An Improved Approximation Algorithm for the K-Supplier Problem with Parameterized Triangle Inequality.

Given an edge-weighted undirected connected graph G = (V, E, ρ , 𝒳, 𝒴) , where ρ : E →R^+∪{ 0 } is an edge-weight function, 𝒳⊂ V is a subset of clients, and 𝒴⊂ V is a subset of candidates, and a positive integer k < |𝒴| , the k-Supplier Problem (k SP) asks for an optimal subset of 𝒴 of cardinality at most k to minimize the radius from 𝒳 to the subset. In this paper, we focus on the case of 𝒳∩𝒴∅ , 𝒳, 𝒴 , and consider the scenario where the shortest path distances d(· , · ) in G satisfy a parameterized triangle inequality between 𝒳 and 𝒴 , i.e., d(x, y) + d(y, z) ≥α· d(x, z), ∀ x, y, z ∈{ u, v, w }, x y, y z, z x , where 1 ≤α≤ 2 is a parameter, for any three distinct vertices, v, u ∈𝒳 and w ∈𝒴 . We present a two-stage dual approximation algorithm ALG for the kSP with parameter triangle inequality between 𝒳 and 𝒴 . If it stops at the end of Stage 1 then it achieves a 2/α -approximation, and if it stops at the end of Stage 2 then it achieves a (2/α ^2 + 1/α) -approximation. ALG runs in a polynomial time and the above two parameterized performance factors of it are both strictly monotonic decreasing with respect to the value of parameter α . For the kSP instances with parameterized triangle inequality having 1 < α≤ 2 , it is implied by 1 ≤2/α < 2 and 1 ≤2/α ^2 + 1/α < 3 that ALG has better approximation ratios than the previously best polynomial-time 3-approximation algorithm of Hochbaum and Shmoys (J. ACM. 33: 533–550, 1986). Furthermore, regardless of the parameterized aspect, ALG achieves a 2-approximation if it stops at the end of Stage 1 and a 3-approximation if it stops at the end of Stage 2, for the general kSP.