We study a general class of bicriteria network design problems. A generic problem in this class is as follows: Given an undirected graph and two minimization objectives (under different cost functions), with a budget specified on the first, find a subgraph from a given subgraph-class that minimizes the second objective subject to the budget on the first. We consider three different criteria - the total edge cost, the diameter and the maximum degree of the network. Here, we present the first polynomial-time approximation algorithms for a large class of bicriteria network design problems for the above mentioned criteria. The following general types of results are presented. First, we develop a framework for bicriteria problems and their approximations. Second, when the two criteria are the same we present a "black box" parametric search technique. This black box takes in as input an (approximation) algorithm for the unicriterion situation and generates an approximation algorit...

This paper deals with the problem of constructing Steiner trees of minimum weight with diameter bounded by d, spanning a given set of k vertices in a graph. Exact solutions or logarithmic ratio approximation algorithms were known before for the cases of d <= 5. Here we give a polynomial time approximation algorithm of ratio O(log k) for constant d, which is asymptotically optimal unless P = NP , and an algorithm of ratio O( k^{\epsilon})), for any fixed 0 < \epsilon < 1, for general d. Keywords: NP-hard problems, approximation algorithms, Steiner trees 1 Introduction 1.1 The problem This paper considers the problem of finding low diameter Steiner trees of minimum weight. Given an n-vertex graph G(V

In numerous practical applications, it is necessary to find the smallest possible tree with a bounded diameter. A diameter-constrained minimum spanning tree (DCMST) of a given undirected, edge-weighted graph, G, is the smallest-weight spanning tree of all spanning trees of G which contain no path with more than k edges, where k is a given positive integer. The problem of finding a DCMST is NP-complete for all values of k; 4 k (n -- 2), except when all edge-weights are identical. A DCMST is essential for the efficiency of various distributed mutual exclusion algorithms, where it can minimize the number of messages communicated among processors per critical section. It is also useful in linear lightwave networks, where it can minimize interference in the network by limiting the traffic in the network lines. Another practical application requiring a DCMST arises in data compression, where some algorithms compress a file utilizing a tree data-structure, and decompress a path in the tree to access a record. A DCMST helps such algorithms to be fast without sacrificing a lot of storage space. We present a survey of the literature on the DCMST problem, study the expected diameter of a random labeled tree, and present five new polynomial-time algorithms for an approximate DCMST. One of our new algorithms constructs an approximate DCMST in a modified greedy fashion, employing a heuristic for selecting an edge to be added to iii the tree in each stage of the construction. Three other new algorithms start with an unconstrained minimum spanning tree, and iteratively refine it into an approximate DCMST. We also present an algorithm designed for the special case when the diameter is required to be no more than 4. Such a diameter-4 tree is also used for evaluating the quality of o...

: In large IR systems, information about word occurrence may be stored in form of a bit matrix, with rows corresponding to different words and columns to documents. Such a matrix is generally very large and very sparse. New methods for compressing such matrices are presented, which exploit possible correlations between rows and between columns. The methods are based on partitioning the matrix into small blocks and predicting the 1-bit distribution within a block by means of various bit generation models. Each block is then encoded using Huffman or arithmetic coding. The methods also use a new way of enumerating subsets of fixed size from a given superset. Preliminary experimental results indicate improvements over previous methods. 1. Introduction The common approach to processing complex boolean queries in large full-text document retrieval systems is to use inverted files: a concordance is accessed via a dictionary, and includes for each different word of the text, the ordered list ...