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.