Olivier Goldschmidt

Polynomial algorithm for the k-cut problem

The k-cut problem is to find a partition of an edge weighted graph into k nonempty components, such that the total edge weight between components is minimum. This problem is NP-complete for arbitrary k and its version involving fixing a vertex in each component is NP hard even for k=3. A polynomial algorithm for the case of a fixed k is presented.<<ETX>>

A Polynomial Algorithm for the k-cut Problem for Fixed k

The k-cut problem is to find a partition of an edge weighted graph into k nonempty components, such that the total edge weight between components is minimum. This problem is NP-complete for an arbitrary k and its version involving fixing a vertex in each component is NP-hard even for k = 3. We present a polynomial algorithm for k fixed, that runs in Onki¾²/2-3k/2+4Tn, m steps, where Tn, m is the running time required to find the minimum s, t-cut on a graph with n vertices and m edges.

SONET/SDH ring assignment with capacity constraints

We consider the problem of interconnecting a set of customer sites using bidirectional SONET rings of equal capacity. Each site is assigned to exactly one ring and a special ring, called the federal ring, interconnects the other rings together. The objective is to minimize the total cost of the network subject to a ring capacity limit where the capacity of a ring is determined by the total bandwidth required between sites assigned to the same ring plus the total bandwidth request between these sites and sites assigned to other rings.We present exact, integer-programming based solution techniques and fast heuristic algorithms for this problem. We compare the results from applying the heuristic algorithms with those produced by the exact methods for real-world as well as randomly generated problem instances. We show that two of the heuristics find solutions that cost at most twice that of an optimal solution. Empirical evidence indicates that in practice the algorithms perform much better than their theoretical bound and often find optimal solutions.

The SONET edge‐partition problem

Motivated by a problem arising in the design of telecommunications networks using the SONET standard, we consider the problem of covering all edges of a graph using subgraphs that contain at most k edges with the objective of minimizing the total number of vertices in the subgraphs. We show that the problem is -hard when k ≥ 3 and present a linear-time n n n n n n n-approximation algorithm. For even k values, we present an approximation scheme with a reduced ratio but with increased complexity.

Note: On the set-union knapsack problem

We consider a generalization of the 0-1 knapsack problem called the set-union knapsack problem (SKP). In the SKP, each item is a set of elements, each item has a nonnegative value, and each element has a nonnegative weight. The total weight of a collection of items is given by the total weight of the elements in the union of the items sets. This problem has applications to data-base partitioning and to machine loading in flexible manufacturing systems. We show that the SKP remains NP-hard, even in very restricted cases. We present an exact, dynamic programming algorithm for the SKP and show sufficient conditions for it to run in polynomial time.

Randomized methods for the number partitioning problem

Randomized versions of Karmarkar and Karps differencing method are introduced for the Number Partitioning problem. The development of these methods and a discussion of their merits are presented. It is shown that these randomized heuristics consistently yield better solutions than those generated by the differencing method.

An efficient graph planarization two-phase heuristic

Given a graph G = (V, E), the graph planarization problem is to find a largest subset F of E, such that H = (V, F) is planar. It is known to be NP-complete. This problem is of interest in automatic graph drawing, in facilities layout, and in the design of printed circuit boards and integrated circuits. We present a two-phase heuristic for solving the planarization problem. The first phase devises a clever ordering of the vertices of the graph on a single line and the second phase tries to find the maximal number of nonintersecting edges that can be drawn above or below this line. Extensive empirical results show that this heuristic outperforms its competitors.

One-half approximation algorithms for the k -partition problem

The k-partition problem seeks to cluster the vertices of an edge-weighted graph, G = (V, E), into k sets of |V|/k vertices each, such that the total weight of the edges having both endpoints in the same cluster is maximized. Bottom-up type heuristics based on matchings are presented for this problem. These heuristics are shown to yield solutions that are at least one-half the weight of the optimal solution for k equal to |V|/3 and |V|/4.

On reliability of graphs with node failures

We consider the reliability of graphs for which nodes fail independently of each other with a constant probability 1 - p. The reliability of a graph is defined to be the probability that the induced subgraph of surviving nodes is connected. A graph is said to be uniformly best when, for all choices of p, it is most reliable in the class of graphs with the same number of nodes and same number of edges. In this paper, we first extend the existing known set of uniformly best graphs. Next, we show that most classes of sparse graphs do not contain a uniformly best graph. Finally, we introduce the important notions of locally best and asymptotically best graphs and illustrate these concepts with a detailed study of graphs having the same number of nodes and edges. 0 1994 John Wiley & Sons, Inc.

Approximation Algorithms for the k -Clique Covering Problem

The problem of covering edges and vertices in a graph (or in a hypergraph) was motivated by a problem arising in the context of the component assembly problem. The problem is as follows: given a graph and a clique size