We consider setting up sleep scheduling in sensor networks. We formulate the problem as an instance of the fractional domatic partition problem and obtain a distributed approximation algorithm by applying linear programming approximation techniques. Our algorithm is an application of the Garg-Könemann (GK) scheme that requires solving an instance of the minimum weight dominating set (MWDS) prob...

Given an integer k ≥ 2, we consider the problem of computing the smallest real number t(k) such that for each set P of points in the plane, there exists a t(k)-spanner for P that has chromatic number at most k. We prove that t(2) = 3, t(3) = 2, t(4) = √ 2, and give upper and lower bounds on t(k) for k > 4. We also show that for any > 0, there exists a (1 + )t(k)-spanner for P that has O(|P |) e...

let $k$ be a positive integer. a subset $s$ of $v(g)$ in a graph $g$ is a $k$-tuple total dominating set of $g$ if every vertex of $g$ has at least $k$ neighbors in $s$. the $k$-tuple total domination number $gamma _{times k,t}(g)$ of $g$ is the minimum cardinality of a $k$-tuple total dominating set of $g$. if$v(g)=v^{0}={v_{1}^{0},v_{2}^{0},ldots ,v_{n}^{0}}$ and $e(g)=e_{0}$, then for any in...

We exploit the close relationship between circular arc graphs and interval graphs to design efficient approximation algorithms for NP-hard Optimization problems on circular arc graphs. The problems considered here are maximum domatic partition and on-line minimum vertex coloring. We present a heuristic for the domatic partition problem with a performance ratio of 4. For on-line coloring, we con...

In this note, we provide a sharp upper bound on the rainbow connection number of tournaments of diameter 2. For a tournament T of diameter 2, we show 2 ≤ − →rc(T ) ≤ 3. Furthermore, we provide a general upper bound on the rainbow k-connection number of tournaments as a simple example of the probabilistic method. Finally, we show that an edge-colored tournament of kth diameter 2 has rainbow k-co...

We consider four different types of multiple domination and provide new improved upper bounds for the kand k-tuple domination numbers. They generalise two classical bounds for the domination number and are better than a number of known upper bounds for these two multiple domination parameters. Also, we explicitly present and systematize randomized algorithms for finding multiple dominating sets...

Suppose K is an unknot lying in the 1-skeleton of a triangulated 3-manifold with t tetrahedra. Hass and Lagarias showed there is an upper bound, depending only on t, for the minimal number of elementary moves to untangle K. We give a simpler proof, utilizing a normal form for surfaces whose boundary is contained in the 1-skeleton of a triangulated 3-manifold. We also obtain a significantly bett...

Using the new approach of braiding sequences we give a proof of the Lin-Wang conjecture, stating that a Vassiliev invariant v of degree k has a value Ov(c(K)k) on a knot K, where c(K) is the crossing number of K and Ov depends on v only. We extend our method to give a quadratic upper bound in k for the crossing number of alternating/positive knots, the values on which suffice to determine uniqu...

We consider the problem of constructing Concurrent Zero Knowledge Proofs [6], in which the fascinating and useful “zero knowledge” property is guaranteed even in situations where multiple concurrent proof sessions are executed with many colluding dishonest verifiers. Canetti et al. [3] show that blackbox concurrent zero knowledge proofs for non-trivial languages require Ω̃(log k) rounds where k ...