Algorithms for the metric ring star problem with fixed edge-cost ratio

We address the metric ring star problem with fixed edge-cost ratio, abbreviated as RSP. Given a complete graph $$G=(V,E)$$ G = ( V , E ) with a specified depot node $$d\in V$$ d ∈ V , a nonnegative cost function $$c\in \mathbb {R}_+^E$$ c ∈ R + E on E which satisfies the triangle inequality, and an edge-cost ratio $$M\ge 1$$ M ≥ 1 , the RSP is to locate a ring $$R=(V',E')$$ R = ( V ′ , E ′ ) in G, a simple cycle through d or d itself, aiming to minimize the sum of two costs: the cost for constructing ring R, i.e., $$M\cdot \sum _{e\in E'}c(e)$$ M · ∑ e ∈ E ′ c ( e ) , and the cost for attaching nodes in $$V{\setminus } V'$$ V \ V ′ to their closest ring nodes (in R), i.e., $$\sum _{u\in V{\setminus } V'}\min _{v\in V'}c(uv)$$ ∑ u ∈ V \ V ′ min v ∈ V ′ c ( u v ) . We show that the singleton ring d is an optimal solution of the RSP when $$M\ge (|V|-1)/2$$ M ≥ ( | V | - 1 ) / 2 . This particularly implies a $$\sqrt{|V|-1}$$ | V | - 1 -approximation algorithm for the RSP with any $$M\ge 1$$ M ≥ 1 . We present randomized 3-approximation algorithm and deterministic 5.06-approximation algorithm for the RSP, by adapting algorithms for the tour-connected facility location problem (tour-CFLP) and single-source rent-or-buy problem due to Eisenbrand et al. and Williamson and van Zuylen, respectively. Building on the PTAS of Eisenbrand et al. for the tour-CFLP, we give a PTAS for the RSP with $$|V|/M=O(1)$$ | V | / M = O ( 1 ) . We also consider the capacitated RSP (CRSP) which puts an upper limit k on the number of leaf nodes that a ring node can serve, and present a $$(10+6M/k)$$ ( 10 + 6 M / k ) -approximation algorithm for this capacitated generalization. Heuristics based on some natural strategies are proposed for both the RSP and CRSP. Simulation results demonstrate that the proposed approximation and heuristic algorithms have good practical performances.