We present a technique for converting RNC algorithms into NC algorithms. Our approach is based on a parallel implementation of the method of conditional probabilities. This method was used to convert probabilistic proofs of existence of combinatorial structures into polynomial time deterministic algorithms. It has the apparent drawback of being extremely sequential in nature. We show certain general conditions under which it is possible to use this technique for devising deterministic parallel algorithms. We use our technique to devise an NC algorithm for the set balancing problem. This problem turns out to be a useful tool for parallel algorithms. Using our de-randomization method and the set balancing algorithm, we provide an NC algorithm for the lattice approximation problem. We also use the lattice approximation problem to bootstrap the set balancing algorithm, and the result is a more processor efficient algorithm. The set balancing algorithm also yields an NC algorithm for near-optimal edge coloring of simple graphs. Our methods also extend to the parallelization of various algorithms in computational geometry that rely upon the random sampling technique of Clarkson. Finally, our methods apply to constructing certain combinatorial structures, e.g. ...

Abstract. Given a network and a set of connection requests on it, we consider the maximum edge-disjoint paths and related generalizations and routing problems that arise in assigning paths for these requests. We present improved approximation algorithms and/or integrality gaps for all problems considered; the central theme of this work is the underlying multi-commodity flow relaxation. Applications of these techniques to approximating families of packing integer programs are also presented. Key words and phrases. Disjoint paths, approximation algorithms, unsplittable flow, rout-ing, packing, integer programming, multicommodity flow, randomized algorithms, rounding, linear programming. 1

Abstract. In this paper, we provide a new class of randomized approximation algorithms for scheduling problems by directly interpreting solutions to so-called time-indexed LPs as probabilities. The most general model we consider is scheduling unrelated parallel machines with release dates (or even network scheduling) so as to minimize the average weighted completion time. The crucial idea for these multiple machine problems is not to use standard list scheduling but rather to assign jobs randomly to machines (with probabilities taken from an optimal LP solution) and to perform list scheduling on each of them. For the general model, we give a (2+ e)-approximation algorithm. The best previously known approximation algorithm has a performance guarantee of 16/3 [HSW96]. Moreover, our algorithm also improves upon the best previously known approximation algorithms for the special case of identical parallel machine scheduling (performance guarantee (2.89 + e) in general [CPS+96] and 2.85 for the average completion time [CMNS97], respectively). A perhaps surprising implication for identical parallel machines is that jobs are randomly assigned to machines, in which each machine is equally likely. In addition, in this case the algorithm has running time O(nlogn) and performance guarantee 2. The same algorithm also is a 2-approximation for the corresponding preemptive scheduling problem on identical parallel machines. Finally, the results for identical parallel machine scheduling apply to both the off-line and the on-line settings with no difference in performance guarantees. In the on-line setting, we are scheduling jobs that continually arrive to be processed and, for each time t, we must construct the schedule until time t without any knowledge of the jobs that will arrive afterwards. 1

We provide a method to produce an efficient algorithm to find an object whose existence is guaranteed by the Lov'asz Local Lemma. We feel that this method will apply to the vast majority of applications of the Local Lemma, unless the application has one of four problematic traits. However, proving that the method applies to a particular application may require proving two (possibly difficult) concentration-like properties.

In this paper, we provide a new class of randomized approximation algorithms for parallel machine scheduling problems. The most general model we consider is scheduling unrelated machines with release dates (or even network scheduling) so as to minimize the average weighted completion time. We introduce an LP relaxation in time-indexed variables for this problem. The crucial idea to derive approximation results is not to use standard list scheduling, but rather to assign jobs randomly to machines (by interpreting LP solutions as probabilities), and to perform list scheduling on each of them. Our main result is a (2 + e)--approximation algorithm for this general model which improves upon performance guarantee 16=3 due to Hall, Shmoys, and Wein. In the absence of nontrivial release dates, we get a (3=2 + e)--approximation. At the same time we prove corresponding bounds on the quality of the LP relaxation. A perhaps surprising implication for identical parallel machines is that jobs are ra...

