We develop a new randomized rounding approach for fractional vectors defined on the edge-sets of bipartite graphs. We show various ways of combining this technique with other ideas, leading to improved (approximation) algorithms for various problems. These include: -low congestion multi-path routing; -richer random-graph models for graphs with a given degree-sequence; -improved approximation algorithms for: (i) throughput-maximization in broadcast scheduling, (ii) delay-minimization in broadcast scheduling, as well as (iii) capacitated vertex cover; and -fair scheduling of jobs on unrelated parallel machines.By the induction hypothesis, Pr [S 1 ∪ S 2 ∪ {x}] ≤ Pr [S 1 ∪ {x}]Pr [S 2 ] and hence (6) follows. An identical argument holds for the case where p 1 , q 1 ∈ S 2 .Case 3. p 1 ∈ S 1 and q 1 ∈ E \ (S 1 ∪ S 2 ). Let S 1 = S 1 \ {p 1 }. Exactly one of two events happens during the iteration.Event A. p 1 gets rounded to zero. In this case,By the induction hypothesis, Pr [S 1 ∪ S 2 ] ≤ Pr [S 1 ]Pr [S 2 ] and hence (6) follows. Event B. p 1 does not get rounded to zero. In this case, Pr[S 1 | T ] = Pr [S 1 ∪ {x}] Pr[S 2 | T ] = Pr [S 2 ] Pr[S 1 ∪ S 2 | T ] = Pr [S 1 ∪ S 2 ∪ {x}]. By the induction hypothesis, Pr [S 1 ∪ S 2 ∪ {x}] ≤ Pr [S 1 ∪ {x}]Pr [S 2 ] and hence (6) follows. Case 4. { p 1 , q 1 } ⊆ E \(S 1 ∪ S 2 ). In this case, Pr[S 1 | T ] = Pr [S 1 ], Pr[S 2 | T ] = Pr [S 2 ] and Pr[S 1 ∪ S 2 | T ] = Pr [S 1 ∪ S 2 ]; we are done by the induction hypothesis. This completes the proof of Theorem 2.4.

No abstract

We study a generalization of covering problems called partial covering. Here we wish to cover only a desired number of elements, rather than covering all elements as in standard covering problems. For example, in k-partial set cover, we wish to choose a minimum number of sets to cover at least k elements. For k-partial set cover, if each element occurs in at most f sets, then we derive a primal-dual f -approximation algorithm (thus implying a 2-approximation for k-partial vertex cover) in polynomial time. Without making any assumption about the number of sets an element is in, for instances where each set has cardinality at most three, we obtain an approximation of 4/3. We also present better-than-2-approximation algorithms for k-partial vertex cover on bounded degree graphs, and for vertex cover on expanders of bounded average degree. We obtain a polynomial-time approximation scheme for k-partial vertex cover on planar graphs, and for covering k points in R d by disks.

Network wide broadcasting is a fundamental operation in ad hoc networks. In broadcasting, a source node sends a message to all the other nodes in the network. In this paper, we consider the problem of collision-free broadcasting in ad hoc wireless networks. Our objective is to minimize the latency and the number of retransmissions in the broadcast. We show that minimum latency broadcasting is NP-hard for ad hoc wireless networks. We also present a simple and distributed collision-free broadcasting algorithm for broadcasting a message. For networks with bounded node transmission ranges, our algorithm simultaneously guarantees that the latency and the number of retransmissions are within O(1) times their respective optimal values. Our algorithm and analysis extends to the case when multiple messages are broadcast from multiple sources. Experimental studies indicate that our algorithms perform much better in practice than the analytical guarantees provided for the worst case.

We study the generalization of covering problems to partial covering. Here we wish to cover only a desired number of elements, rather than covering all elements as in standard covering problems. For example, in k-set cover, we wish to choose a minimum number of sets to cover at least k elements. For k-set cover, if each element occurs in at most f sets, then we derive a primal-dual f-approximation algorithm (thus implying a 2-approximation for k-vertex cover) in polynomial time. In addition to its simplicity, this algorithm has the advantage of being parallelizable. For instances where each set has cardinality at most three, we obtain an approximation of 4=3. We also present better-than-2-approximation algorithms for k-vertex cover on bounded degree graphs, and for vertex cover on expanders of bounded average degree. We obtain a polynomial-time approximation scheme for k-vertex cover on planar graphs, and for covering points in R d by disks.