Width Hierarchy for k-OBDD of Small Width

In this paper was explored well known model k-OBDD. There are proven width based hierarchy of classes of boolean functions which computed by k-OBDD. The proof of hierarchy is based on sufficient condition of Boolean function's non representation as k-OBDD and complexity properties of Boolean function SAF. This function is modification of known Pointer Jumping (PJ) and Indirect Storage Access (ISA) functions.Comment: 8 page

Finding the Median (Obliviously) with Bounded Space

We prove that any oblivious algorithm using space $S$ to find the median of a list of $n$ integers from $\{1,...,2n\}$ requires time $\Omega(n \log\log_S n)$. This bound also applies to the problem of determining whether the median is odd or even. It is nearly optimal since Chan, following Munro and Raman, has shown that there is a (randomized) selection algorithm using only $s$ registers, each of which can store an input value or $O(\log n)$-bit counter, that makes only $O(\log\log_s n)$ passes over the input. The bound also implies a size lower bound for read-once branching programs computing the low order bit of the median and implies the analog of $P \ne NP \cap coNP$ for length $o(n \log\log n)$ oblivious branching programs

A lower bound on branching programs reading some bits twice

AbstractBy (1, + k(n))-branching programs (b.p.'s) we mean those b.p.'s which during each of their computations are allowed to test at most k(n) input bits repeatedly. For a Boolean function computable within polynomial time a trade-off is presented between the size and the number of repeatedly tested input bits of any b.p. P computing the function. Namely, if at most k(n) repeated tests are allowed, where log2 n ⩽ k(n) ⩽ n(1000 log2 n), then the size of P is at least exp(Ω(n(k(n)log2 n))12). This is exponential whenever k(n) ⩽ nα for a fixed α < 1 and superpolynomial whenever k(n) = o(nlog32 n).The presented result is a step towards a superpolynomial lower bound for 2-b.p.'s which is an open problem since 1984 when the first superpolynomial lower bounds for 1-b.p.'s were proven (Wegener, 1988; Žák, 1984). The present result is an improvement on (Žák, 1995)

No small nondeterministic read-once branching programs for CNFs of bounded treewidth

In this paper, given a parameter k, we demonstrate an infinite class of cnfs of treewidth at most k of their primal graphs such that equivalent nondeterministic read-once branching programs (nrobps) are of size at least nck for some universal constant c. Thus we rule out the possibility of fixed-parameter tractable space complexity of nrobps parameterized by the smallest treewidth of equivalent cnfs

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

The satisfiability problem for probabilistic ordered branching programs

We show that the satisfiability problem for bounded-error probabilistic ordered branching programs is \NP -complete. If the error is very small, however (more precisely, if the error is bounded by the reciprocal of the width of the branching program), then we have a polynomial-time algorithm for the satisfiability problem