Showing papers by "Aravind Srinivasan published in 2001"

Distributions on level-sets with applications to approximation algorithms

Abstract: We consider a family of distributions on fixed-weight vectors in {0, 1}/sup t/; these distributions enjoy certain negative correlation properties and also satisfy pre-specified conditions on their marginal distributions. We show the existence of such families, and present a linear-time algorithm to sample from them. This yields improved approximation algorithms for the following problems: (a) low-congestion multi-path routing; (b) maximum coverage versions of set cover; (c) partial vertex cover problems for bounded-degree graphs; and (d) the Group Steiner Tree problem. For (a) and (b), the improvement is in the approximation ratio; for (c), we show how to speedup existing approximation algorithms while preserving the best-known approximation ratio; we also improve the approximation ratio for certain families of instances of unbounded degree. For (d), we derive an approximation algorithm whose approximation guarantee is at least as good as what is known; our algorithm is shown to have a better approximation guarantee for the worst known input families for existing algorithms.

Approximation Algorithms for Partial Covering Problems

Abstract: 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 Rd by disks.

New approaches to covering and packing problems

Abstract: Covering and packing integer programs model a large family of combinatorial optimization problems. The current-best approximation algorithms for these are an instance of the basic probabilistic method: showing that a certain randomized approach produces a good approximation with positive probability. This approach seems inherently sequential; by employing the method of alteration we present the first RNC and NC approximation algorithms that match the best sequential guarantees. Extending our approach, we get the first RNC and NC approximation algorithms for certain multi-criteria versions of these problems. We also present the first NC algorithms for two packing and covering problems that are not subsumed by the above result: finding large independent sets in graphs, and rounding fractional Group Steiner solutions on trees.

Better Approximation Guarantees for Job-Shop Scheduling

Abstract: Job-shop scheduling is a classical NP-hard problem. Shmoys, Stein, and Wein presented the first polynomial-time approximation algorithm for this problem that has a good (polylogarithmic) approximation guarantee. We improve the approximation guarantee of their work and present further improvements for some important NP-hard special cases of this problem (e.g., in the preemptive case where machines can suspend work on operations and later resume). We also present NC algorithms with improved approximation guarantees for some NP-hard special cases.

A Constant-Factor Approximation Algorithm for Packet Routing and Balancing Local vs. Global Criteria

Abstract: We present the first constant-factor approximation algorithm for a fundamental problem: the store-and-forward packet routing problem on arbitrary networks. Furthermore, the queue sizes required at the edges are bounded by an absolute constant. Thus, this algorithm balances a global criterion (routing time) with a local criterion (maximum queue size) and shows how to get simultaneous good bounds for both. For this particular problem, approximating the routing time well, even without considering the queue sizes, was open. We then consider a class of such local vs. global problems in the context of covering integer programs and show how to improve the local criterion by a logarithmic factor by losing a constant factor in the global criterion.

The one-inclusion graph algorithm is near-optimal for the prediction model of learning

Abstract: Haussler, Littlestone and Warmuth (1994) described a general-purpose algorithm for learning according to the prediction model, and proved an upper bound on the probability that their algorithm makes a mistake in terms of the number of examples seen and the Vapnik-Chervonenkis (VC) dimension of the concept class being learned We show that their bound is within a factor of 1+o(1) of the best possible such bound for any algorithm

Finding large independent sets of hypergraphs in parallel

Abstract: 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 efficien 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.

Experimental Analysis of Algorithms for Bilateral-Contract Clearing Mechanisms Arising in Deregulated Power Industry

Abstract: We consider the bilateral contract satisfaction problem arising from electrical power networks due to the proposed deregulation of the electric utility industry in the USA. Given a network and a (multi)set of pairs of vertices (contracts) with associated demands, the goal is to find the maximum number of simultaneously satisfiable contracts. We study how four different algorithms perform in fairly realistic settings; we use an approximate electrical power network from Colorado. Our experiments show that three heuristics outperform a theoretically better algorithm. We also test the algorithms on four types of scenarios that are likely to occur in a deregulated marketplace. Our results show that the networks that are adequate in a regulated marketplace might be inadequate for satisfying all the bilateral contracts in a deregulated industry.

Efficient algorithms for location and sizing problems in network design

Abstract: Large-scale location, sizing and homing problems of distributed network elements, have received much attention recently due to the massive deployment of broadband communication networks for services like Internet telephony and Web caching. Key considerations in designing these networks include modularity of capacity, economies of scale in cost, and reliability. We formulate a general class of such network design problems as Mixed-Integer Programs. These problems are computationally intractable in general; under various asymptotic conditions, we show how to compute near-optimal solutions. To deal with arbitrary instances, we develop new algorithms based on linear programming, as well as greedy randomized adaptive search. These algorithms achieved near-optimal solutions with reasonable computation time for our experiments.

Domatic partitions and the Lovász local lemma

Abstract: We resolve the problem posed as the main open question in [4]: letting δ(G), ∆(G) and D(G) respectively denote the minimum degree, maximum degree, and domatic number (defined below) of an undirected graph G = (V,E), we show that D(G) ≥ (1−o(1))δ(G)/ ln(∆(G)), where the “o(1)” term goes to zero as ∆(G) → ∞. A dominating set of G is any set S ⊆ V such that for all v ∈ V , either v ∈ S or some neighbor of v is in S. A domatic partition of V is a partition of V into dominating sets, and the number of these dominating sets is called the size of such a partition. The domatic number D(G) of G is the maximum size of a domatic partition; it is NP-hard to find a maximumsized domatic partition. This is a very well-studied problem especially for various special classes of (perfect) graphs: see, e.g., [2, 6, 7] and the references in [4]. Recent interesting work of [4] has given the first non-trivial approximation algorithm for the domatic partition problem, whose approximation guarantee is also shown to be essentially best-possible in [4]. Let n = |V |, δ = δ(G), and ∆ = ∆(G). It is easy to check that D(G) ≤ δ + 1. An efficient algorithm to find a domatic partition of size (1 − o(1))δ/ lnn is shown in [4], where the o(1) term goes to zero as n increases; thus, this is a (1 + o(1)) lnn approximation. It is also shown in [4] that for any fixed > 0, an (1 − ) lnnapproximation algorithm for D(G) would imply that NP ⊆ DTIME[n log ]; hence such an algorithm appears unlikely. An interesting point is that this seems to be the first natural maximization problem proven to have a Θ(logn) approximation threshold. Can we say something better for sparse graphs? It is shown in [4] that D(G) ≥ (1 − o(1))δ/(3 ln ∆), where the o(1) term is a function of ∆ that goes to zero as ∆ increases. (Among the very few such lower bounds known before was that D(G) ≥ dn/(n− δ)e [8]. This is relevant primarily for very dense graphs. For instance, when 1 ≤ δ ≤ n/2, this bound says that D(G) ≥ 2; however, D(G) ≥ 2 is readily seen to hold if (and only