Approximation Schemes for Degree-Restricted MST and Red–Blue Separation Problems

Abstract

We develop a quasi-polynomial time approximation scheme for the Euclidean version of the Degree-Restricted MST Problem by adapting techniques used previously by Arora for approximating TSP. Given n points in the plane, d = 3 or 4, and ε > 0, the scheme finds an approximation with cost within 1 + ε of the lowest cost spanning tree with the property that all nodes have degree at most d.We also develop a polynomial time approximation scheme for the Euclidean version of the Red-Blue Separation Problem, again extending Arora's techniques. Given ε > 0, the scheme finds an approximation with cost within 1 + ε of the cost of the optimum separating polygon of the input nodes, in nearly linear time. Introduction.In the Degree-Restricted Minimum Spanning Tree (DRMST) problem we are given n points in R 2 (more generally, R k ) and a degree bound d ≥ 2, and have to find the spanning tree of lowest cost in which every node has degree at most d. The case d = 2 is equivalent to traveling salesman and hence NP-hard. Papadimitriou and Vazirani [15] showed NP-hardness for d = 3 in the plane, and conjectured that the problem remains NP-hard for d = 4. The problem can be solved in polynomial time for d = 5 since there always exists a minimum spanning tree with degree at most 5. We are interested in approximation algorithms for the difficult cases, both in R 2 and R k . (In R k the degree bound on the optimum spanning tree is of the form exp( (k)), so all degrees less than that are interesting cases for the problem.) This problem is the most basic of a family of well-studied problems about finding degree-constrained structures; see Raghavachari's survey [16].An approximation scheme for an NP-minimization problem is an algorithm that can, for every ε > 0, compute a solution whose cost is at most (1+ε) times the optimum. If the running time is polynomial for every fixed ε, then we say that the approximation scheme is a Polynomial Time Approximation Scheme (PTAS), and if the running time is not quite polynomial but n poly(log n) , then we say the approximation scheme is a Quasi-Polynomial Time Approximation Scheme (QPTAS).

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