A Constant Factor Approximation Algorithm for the Storage Allocation Problem

Abstract

We study the storage allocation problem (SAP) which is a variant of the unsplittable flow problem on paths (UFPP). A SAP instance consists of a path $P = (V,E)$ and a set J of tasks. Each edge $e \in E$ has a capacity $c_e$ and each task $j \in J$ is associated with a path $I_j$ in P, a demand $d_j$ and a weight $w_j$. The goal is to find a maximum weight subset $S \subseteq J$ of tasks and a height function $h:S \rightarrow \mathbb {R}^+$ such that (i) $h(j)+d_j \le c_e$, for every $e \in I_j$; and (ii) if $j,i \in S$ such that $I_j \cap I_i \ne \emptyset$ and $h(j) \ge h(i)$, then $h(j) \ge h(i) + d_i$. SAP can be seen as a rectangle packing problem in which rectangles can be moved vertically, but not horizontally. We present a polynomial time $(9+\varepsilon )$-approximation algorithm for SAP. Our algorithm is based on a variation of the framework for approximating UFPP by Bonsma et al. [FOCS 2011] and on a $(4+\varepsilon )$-approximation algorithm for $\delta$-small SAP instances (in which $d_j \le \delta \cdot c_e$, for every $e \in I_j$, for a sufficiently small constant $\delta >0$). In our algorithm for $\delta$-small instances, tasks are packed carefully in strips in a UFPP manner, and then a $(1+\varepsilon )$ factor is incurred by a reduction from SAP to UFPP in strips. The strips are stacked to form a SAP solution. Finally, we provide a $(10+\varepsilon )$-approximation algorithm for SAP on ring networks.

Appendix: A Local Ratio Algorithm for Packing Small Tasks in a Strip

In this section we provide a local ratio algorithm that computes $\frac{1}{2}B$-packable $(5+\varepsilon )$-approximate solutions for $\delta$-small SAP instances in which edge capacities are between B and 2B, for some constant $\delta > 0$ (depending on $\varepsilon$).

The local ratio technique [5, 8] is based on the Local Ratio Lemma. We use the maximization version of the lemma which applies to optimization problems in which the input is a profit vector $w \in (\mathbb {Q}^+)^n$ and a set of feasibility constraints $\mathcal {F}$. The problem is to find a solution vector $x \in \mathbb {Q}^n$ that maximizes the inner product $p \cdot x$ subject to the constraints $\mathcal {F}$.

Lemma 19

(Local Ratio [5]). Let $\mathcal F$ be a set of constraints and let $w,w_1$, and $w_2$ be profit functions such that $w = w_1 + w_2$. Then, if x is r-approximate both with respect to $(\mathcal {F},w_1)$ and with respect to $(\mathcal {F},w_2)$, for some r, then x is also an r-approximate solution with respect to $(\mathcal {F},w)$.

We are now ready to present our local ratio algorithm.

Sorting the tasks according to their right end-point can be done in $O(n \log n)$. There are O(n) recursive calls, each requiring linear time. Hence the running time of Algorithm Strip is $O(n^2)$.

We show that Algorithm Strip computes approximate solutions whose load on any edge is at most $\frac{1}{2}B$.

Lemma 20

Given a $\delta$-small SAP instance in which $b(j) \in [B,2B)$, for every $j \in J$, Algorithm Strip computes a $\frac{1}{2}B$-packable UFPP solution S. Furthermore, $w(S) \ge \frac{5}{1-4\delta } \cdot \textsc {opt} _{\textsc {SAP}}(J)$.

Proof

We first prove that S is $\frac{1}{2}B$-packable, for every e, by induction on the number of recursive calls. In the base case $S=\emptyset$ and we are done. For the inductive step, assume that $d(S'(e)) \le \frac{1}{2}B$, for every $e \in E$. First, $d(S(e)) = d(S'(e)) \le \frac{1}{2}B$, for every $e \not \in I_{j^*}$. For $e \in I_{j^*}$, observe that $d(S(e)) \le d(S(e^*)) \le \frac{1}{2}B$.

We prove that S is $\frac{5}{1-4\delta }$-approximate also by induction on the number of recursive calls. In the base case $S=\emptyset$ is optimal. For the inductive step, assume that $S'$ is $\frac{5}{1-4\delta }$-approximate with respect to $J^+$ and $w_{2}.$ Since $w_2(j^*)=0$, S is also $\frac{5}{1-4\delta }$-approximate with respect to J and $w_2$, We show that S is also $\frac{5}{1-4\delta }$-approximate with respect to J and $w_1$. This completes the proof, since by the Local Ratio Lemma we get that S is $\frac{5}{1-4\delta }$-approximate with respect to J and w as well.

It remains to show that S is $\frac{5}{1-4\delta }$-approximate with respect to J and $w_1$. Notice that either $j^* \in S$ or $d(S(e^*)) + d_{j^*} > \frac{1}{2}B$. If $j^* \in S$, then $w_1(S) \ge w(j^*)$. Otherwise,

$$\begin{aligned} w_1(S)>w(j^*) \cdot 2 \cdot \frac{B/2 - d_{j^*}}{B} \ge w(j^*) \cdot 2 \cdot \frac{B/2 - 2\delta B}{B} =w(j^*) \cdot (1 - 4 \delta ) ~. \end{aligned}$$

On the other hand, for a feasible SAP solution T we have that

$$\begin{aligned} w_1(T) =w_1(T(e^*)) \le w(j^*) + w(j^*) \cdot 2 \cdot 2B/B =5 w(j^*), \end{aligned}$$

due to Observation 1. Therefore w(S) is $\frac{5}{1-4\delta }$-approximate with respect to J and $w_1$. $\square$

The following lemma replaces Lemma 5.

Lemma 21

There exists a polynomial time algorithm such that for every constant $\varepsilon >0$, there exists a constant $\delta >0$, such that the algorithm computes $\frac{1}{2}B$-packable $(5+\varepsilon )$-approximate solutions for $\delta$-small SAP instances in which $b(j) \in [B,2B)$, for every $j \in J$.

Proof

First, execute Algorithm Strip to compute a $\frac{1}{2}B$-packable $\frac{5}{1-4\delta }$-approximate UFPP solution S. By Lemma 4, S can be transformed into a $\frac{1}{2}B$-packable SAP solution $(S',h')$ such that $w(S') \ge (1-4\delta ) w(S)$ in polynomial time. It follows that

$$\begin{aligned} w(S')\ge \frac{(1-4\delta )^2}{5} \cdot \textsc {opt} _{\textsc {SAP}}(J) \ge \frac{1-8\delta }{5} \cdot \textsc {opt} _{\textsc {SAP}}(J)~. \end{aligned}$$

The lemma follows by setting $\delta$ such that $\delta < \delta _0$ and $\frac{5}{1-8\delta } \le 5 + \varepsilon$. $\square$