Search

A monotone Boolean circuit is composed of OR gates, AND gates and input gates corresponding to the input variables and the Boolean constants. It is $q$-multilinear if for each its output gate $o$ and for each prime implicant $s$ of the function computed at $o$, the arithmetic version of the circuit resulting from the replacement of OR and AND gates by addition and multiplication gates, respectively, computes a polynomial at $o$ which contains a monomial including the same variables as $s$ and each of the variables in $s$ has degree at most $q$ in the monomial. First, we study the complexity of computing semi-disjoint bilinear Boolean forms in terms of the size of monotone $q$-multilinear Boolean circuits. In particular, we show that any monotone $1$-multilinear Boolean circuit computing a semi-disjoint Boolean form with $p$ prime implicants includes at least $p$ AND gates. We also show that any monotone $q$-multilinear Boolean circuit computing a semi-disjoint Boolean form with $p$ prime implicants has $\Omega(\frac p {q^4})$ size. Next, we study the complexity of the monotone Boolean function $Isol_{k,n}$ that verifies if a $k$-dimensional Boolean matrix has at least one $1$ in each line (e.g., each row and column when $k=2$), in terms of monotone $q$-multilinear Boolean circuits. We show that that any $\Sigma_3$ monotone Boolean circuit for $Isol_{k,n}$ has an exponential in $n$ size or it is not $(k-1)$-multilinear., Comment: 15 pages, preliminary version in proceedings of SOFSEM 2023

We consider the fundamental problem of detecting/counting copies of a fixed pattern graph in a host graph. The recent progress on this problem has not included complete pattern graphs, i.e., cliques (and their complements, i.e., edge-free pattern graphs, in the induced setting). The fastest algorithms for the aforementioned patterns are based on a straightforward reduction to triangle detection/counting. We provide an alternative method of detection/counting copies of fixed size cliques based on a multi-dimensional matrix product. It is at least as time efficient as the triangle method in cases of $K_4$ and $K_5.$ The complexity of the multi-dimensional matrix product is of interest in its own rights. We provide also another alternative method for detection/counting $K_r$ copies, again time efficient for $r\in \{ 4,\ 5 \}$., Comment: 12 pages

The {\em parallel time} of a population protocol is defined as the average number of required interactions that an agent in the protocol participates, i.e., the quotient between the total number of interactions required by the protocol and the total number $n$ of agents, or just roughly the number of required rounds with $n$ interactions. This naming triggers an intuition that at least on the average a round of $n$ interactions can be implemented in $O(1)$ parallel steps. We show that when the transition function of a population protocol is treated as a black box then the expected maximum number of parallel steps necessary to implement a round of $n$ interactions is $\Omega (\frac {\log n}{\log \log n})$. We also provide a combinatorial argument for a matching upper bound on the number of parallel steps in the average case under additional assumptions., Comment: 8 pages

We present implications from the known conjectures like APSP, 3SUM and ETH in a form of a negated containment of a linear-time with a non-deterministic logarithmic-bit oracle in a respective deterministic bounded-time class They are different for different conjectures and they exhibit in particular the dependency on the input range parameters., Comment: The section on range reduction in a previous version contained a flaw in a proof and therefore it has been removed

The number of triangulations of a planar n point set is known to be $c^n$, where the base $c$ lies between $2.43$ and $30.$ The fastest known algorithm for counting triangulations of a planar n point set runs in $O^*(2^n)$ time. The fastest known arbitrarily close approximation algorithm for the base of the number of triangulations of a planar n point set runs in time subexponential in $n.$ We present the first quasi-polynomial approximation scheme for the base of the number of triangulations of a planar point set.

The exact complexity of geometric cuts and bisections is the longstanding open problem including even the dimension one. In this paper, we resolve this problem for dimension one (the real line) by designing an exact polynomial time algorithm. Our results depend on a new technique of dealing with metric equalities and their connection to dynamic programming. The method of our solution could be also of independent interest.

We consider the problem of computing all-pairs shortest paths in a directed graph with real weights assigned to vertices. For an $n\times n$ 0-1 matrix $C,$ let $K_{C}$ be the complete weighted graph on the rows of $C$ where the weight of an edge between two rows is equal to their Hamming distance. Let $MWT(C)$ be the weight of a minimum weight spanning tree of $K_{C}.$ We show that the all-pairs shortest path problem for a directed graph $G$ on $n$ vertices with nonnegative real weights and adjacency matrix $A_G$ can be solved by a combinatorial randomized algorithm in time $$\widetilde{O}(n^{2}\sqrt {n + \min\{MWT(A_G), MWT(A_G^t)\}})$$ As a corollary, we conclude that the transitive closure of a directed graph $G$ can be computed by a combinatorial randomized algorithm in the aforementioned time. $\widetilde{O}(n^{2}\sqrt {n + \min\{MWT(A_G), MWT(A_G^t)\}})$ We also conclude that the all-pairs shortest path problem for uniform disk graphs, with nonnegative real vertex weights, induced by point sets of bounded density within a unit square can be solved in time $\widetilde{O}(n^{2.75})$.

In this paper, we establish two necessary conditions for a joint triangulation of two sets of $n$ points in the plane and conjecture that they are sufficient. We show that these necessary conditions can be tested in $O(n^3)$ time. For the problem of a joint triangulation of two simple polygons of $n$ vertices, we propose an $O(n^3)$ time algorithm for constructing a joint triangulation using dynamic programming., Comment: The extended abstrat of this paper appeared in the Proceedings of India-Taiwan Conference on Discrete Mathematics, Taipei, pp. 34-43, 2009

First, we study geometric variants of the standard set cover motivated by assignment of directional antenna and shipping with deadlines, providing the first known polynomial-time exact solutions. Next, we consider the following general capacitated set cover problem. There is given a set of elements with real weights and a family S of sets of elements. One can use a set if it is a subset of one of the sets on our lists and the sum of weights is at most one. The goal is to cover all the elements with the allowed sets.We show that any polynomial-time algorithm that approximates the un-capacitated version of the set cover problem with ratio r can be converted to an approximation algorithm for the capacitated version with ratio r + 1.357.In particular, the composition of these two results yields a polynomial-time approximation algorithm for the problem of covering a set of customers represented by a weighted n-point set with a minimum number of antennas of variable angular range and fixed capacity with ratio 2.357. Finally, we provide a PTAS for the dual problem where the number of sets (e.g., antennas) to use is fixed and the task is to minimize the maximum set load, in case the sets correspond to line intervals or arcs.