In this paper we study approximation algorithms for solving a general covering integer program. An n-vector x of nonnegative integers is sought, which minimizes c T x; subject to Ax b; x d: The entries of A; b; c are nonnegative. Let m be the number of rows of A: Covering problems have been heavily studied in combinatorial optimization. We focus on the effect of the multiplicity constraints, x d; on approximability. Two longstanding open questions remain for this general formulation with upper bounds on the variables. (i) The integrality gap of the standard LP relaxation is arbitrarily large. Existing approximation algorithms that achieve the well-known O(log m)-approximation with respect to the LP value do so at the expense of violating the upper bounds on the variables by the same O(log m) multiplicative factor. What is the smallest possible violation of the upper bounds that still achieves cost within O(log m) of the standard LP optimum ? (ii) The best known approximation ratio for the problem has been O(log(max j P i A ij )) since 1982. This bound can be as bad as polynomial in the input size. Is an O(log m)-approximation, like the one known for the special case of Set Cover, possible? We settle these two open questions. To answer the first question we give an algorithm based on the relatively simple new idea of randomly rounding variables to smaller-thaninteger units. To settle the second question we give a reduction from approximating the problem while respecting multiplicity constraints to approximating the problem with a bounded violation of the multiplicity constraints. 1 Research partially supported by NSERC Grant 227809-00 and a CFI New Opportunities Award 1.

Koller and Megiddo introduced the paradigm of constructing compact distributions that satisfy a given set of constraints, and showed how it can be used to efficiently derandomize certain types of algorithm. In this paper, we significantly extend their resdts in two ways. First, we show how their approach can be applied to deal with more general expectation constraints. More importantly, we provide the first parallel (Ne) algorithm for constructing a compact distribution that satisfies the constraints up to a small relative error. This algorithm deals with constraints over any event that can be verified by finite automata, including all inde-pendence constraints as well as constraints over events relating to the parity OT sum of a certain set of variables. OUT construction relies on a new and independently interesting parallel algorithm for converting a solution to a linear system into an almost basic approximate solution to the same system. We use these techniques in the first AfC derandomization of an algorithm for constructing large independent sets in d-uniform hypergraphs for arbitrary d. We also show how the linear programming perspective suggests new proof techniques which might be useful in general probabilistic analysis.

We survey techniques for replacing randomized algorithms in computational geometry by deterministic ones with a similar asymptotic running time. 1 Randomized algorithms and derandomization A rapid growth of knowledge about randomized algorithms stimulates research in derandomization, that is, replacing randomized algorithms by deterministic ones with as small decrease of efficiency as possible. Related to the problem of derandomization is the question of reducing the amount of random bits needed by a randomized algorithm while retaining its efficiency; the derandomization can be viewed as an ultimate case. Randomized algorithms are also related to probabilistic proofs and constructions in combinatorics (which came first historically), whose development has similarly been accompanied by the effort to replace them by explicit, non-random constructions whenever possible. Derandomization of algorithms can be seen as a part of an effort to map the power of randomness and explain its role. ...

Given matrices A and B and vectors a, b, c and d, all with non-negative entries, we consider the problem of computing min{cT x: x ∈ Z n +,Ax�a, Bx �b, x �d}. We give a bicriteria-approximation algorithm that, given � ∈ (0, 1], finds a solution of cost O(ln(m)/ � 2) times optimal, meeting the covering constraints (Ax �a) and multiplicity constraints (x �d), and satisfying Bx�(1 + �)b + �, where � is the vector of row sums �i = � j Bij. Here m denotes the number of rows of A. This gives an O(ln m)-approximation algorithm for CIP—minimum-cost covering integer programs with multiplicity constraints, i.e., the special case when there are no packing constraints Bx�b. The previous best approximation ratio has been O(ln(maxj iAij)) since 1982. CIP contains the set cover problem as a special case, so O(ln m)-approximation is the best possible unless P = NP.

Approximation algorithms provide a natural way to approach computationally hard problems. There are currently many known paradigms in this area, including greedy algorithms, primal-dual methods, methods based on mathematical programming (linear and semidefinite programming in particular), local improvement, and "low distortion" embeddings of general metric spaces into special families of metric spaces. Randomization is a useful ingredient in many of these approaches, and particularly so in the form of randomized rounding of a suitable relaxation of a given problem. We survey this technique here, with a focus on correlation inequalities and their applications.