951 research outputs found
Time-space trade-offs for branching programs
AbstractBranching program depth and the logarithm of branching program complexity are lower bounds on time and space requirements for any reasonable model of sequential computation. In order to gain more insight to the complexity of branching programs and to the problems of time-space trade-offs one considers, on one hand, width-restricted and, on the other hand, depth-restricted branching programs. We present these computation models and the trade-off results already proved. We prove a new result of this type by presenting an effectively defined Boolean function whose complexity in depth-restricted one-time-only branching programs is exponential while its complexity even in width-2 branching programs is polynomial
An exponential lower bound for real-time branching programs
Branching programs are a general model of sequential computation. One of their computational features is their possibility to question (repeatedly) the information from each input bit. Real-time branching programs make at most n questions when computing on an input of length n. The restriction “real-time” allows to find a simple language which requires the lower bound 2√2n/8 on memory (= the state space)
Yet harder knapsack problems
AbstractAlready 30 years ago, Chvátal has shown that some instances of the zero-one knapsack problem cannot be solved in polynomial time using a particular type of branch-and-bound algorithms based on relaxations of linear programs together with some rudimentary cutting-plane arguments as bounding rules. We extend this result by proving an exponential lower bound in a more general class of branch-and-bound and dynamic programming algorithms which are allowed to use memoization and arbitrarily powerful bound rules to detect and remove subproblems leading to no optimal solution
On uncertainty versus size in branching programs
AbstractWe propose an information-theoretic approach to proving lower bounds on the size of branching programs. The argument is based on Kraft type inequalities for the average amount of uncertainty about (or entropy of) a given input during the various stages of computation. The uncertainty is measured by the average depth of so-called ‘splitting trees’ for sets of inputs reaching particular nodes of the program.We first demonstrate the approach for read-once branching programs. Then, we introduce a strictly larger class of so-called ‘balanced’ branching programs and, using the suggested approach, prove that some explicit Boolean functions cannot be computed by balanced programs of polynomial size. These lower bounds are new since some explicit functions, which are known to be hard for most previously considered restricted classes of branching programs, can be easily computed by balanced branching programs of polynomial size
Parameterized Compilation Lower Bounds for Restricted CNF-formulas
We show unconditional parameterized lower bounds in the area of knowledge
compilation, more specifically on the size of circuits in decomposable negation
normal form (DNNF) that encode CNF-formulas restricted by several graph width
measures. In particular, we show that
- there are CNF formulas of size and modular incidence treewidth
whose smallest DNNF-encoding has size , and
- there are CNF formulas of size and incidence neighborhood diversity
whose smallest DNNF-encoding has size .
These results complement recent upper bounds for compiling CNF into DNNF and
strengthen---quantitatively and qualitatively---known conditional low\-er
bounds for cliquewidth. Moreover, they show that, unlike for many graph
problems, the parameters considered here behave significantly differently from
treewidth
A Note on Amortized Branching Program Complexity
In this paper, we show that while almost all functions require exponential size branching programs to compute, for all functions f there is a branching program computing a doubly exponential number of copies of f which has linear size per copy of f. This result disproves a conjecture about non-uniform catalytic computation, rules out a certain type of bottleneck argument for proving non-monotone space lower bounds, and can be thought of as a constructive analogue of Razborov\u27s result that submodular complexity measures have maximum value O(n)
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