Showing papers by "Madhav V. Marathe published in 1993"

Many birds with one stone: multi-objective approximation algorithms

Abstract: We study network-design problems with multiple design objectives. In particular, we look at two cost measures to be minimized simultaneously: the total cost of the network and the maximum degree of any node in the network. Our main result can be roughly stated as follows: given an integer $b$, we present approximation algorithms for a variety of network-design problems on an $n$-node graph in which the degree of the output network is $O(b \log (\frac{n}{b}))$ and the cost of this network is $O(\log n)$ times that of the minimum-cost degree-$b$-bounded network. These algorithms can handle costs on nodes as well as edges. Moreover, we can construct such networks so as to satisfy a variety of connectivity specifications including spanning trees, Steiner trees and generalized Steiner forests. The performance guarantee on the cost of the output network is nearly best-possible unless $NP = \tilde{P}$. We also address the special case in which the costs obey the triangle inequality. In this case, we obtain a polynomial-time approximation algorithm with a stronger performance guarantee. Given a bound $b$ on the degree, the algorithm finds a degree-$b$-bounded network of cost at most a constant time optimum. There is no algorithm that does as well in the absence of triangle inequality unless $P = NP$. We also show that in the case of spanning networks, we can simultaneously approximate within a constant factor yet another objective: the maximum cost of any edge in the network, also called the bottleneck cost of the network. We extend our algorithms to find TSP tours and $k$-connected spanning networks for any fixed $k$ that simultaneously approximate all these three cost measures.

The Complexity of Approximating PSPACE-Complete Problems for Hierarchical Specifications (Extended Abstract)

Abstract: We extend the concept of polynomial time approximation algorithms to apply to problems for hierarchically specified graphs, many of which are PSPACE-complete. We present polynomial time approximation algorithms for several standard PSPACE-hard problems considered in the literature. In contrast, we prove that finding e-approximations for any e > 0, for several other problems when the instances are specified hierarchically, is PSPACE-hard. We present polynomial time approximation algorithms for the following problems when the graphs are specified hierarchically: minimum vertex cover, maximum 3SAT, weighted max cut, minimum maximal matching, and bounded degree maximum independent set.In contrast, we show that for any e > 0, obtaining e-approximations for the following problems when the instances are specified hierarchically is PSPACE-hard: the number of true gates in a monotone acyclic circuit when all input values are specified and the optimal value of the objective function of a linear program. It is also shown that obtaining a performance guarantee of less than 2 is PSPACE-hard for the following problems when the instances are specified hierarchically: high degree subgraph, k-vertex connected subgraph and k-edge connected subgraph.

Compact Location Problems

Abstract: We consider the problem of placing a specified number (p) of facilities on the nodes of a network so as to minimize some measure of the distances between facilities. This formulation models a number of problems arising in facility location, statistical clustering, pattern recognition, and also a processor allocation problem in multiprocessor systems.

Hierarchical Specified Unit Disk Graphs (Extended Abstract)

Abstract: We characterize the complexity of several basic optimization problems for unit disk graphs specified hierarchically as in [LW87a, Le88, LW92]. Both PSPACE-hardness results and polynomial time approximations are presented for most of the problems considered. These problems include minimum vertex coloring, maximum independent set, minimum clique cover, minimum dominating set and minimum independent dominating set.

Efficient approximation algorithms for domatic partition and on-line coloring of circular arc graphs

Abstract: Exploits the close relationship between circular arc graphs and interval graphs to design efficient approximation algorithms for NP-hard optimization problems on circular arc graphs. The problems considered are maximum domatic partition and online minimum vertex coloring. We present a heuristic for the domatic partition problem with a performance ratio of 4. For online coloring, we consider two different online models. In the first model, arcs are presented in the increasing order of their left endpoints. For this model, our heuristic guarantees a solution which is within a factor of 2 of the optimal (off-line) value; and we show that no online coloring algorithm can achieve a performance guarantee of less than 3/2. In the second online model, arcs are presented in an arbitrary order; and it is known that no online coloring algorithm can achieve a performance guarantee of less than 3. For this model, we present a heuristic which provides a performance guarantee of 4. >