Approximation Algorithms for Generalized MST and TSP in Grid Clusters

Abstract

Abstract. We consider a special case of the generalized minimum spanning tree problem (GMST) and the generalized travelling salesman problem (GTSP) where we are given a set of points inside the integer grid (in Euclidean plane) where each gride cell is 1 × 1. In the MST version of the problem, the goal is to find a minimum tree that contains exactly one point from each non-empty grid cell (cluster). Similarly, in the TSP version of the problem, the goal is to find a minimum weight cycle containing one point from each non-empty grid cell. We give a (1 + 4 √ 2 + ) and (1.5 + 8 √ 2 + )-approximation algorithm for these two problems in the described setting, respectively. Our motivation is based on the problem posed in [7] for a constant approximation algorithm. The authors designed a PTAS for the more special case of the GMST where non-empty cells are connected end dense enough. However, their algorithm heavily relies on this connectivity restriction and is unpractical. Our results develop the topic further.

Approximating bottleneck spanning trees on partitioned tuples of points

“…The metric version of the k-GBST can be approximated by a ratio of 2k using linear programming [36] combined with the so-called parsimonious property [18]. Related work [6,34,35] also addresses the generalized traveling salesperson problem (TSP) in which the tour must contain exactly one point from each cluster. The group Steiner tree is another related problem which asks for a shortest tree that contains at least one point from each cluster.…”

Red-Blue-Partitioned MST, TSP, and Matching

“…Subscripts of u nodes are omitted for clarity. For each variable x i appearing (in some position k ∈ [3]) within a clause C j , we draw a connection gadget between x i 's e i F and C j 's f j k . First observe the following, which can be verified by inspection:…”

“…(-) (-) optimizing a path visiting at most one node from each pair [8], generalized MST [16,18,19,3], generalized TSP [3], constrained forest problems [9], adding conflict constraints to MST [20,13,6] and to perfect matching [17,6], and balanced partition of MSTs [1].…”

Network Optimization on Partitioned Pairs of Points

“…They show that this problem is NP-hard and that no constant-factor approximation algorithm exists for this problem unless P = N P . Related work addresses the generalized traveling salesperson problem [5,16,17,18], in which a tour must visit one point from each of the given subsets. Arkin et al [3] studied the problem in which one is given a set V and a set of subsets of V , and one wants to select at least one element from each subset in order to minimize the diameter of the chosen set.…”