy Result for Partitioned BDDs

One of the great challenges of complexity theory is the problem of analyzing the dependence of the complexity of Boolean functions on the resources nondeterminism and randomness. So far this problem could be solved only for very few models of computation. For so-called partitioned binary decision diagrams , which are a restricted variant of nondeterministic read-once branching programs, Bollig and Wegener have proven an astonishing hierarchy result which shows that the smallest possible decrease of the available amount of nondeterminism may incur an exponential blow-up of the branching program size. They have shown that k -partitioned BDDs which may nondeterministically choose between k alternative subprograms may be exponentially larger than (k+1) -partitioned BDDs for the same function if k = o(( log n / log log n) 1/2 ) , where n is the input size. In this paper an improved hierarchy result is established which still works if the number of nondeterministic decisions is O((n/ log 1+e n) 1/4 ) , where e > 0 is an arbitrary small constant.

On P versus NP $ \cap $ co-NP for decision trees and read-once branching programs

It is known that if a Boolean function f in n variables has a DNF and a CNF of size $ \le N $ then f also has a (deterministic) decision tree of size exp(O(log n log2 N)). We show that this simulation cannot be made polynomial: we exhibit explicit Boolean functions f that require deterministic trees of size exp $ (\Omega({\rm log^2} N)) $ where N is the total number of monomials in minimal DNFs for f and ¬f. Moreover, we exhibit new examples of explicit Boolean functions that require deterministic read-once branching programs of exponential size whereas both the functions and their negations have small nondeterministic read-once branching programs. One example results from the Bruen—Blokhuis bound on the size of nontrivial blocking sets in projective planes: it is remarkably simple and combinatorially clear. Other examples have the additional property that f is in AC0.