Beate Bollig

On the effect of local changes in the variable ordering of ordered decision diagrams

Ordered binary and functional decision diagrams (OBDDs and OFDDs) have turned out to be useful representations of Boolean functions with many applications. Since it is NP-hard to compute an optimal variable ordering or to improve a given variable ordering, local search and simulated annealing algorithms are used to improve variable orderings. The effect of local changes in variable orderings is studied and a simulated annealing algorithm for the computation of good variable orderings is presented. For several benchmark circuits the best known variable orderings are improved significantly.

Hierarchy theorems for k OBDDs and k IBDDs

Almost the same types of restricted branching programs (or binary decision diagrams BDDs) are considered in complexity theory and in applications like hardware verification. These models are read-once branching programs (free BDDs) and certain types of oblivious branching programs (ordered and indexed BDDs with k layers). The complexity of the satisfiability problem for these restricted branching programs is investigated and tight hierarchy results are proved for the classes of functions representable by k layers of ordered or indexed BDDs of polynomial size.

Parity graph-driven 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. Exponential lower bounds for 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 the so-called 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 parity nonoblivious read-once branching program model for an explicitly defined Boolean function. In addition, more insight into the structure of integer multiplication is yielded.

On the complexity of the hidden weighted bit function for various BDD models

Ordered binary decision diagrams (OBDDs) and several more general BDD models have turned out to be representations of Boolean functions which are useful in applications like verification, timing analysis, test pattern generation or combinatorial optimization. The hidden weighted bit function (HWB) is of particular interest, since it seems to be the simplest function with exponential OBDD size. The complexity of this function with respect to different circuit models, formulas, and various BDD models is discussed.

A very simple function that requires exponential size read-once branching programs

Abstract A Boolean function in n variables is presented that is computable in depth 2 monotone AC0 and has prime implicants of length 2 only but requires 2 Ω(√n) size read-once branching programs. The function considered is defined using (1, 1)-disjoint Boolean sums and the solution of the famous problem of Zarankiewicz.

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.

On the OBDD complexity of the most significant bit of integer multiplication

Integer multiplication as one of the basic arithmetic functions has been in the focus of several complexity theoretical investigations. Ordered binary decision diagrams (OBDDs) are one of the most common dynamic data structures for boolean functions. Among the many areas of application are verification, model checking, computer-aided design, relational algebra, and symbolic graph algorithms. In this paper it is shown that the OBDD complexity of the most significant bit of integer multiplication is exponential answering an open question posed by Wegener (2000) [18].

Implicit computation of maximum bipartite matchings by sublinear functional operations

The maximum bipartite matching problem, an important problem in combinatorial optimization, has been studied for a long time. In order to solve problems for very large structured graphs in reasonable time and space, implicit algorithms have been investigated. Any object to be manipulated is binary encoded and problems have to be solved mainly by functional operations on the corresponding Boolean functions. OBDDs are a popular data structure for Boolean functions, therefore, OBDD-based algorithms have been used as a heuristic approach to handle large input graphs. Here, two OBDD-based maximum bipartite matching algorithms are presented, which are the first ones using only a sublinear number of operations (with respect to the number of vertices of the input graph) for a problem unknown to be in NC, the complexity class that contains all problems computable in deterministic polylogarithmic time with polynomially many processors. Furthermore, the algorithms are experimentally evaluated.

Complexity Theoretical Aspects of OFDDs

Experimental results have shown that OFDDs (ordered functional decision diagrams) are a representation of Boolean functions which are sometimes superior to OBDDs (ordered binary decision diagrams). Most of the complexity theoretical problems have been solved for OBDDs. Here some results for OFDDs are proved. It is NP-complete to decide whether a function represented by some OFDD can be represented by an OFDD of size s using another variable ordering. Given an OFDD representation for an incompletely specified function, it is NP-hard to compute an optimal OFDD cover for this function respecting the same variable ordering. The replacement of variables by constants may cause an exponential blow-up of the OFDD size. Finally, it is investigated how a local change of the variable ordering may change the OFDD size. This leads to simulated annealing algorithms to improve variable orderings.

Larger lower bounds on the OBDD complexity of integer multiplication

Integer multiplication as one of the basic arithmetic functions has been in the focus of several complexity theoretical investigations and ordered binary decision diagrams (OBDDs) are one of the most common dynamic data structures for Boolean functions. Recently, the question whether the OBDD complexity of the most significant bit of integer multiplication is exponential has been answered affirmatively. In this paper a larger general lower bound is presented using a simpler proof. Furthermore, we prove a larger lower bound for the variable order assumed to be one of the best ones for the most significant bit. Moreover, the best known lower bound on the OBDD complexity for the so-called graph of integer multiplication is improved.