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

Abstract

We address the metric ring star problem with fixed edge-cost ratio, abbreviated as RSP. Given a complete graph $G=(V,E)$ with a specified depot node $d\in V$, a nonnegative cost function $c\in \mathbb {R}_+^E$ on E which satisfies the triangle inequality, and an edge-cost ratio $M\ge 1$, the RSP is to locate a ring $R=(V',E')$ 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., $M\cdot \sum _{e\in E'}c(e)$, and the cost for attaching nodes in $V{\setminus } V'$ to their closest ring nodes (in R), i.e., $\sum _{u\in V{\setminus } V'}\min _{v\in V'}c(uv)$. We show that the singleton ring d is an optimal solution of the RSP when $M\ge (|V|-1)/2$. This particularly implies a $\sqrt{|V|-1}$-approximation algorithm for the RSP with any $M\ge 1$. 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 $|V|/M=O(1)$. 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 $(10+6M/k)$-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.

Appendix: Randomized 3-approximation for the RSP

Let Z, R and K be the total, ring and assignment cost of the solution $S=Q\oplus A$ output by Algorithm 1.3, respectively. And Let $Z^*,R^*$ and $K^*$ be the total, ring and assignment cost of an optimal solution $S^*$, respectively. In $S^*$, $Q^*$ is the cycle, $F^*=V(Q^*)$ is the node set of the cycle, and $\sigma ^*$ is the assignment function.

Lemma 8

The ring cost of S satisfies $E[R]\le \rho (R^*+2\alpha \cdot K^*)$.

Proof

Consider a Hamilton cycle over F: based on the cycle $Q^*$ over $F^*$, add two edges between every node $i\in F$ and its closest node in $F^*$, and then compute an Eulerian tour on the resulting multigraph and shortcut it into a Hamilton tour T over F. We have

$$\begin{aligned} E[c(T)]\le \frac{R^*}{M}+2\sum _{j\in V}\frac{\alpha }{M}c(j,F^*)=\frac{R^*}{M}+\frac{2\alpha }{M}K^*. \end{aligned}$$

And the expected cost of the $\rho$-approximation cycle Q satisfies

$$\begin{aligned} E[c(Q)]\le \rho \cdot E[c(T)]\le \rho \cdot \left( \frac{R^*}{M}+\frac{2\alpha }{M}K^*\right) . \end{aligned}$$

Altogether we obtain

$$\begin{aligned} E[R]=M\cdot E[c(Q)]\le \rho (R^*+2\alpha \cdot K^*). \end{aligned}$$

We consider the following cycle-core connection game [see Eisenbrand et al. (2010)]: let the optimal ring $Q^*$ be the cycle-core in the game, and $\mathcal {C}$ be the set of edges in $Q^*$. Let $\mathcal {D}\subset V$ be the set of nodes outside the cycle-core $Q^*$. The function $\sigma ^*$ assigns every node $j\in V$ to its closest node in $F^*$. Each node $j\in \mathcal {D}$ has two oppositely directed edges $(j,\sigma ^*(j))$ and $(\sigma ^*(j),j)$, and let $\mathcal {H}$ be the set of such directed edges. Let $\mathcal {G}=(V,\mathcal {E}=\mathcal {C}\cup \mathcal {H})$ be the resulting graph, and $c:\mathcal {E}\rightarrow \mathbb {Q}^+$ the weight function on the edges in $\mathcal {G}$. Recall that $F\subseteq V$ is the set of marked nodes. Now, every node $j\in \mathcal {D}$ sends one unit of (unsplittable) flow to its closet marked node (with respect to c). We can bound the cost of the total flow in this game in the following lemma.

Lemma 9

The cost X of the flow in the cycle-core connection game satisfies $E[K]\le E[X]\le \frac{M\cdot c(Q^*)}{2\alpha }+2K^*=\frac{R^*}{2\alpha }+2K^*$.

Fig. 4

A cycle-core connection game. Marked nodes are drawn in bold. The flow of j in the routing scheme is indicated by the bold path

Proof

For the cycle-core connection game defined above, we can reasonably assume that all marked nodes are outside the cycle-core, and each cycle-node has exactly one node assigned to it (by adding duplicates of cycle-nodes and edges with zero cost between duplicates), as Fig. 4 shows. And we can additionally assume $\mathcal {D}=V$ is the set of nodes outside the cycle-core (by the same approach). In the solution $S=(F,Q,\sigma )$, $\sigma$ assigns each $j\in V$ to its closest node in F directly with cost $c(j,\sigma (j))$, while in the game j sends one unit flow to F via a path in $\mathcal {G}$. Thus the first inequality $E[K]\le E[X]$ follows.

We consider the following sub-optimal flow routing scheme: each node $j\in \mathcal {D}$ sends its flow unit to the closest marked one with respect to unit edge length. Let f(e) be the load on edge $e\in \mathcal {E}$ in such flow, and denote by Y be total cost of this flow (with respect to c). Clearly, $E[X]\le E[Y]$. Additionally, the expected cost of this flow is

$$\begin{aligned} E[Y]=\sum _{e\in \mathcal {H}}E[f(e)]\cdot c(e)+\sum _{e\in \mathcal {C}}E[f(e)]\cdot c(e). \end{aligned}$$

It is easy to see $E[f(e)]\le 1$ for any $e\in \mathcal {H}$, by the symmetry of edges in $\mathcal {H}$.

Consider an arbitrary edge $e\in \mathcal {C}$. Let $X_j$ be the number of edges in $\mathcal {C}$ passed by the flow-path of node $j\in \mathcal {D}$. Clearly,

$$\begin{aligned} \sum _{e\in \mathcal {C}}f(e)=\sum _{j\in \mathcal {D}}X_j. \end{aligned}$$

Since $|\mathcal {C}|=|\mathcal {D}|$, we can conclude that $E[f(e)]=E[X_j]$. We call a cycle node $i=\sigma (j)$ by-marked if $j\in \mathcal {D}$ is marked. We now observe that $X_j>k$ if and only if j’s closest $2k+1$ nodes in the cycle-core $Q^*$ of $\mathcal {G}$ (with respect to unit edge weights) are not by-marked. As a consequence, we have

$$\begin{aligned} Pr(X_j>k)=\left( 1-\frac{2k+1}{n}\right) \left( 1-\frac{\alpha }{M}\right) ^{2k+1}<\left( 1-\frac{\alpha }{M}\right) ^{2k+1}. \end{aligned}$$

We conclude that

$$\begin{aligned} E[f(e)]=E[X_j]=\sum _{k\ge 0}Pr(X_j>k)\le \frac{1-\alpha /M}{1-(1-\alpha /M)^2}\le \frac{M}{2\alpha }, \end{aligned}$$

which follows that

$$\begin{aligned} E[X]\le E[Y]\le \sum _{e\in \mathcal {H}}c(e)+\frac{M}{2\alpha }\cdot \sum _{e\in \mathcal {C}}\cdot c(e)=\frac{M}{2\alpha }\cdot c(Q^*)+2K^*. \end{aligned}$$

$\square$

Theorem 6

There is an expected 3-approximation algorithm for the RSP.

Proof

By Lemmas 8 and 9, we have

$$\begin{aligned} E[Z]=E[R]+E[K]&\le 1.5(R^*+2\alpha \cdot K^*)+\frac{R^*}{2\alpha }+2K^*\\&=\left( 1.5+\frac{1}{2\alpha }\right) R^*+(2+3\alpha )K^*. \end{aligned}$$

We obtain that $E[Z]\le 3(R^*+K^*)=3Z^*$ when $\alpha =1/3$, which gives the claimed approximation ratio 3. $\square$