A Push-Relabel Algorithm for Approximating Degree Bounded MSTs

Abstract

AbstractGiven 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.KeywordsSpan TreeMaximum DegreeMinimum Span TreeTree EdgeDegree BoundThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.