Dorit S. Hochbaum

Approximation schemes for covering and packing problems in image processing and VLSI

A unified and powerful approach is presented for devising polynomial approximation schemes for many strongly NP-complete problems. Such schemes consist of families of approximation algorithms for each desired performance bound on the relative error ε > &Ogr;, with running time that is polynomial when ε is fixed. Though the polynomiality of these algorithms depends on the degree of approximation ε being fixed, they cannot be improved, owing to a negative result stating that there are no fully polynomial approximation schemes for strongly NP-complete problems unless NP = P. The unified technique that is introduced here, referred to as the shifting strategy, is applicable to numerous geometric covering and packing problems. The method of using the technique and how it varies with problem parameters are illustrated. A similar technique, independently devised by B. S. Baker, was shown to be applicable for covering and packing problems on planar graphs.

A Best Possible Heuristic for the k-Center Problem

In this paper we present a 2-approximation algorithm for the k-center problem with triangle inequality. This result is “best possible” since for any δ < 2 the existence of δ-approximation algorithm would imply that P = NP. It should be noted that no δ-approximation algorithm, for any constant δ, has been reported to date. Linear programming duality theory provides interesting insight to the problem and enables us to derive, in O|E| log |E| time, a solution with value no more than twice the k-center optimal value. A by-product of the analysis is an O|E| algorithm that identifies a dominating set in G2, the square of a graph G, the size of which is no larger than the size of the minimum dominating set in the graph G. The key combinatorial object used is called a strong stable set, and we prove the NP-completeness of the corresponding decision problem.

Using dual approximation algorithms for scheduling problems theoretical and practical results

The problem of scheduling a set of n jobs on m identical machines so as to minimize the makespan time is perhaps the most well-studied problem in the theory of approximation algorithms for NP-hard optimization problems. In this paper we present the strongest possible type of result for this problem, a polynomial approximation scheme. More precisely, for each ε, we give an algorithm that runs in time O((n/ε)1/ε2) and has relative error at most ε. For algorithms that are polynomial in n and m, the strongest previously-known result was that the MULTIFIT algorithm delivers a solution with no worse than 20% relative error. In addition, we present a refinement of our scheme in the case where the performance guarantee is equal to that of MUL-TIFIT, that yields an algorithm that is both more efficient and easier to analyze than MULTIFIT. In this case, in order to guarantee a maximum relative error of 1/5+2-k, the algorithm runs in O(n(k+logn)) time. The scheme is based on a new approach to constructing approximation algorithms, which we call dual approximation algorithms, where the aim is find superoptimal, but infeasible solutions, and the performance is measured by the degree of infeasibility allowed. This notion should find wide applicability in its own right, and should be considered for any optimization problem where traditional approximation algorithms have been particularly elusive.

Approximation Algorithms for the Set Covering and Vertex Cover Problems

We propose a heuristic that delivers in

A polynomial approximation scheme for scheduling on uniform processors: Using the dual approximation approach

O(n^3 )

Efficient bounds for the stable set, vertex cover and set packing problems

steps a solution for the set covering problem the value of which does not exceed the maximum number of sets covering an element times the optimal value.

Heuristics for the fixed cost median problem

We present a polynomial approximation scheme for the minimum makespan problem on uniform parallel processors. More specifically, the problem is to find a schedule for a set of independent jobs on a collection of machines of different speeds so that the last job to finish is completed as quickly as possible. We give a family of polynomial-time algorithms

A unified approach to approximation algorithms for bottleneck problems

\{ {A_\varepsilon } \}

Convex separable optimization is not much harder than linear optimization

such that

An efficient algorithm for Co-segmentation

A_\varepsilon