This paper studies an optimization problem that arises in the context of distributed resource allocation: Given a conflict graph that represents the competition of processors over resources, we seek an allocation under which no two jobs with conflicting requirements are executed simultaneously. Our objective is to minimize the average response time of the system. In alternative formulation this is known as the Minimum Color Sum (MCS) problem [24]. We show, that the algorithm based on finding iteratively a maximum independent set (MaxIS) is a 4-approximation to the MCS. This bound is tight to within a factor of 2. We give improved ratios for the classes of bipartite, bounded-degree, and line graphs. The bound generalizes to a 4ae-approximation of MCS for classes of graphs for which the maximum independent set problem can be approximated within a factor of ae. On the other hand, we show that an n1 \Gamma ffl-approximation is NP-hard, for some ffl? 0. For some instances of the resource allocation problem, such as the Dining Philosophers, an efficient solution requires edge coloring of the conflict graph. We introduce the Minimum Edge Color Sum (MECS) problem which is shown to be NP-hard. We show that a 2-approximation to MECS(G) can be obtained distributively using compact coloring within O(log² n) communication rounds.

... for short, is one of the ~implest, most efficient, and most thoroughly studied methods for finding independent sets in graphs. We show that it surprisingly achieves a performance ratio of (A+ 2)/3 for approximating inde-pendent sets in graphs with degree bounded by A. The analysis directs us towards a simple parallel and dis-tributed algorithm with identical performance, which on constant-degree graphs runs in O(log ” n) time us-ing linear number of processors. We also analyze the Greedy algorithm when run in combination with a frac-tional relaxation technique of Nemhauser and Trotter, and obtain an improved (2Z + 3)/5 performance ratio on graphs with average degree ~. Finally, we introduce a generally applicable technique for improving the approximation ratios of independent set algorithms, and illustrate it by improving the per-formance ratio of Greedy for large ∆.

We consider the problem of approximating the independence number and the chromatic number of k-uniform hypergraphs on n vertices. For xed integers k 2, we obtain for both problems that one can achieve in polynomial time approximation ratios of at most O(n=(log (k,1) n) 2). This extends results of Boppana and Halldorsson [5] who showed for the graph case that an approximation ratio of O(n=(log n) 2) can be achieved in polynomial time. On the other hand, assuming NP 6 = ZPP, one cannot obtain in polynomial time for the independence number and the chromatic number of k-uniform hypergraphs an approximation ratio of n 1, for xed>0. 1

Abstract. The paper considers broadcasting protocols in radio networks with known topology that are efficient in both time and energy. The radio network is modelled as an undirected graph G = (V, E) where |V | = n. It is assumed that during execution of the communication task every node in V is allowed to transmit at most once. Under this assumption it is shown that any radio broadcast protocol requires D + Ω ( √ n − D) transmission rounds, where D is the diameter of G. This lower bound is complemented with an efficient construction of a deterministic protocol that accomplishes broadcasting in D + O ( √ n log n) rounds. Moreover, if we allow each node to transmit at most k times, the lower bound D + Ω((n − D) 1/(2k) ) on the number of transmission rounds holds. We also provide a randomised protocol that accomplishes broadcasting in D + O(kn 1/(k−2) log 2 n) rounds. The paper concludes with a discussion of several other strategies for energy efficient radio broadcasting and a number of open problems in the area.

A basic problem in hypergraphs is that of finding a large independent set–one of guaranteed size–in a given hypergraph. Understanding the parallel complexity of this and related independent set problems on hypergraphs is a fundamental open issue in parallel computation. Caro and Tuza (J. Graph Theory, Vol. 15, pp. 99–107, 1991) have shown a certain lower bound αk(H) on the size of a maximum independent set in a given k-uniform hypergraph H, and have also presented an efficient sequential algorithm to find an independent set of size αk(H). They also show that αk(H) is the size of the maximum independent set for various hypergraph families. Here, we develop the first RNC algorithm to find an independent set of size αk(H), and also derandomize it for various special cases. We also present lower bounds on independent set size and corresponding RNC algorithms for non-uniform hypergraphs.

Abstract The paper considers broadcasting protocols in radio networks with known topology that are efficient in both time and energy. The radio network is modelled as an undirected graph G = (V, E) where |V |=n. It is assumed that during execution of the communication task every node in V is allowed to transmit at most once. Under this assumption it is shown that any radio broadcast protocol requires D + � ( √ n − D) transmission rounds, where D is the diameter of G. This lower bound is complemented with an efficient construction of a deterministic protocol that accomplishes broadcasting in D + O ( √ n log n) rounds. Moreover, if we allow each node to transmit at most k times, the lower bound D + �((n − D) 1/(2k) ) on the number of transmission rounds holds. We also provide a randomised protocol that accomplishes broadcasting in D + O(kn1/(k−2) log2 n) rounds. The paper concludes with a number of open problems in the area.