LP-Based Approximation Algorithms for Facility Location in Buy-at-Bulk Network Design

Abstract

We study problems that integrate buy-at-bulk network design into the classical (connected) facility location problem. In such problems, we need to open facilities, build a routing network, and route every client demand to an open facility. Furthermore, capacities of the edges can be purchased in discrete units from K different cable types with costs that satisfy economies of scale. We extend the linear programming framework of Talwar (IPCO 2002) for the single-source buy-at-bulk problem to these variants and prove integrality gap upper bounds for both facility location and connected facility location buy-at-bulk problems. For the unconnected variant we prove an integrality gap bound of O(K), and for the connected version, we get the first LP-based bound of O(1).

Appendix: A Naive Model for DDCFL

In this section, we show that an alternative, but perhaps more natural IP formulation for DDCFL has unbounded integrality gap. Consider the following integer programming formulation:

$$\begin{aligned} \min \quad C^\text {op}+ C^\text {core}+&C^\text {fixed}+ C^\text {route}&\end{aligned}$$

(IP-DDCFL-2)

$$\begin{aligned} g_{[K]}^{j}(j)&\le -1&\forall \ j \in D \end{aligned}$$

(2)

$$\begin{aligned} g_{[K]}^{j}(v)&= 0&\forall \ j \in D,\ v \in V {\setminus } (F\cup \{j\}) \end{aligned}$$

(3)

$$\begin{aligned} f_{(u,v);k}^{j} + f_{(v,u);k}^{j}&\le x_{uv}^{k}&\forall \ j \in D,\ k \in [K],uv\in E \end{aligned}$$

(6)

$$\begin{aligned} y_r&= 1&\end{aligned}$$

(8)

$$\begin{aligned} g_{[q,K]}^j(v)&\le 0&\forall \ j \in D, \ v \in V {\setminus } F,\ 1\le q \le K \end{aligned}$$

(9)

$$\begin{aligned} g_{[q,K]}^j(i) - \sum _{\bar{e}\in \delta (i)} z_{\bar{e}}&\le 0&\forall \ j \in D,\ i \in F {\setminus } \{r\},\ 1\le q \le K \end{aligned}$$

(10)

$$\begin{aligned} g_{[K]}^j(i)&\le y_i&\forall \ j \in D,\ i \in F \end{aligned}$$

(22)

$$\begin{aligned} y_i - \sum _{\bar{e}\in \delta (S)}z_{\bar{e}}&\le 0&\forall \ S \subseteq V {\setminus } \{r\}:S \cap F \ne \emptyset ,\ i \in S \nonumber \\ x_{\bar{e}}^{k}, f_{e;k}^{j}, y_{i},z_{\bar{e}}&\in \{0,1\}&\end{aligned}$$

(23)

The difference between this formulation and (IP-DDCFL) is that (IP-DDCFL-2) does not have variables h. We replace constraints (4) and (7) by (22) and (23) respectively.

Theorem 4

The integrality gap of (IP-DDCFL-2) can be arbitrarily large.

Proof

Consider the instance described in Fig. 6, where the square nodes represent facilities and the circle nodes represent clients. In this instance, $K=1$, i.e. we have a unique access cable type, and we set $\sigma = \delta = 1$. The core cable has a cost (per unit length) equal to M with $1<M < q$. For every facility $i \in \{1, \cdots p\}$, we set an opening cost of $\mu _{i} = 1$. We also set $\mu _{n} = \infty$. The root facility r, which must be opened, has an opening cost of 0. The distances are given by the metric completion of the edge costs depicted in the figure, where $L \gg 1$ is a constant larger than any other finite parameter of the instance.

The optimal integral solution to this instance can connect all the clients to a fixed facility $i^* \in \{1, \ldots p\}$ via access links; note that facilities $\{1, \ldots p\}$ are (almost) collocated. Then this open facility is connected to the root node via (unopened) facility n using core links.

However, the LP relaxation of (IP-DDCFL-2) can cheat and open all facilities $i \in \{1, \ldots p\}$ to the extends of 1 / p to serve clients demands. Then it can install core links to the extends of 1 / p on the edges connecting them (via node n) to the root node. This means that LP only pays $M\cdot L /p$ for the core link along edge nr, while the integral solution pays a cost of $M\cdot L$ on that for the same edge. Since L was chosen as an arbitrarily large constant, this is the only relevant value to compare. Hence, the integrality gap is proportional to p and thus it can be made arbitrarily large. $\square$

Fig. 6

An instance of DDCFL with q clients of unit demands and $(p+2)$ potential facilities with facility r as the root node