We consider so-called ``incremental'' dynamic programming (DP) algorithms, and are interested in the number of subproblems produced by them. The standard DP algorithm for the n-dimensional Knapsack problem is incremental, and produces nK subproblems, where K is the capacity of the knapsack. We show that any incremental algorithm for this ... more >>>

Motivated by its relation to the length of cutting plane proofs for the Maximum Biclique problem, here we consider the following communication game on a given graph G, known to both players. Let K be the maximal number of vertices in a complete bipartite subgraph of G (which is not ... more >>>

Khrapchenko's classical lower bound $n^2$ on the formula size of the
parity function~$f$ can be interpreted as designing a suitable
measure of subrectangles of the combinatorial rectangle
$f^{-1}(0)\times f^{-1}(1)$. Trying to generalize this approach we
arrived at the concept of \emph{convex measures}. We prove the
... more >>>

A completion of an m-by-n matrix A with entries in {0,1,*} is obtained
by setting all *-entries to constants 0 or 1. A system of semi-linear
equations over GF(2) has the form Mx=f(x), where M is a completion of
A and f:{0,1}^n --> {0,1}^m is an operator, the i-th coordinate ... more >>>

In this note we consider unbounded fanin depth-2 circuits with arbitrary boolean functions as gates.

We define the entropy of an operator f:{0,1}^n --> {0,1}^m is as the logarithm of the maximum number of vectors distinguishable by at least one special subfunction of f. Then we prove that every depth-2 ... more >>>

The \emph{replication number} of a branching program is the minimum
number R such that along every accepting computation at most R
variables are tested more than once. Hence 0\leq R\leq n for every
branching program in n variables. The best results so far were
exponential lower ... more >>>

We consider the minimal number of AND and OR gates in monotone
circuits for quadratic boolean functions, i.e. disjunctions of
length-$2$ monomials. The single level conjecture claims that
monotone single level circuits, i.e. circuits which have only one
level of AND gates, for quadratic functions are ... more >>>

We consider the P versus NP\cap coNP question for the classical two-party communication protocols: if both a boolean function and its negation have small nondeterministic communication complexity, what is then its deterministic and/or probabilistic communication complexity? In the fixed (worst) partition case this question was answered by Aho, Ullman and ... more >>>

A boolean circuit $f(x_1,\ldots,x_n)$ \emph{represents} a graph $G$
on $n$ vertices if for every input vector $a\in\{0,1\}^n$ with
precisely two $1$'s in, say, positions $i$ and $j$, $f(a)=1$
precisely when $i$ and $j$ are adjacent in $G$; on inputs with more
or less than two $1$'s ... more >>>

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 ... more >>>

In 1957 Markov proved that every circuit in $n$ variables
can be simulated by a circuit with at most $\log(n+1)$ negations.
In 1974 Fischer has shown that this can be done with only
polynomial increase in size.

In this note we observe that some explicit monotone functions
more >>>

We show that recognizing the $K_3$-freeness and $K_4$-freeness of
graphs is hard, respectively, for two-player nondeterministic
communication protocols with exponentially many partitions and for
nondeterministic (syntactic) read-$s$ times branching programs.

The key ingradient is a generalization of a coloring lemma, due to
Papadimitriou and Sipser, which says that for every ... more >>>

We propose an information-theoretic approach to proving lower
bounds on the size of branching programs. The argument is based on
Kraft-McMillan 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
more >>>

We consider a general model of monotone circuits, which
we call d-local. In these circuits we allow as gates:
(i) arbitrary monotone Boolean functions whose minterms or
maxterms (or both) have length at most <i>d</i>, and
(ii) arbitrary real-valued non-decreasing functions on ... more >>>

We propose an information-theoretic approach to proving
lower bounds on the size of branching programs (b.p.). The argument
is based on Kraft-McMillan type inequalities for the average amount of
uncertainty about (or entropy of) a given input during various
stages of the computation.

We ... more >>>

In a semantic resolution proof we operate with clauses only
but allow {\em arbitrary} rules of inference:

C_1 C_2 ... C_m
__________________
C

Consistency is the only requirement. We prove a very simple
exponential lower bound for the size of bounded fanin semantic
... more >>>

We first consider so-called {\em $(1,+s)$-branching programs}
in which along every consistent path at most $s$ variables are tested
more than once. We prove that any such program computing a
characteristic function of a linear code $C$ has size at least
... more >>>

We prove a general combinatorial lower bound on the
size of monotone circuits. The argument is different from
Razborov's method of approximation, and is based on Sipser's
notion of `finite limit' and Haken's `counting bottlenecks' idea.
We then apply this criterion to the ... more >>>

We introduce the model of conservative one-way multiparty complexity
and prove lower and upper bounds on the complexity of pointer jumping.

The pointer jumping function takes as its input a directed layered
graph with a starting node and layers of nodes, and a single edge
more >>>

A syntactic read-k times branching program has the restriction
that no variable occurs more than k times on any path (whether or not
consistent). We exhibit an explicit Boolean function f which cannot
be computed by nondeterministic syntactic read-k times branching programs
of size less than exp(\sqrt{n}}k^{-2k}), ... more >>>