Approximating Min-Cost Chain-Constrained Spanning Trees: A Reduction from Weighted to Unweighted Problems

Abstract

We study the min-cost chain-constrained spanning-tree (abbreviated MCCST) problem: find a min-cost spanning tree in a graph subject to degree constraints on a nested family of node sets. We devise the first polytime algorithm that finds a spanning tree that (i) violates the degree constraints by at most a constant factor and (ii) whose cost is within a constant factor of the optimum. Previously, only an algorithm for unweighted CCST was known [13], which satisfied (i) but did not yield any cost bounds. This also yields the first result that obtains an O(1)-factor for both the cost approximation and violation of degree constraints for any spanning-tree problem with general degree bounds on node sets, where an edge participates in multiple degree constraints.

A notable feature of our algorithm is that we reduce MCCST to unweighted CCST (and then utilize [13]) via a novel application of Lagrangian duality to simplify the cost structure of the underlying problem and obtain a decomposition into certain uniform-cost subproblems.

We show that this Lagrangian-relaxation based idea is in fact applicable more generally and, for any cost-minimization problem with packing side-constraints, yields a reduction from the weighted to the unweighted problem. We believe that this reduction is of independent interest. As another application of our technique, we consider the k-budgeted matroid basis problem, where we build upon a recent rounding algorithm of [4] to obtain an improved $n^{O(k^{1.5}/\epsilon )}$-time algorithm that returns a solution that satisfies (any) one of the budget constraints exactly and incurs a $(1+\epsilon )$-violation of the other budget constraints.

A full version of the paper is available on the CS arXiv.

A. Linhares and C. Swamy—Research supported partially by NSERC grant 327620-09 and the second author’s Discovery Accelerator Supplement Award, and Ontario Early Researcher Award.