Given a graph G and degree bound B on its nodes, the bounded-degree minimum spanning tree (BDMST) problem is to \ufb01nd 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 (\u03b1, f(B))-approximation algorithm for the BDMST problem produces a spanning tree that has maximum degree f(B) and cost within a factor \u03b1 of the optimal cost. K\u00a8one mann and Ravi [13,14] give a polynomial time (1 + 1 \u03b2, bB(1 + \u03b2) + log b n)-approximation algorithm for any b>1, \u03b2>0. In a recent paper [2], Chaudhuri et al. improved these results with a (1, bB+\u221ablogb n)-approximation for any b>1. In this paper, we present a (1+1 \u03b2 , 2B(1+\u03b2)+o(B(1+\u03b2)))-approximation polynomial-time algorithm. That is, we give the \ufb01rst 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 \ufb01nding 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 \u0394opt, our algorithm obtains in polynomial time an MST of G of degree at most 2\u0394opt + o(\u0394opt). This result improves on a previous result of Fischer [4] that \ufb01nds an MST of G of degree at most b\u0394opt + log b n 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 \ufb02ow problem. To our knowledge, this is the \ufb01rst 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 + \u0005)factor of the optimal solution to TSPP and of maximum degree O(1 \u0003) for any \u0005>0, even on graphs not satisfying the triangle inequality.<\/p>\n","protected":false},"excerpt":{"rendered":"
Given a graph G and degree bound B on its nodes, the bounded-degree minimum spanning tree (BDMST) problem is to \ufb01nd 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). 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