A Push-Relabel Algorithm for Approximating Degree Bounded MSTs

Abstract

Given a graph G and degree bound B on its nodes, the bounded-degree minimum spanning tree (BDMST) problem is to find a minimum cost spanning tree among the spanning trees with maximum degree B. This bi-criteria optimization problem generalizes several combinatorial problems, including the Traveling Salesman Path Problem (TSPP).

An $(\alpha,\:f(B))$-approximation algorithm for the BDMST problem produces a spanning tree that has maximum degree f(B) and cost within a factor α of the optimal cost. Könemann and Ravi [13,14] give a polynomial-time $({1+\frac{1}{\beta}},\:{bB(1+\beta) + \log_bn})$-approximation algorithm for any b > 1, β> 0. In a recent paper [2], Chaudhuri et al. improved these results with a $({1},\:{bB+\sqrt{b}\log_bn})$-approximation for any b > 1. In this paper, we present a $({1+\frac{1}{\beta}},\:{2B(1+ \beta) + o(B(1+\beta))})$-approximation polynomial-time algorithm. That is, we give the first algorithm that approximates both degree and cost to within a constant factor of the optimal. These results generalize to the case of non-uniform degree bounds.

The crux of our solution is an approximation algorithm for the related problem of finding a minimum spanning tree (MST) in which the maximum degree of the nodes is minimized, a problem we call the minimum-degree MST (MDMST) problem. Given a graph G for which the degree of the MDMST solution is $\Delta_{\mbox{\sc{opt}}}$, our algorithm obtains in polynomial time an MST of G of degree at most $2\Delta_{\mbox{\sc{opt}}} + o(\Delta_{\mbox{\sc{opt}}})$. This result improves on a previous result of Fischer [4] that finds an MST of G of degree at most $b\Delta_{\mbox{\sc{opt}}} + \log_bn$ for any b > 1, and on the improved quasipolynomial algorithm of [2].

Our algorithm uses the push-relabel framework developed by Goldberg [7] for the maximum flow problem. To our knowledge, this is the first instance of a push-relabel approximation algorithm for an NP-hard problem, and we believe these techniques may have larger impact. We note that for B = 2, our algorithm gives a tree of cost within a (1+ε)-factor of the optimal solution to TSPP and of maximum degree $O(\frac{1}{\epsilon})$ for any ε> 0, even on graphs not satisfying the triangle inequality.