Improved Approximation Algorithms for the Demand Routing and Slotting Problem with Unit Demands on Rings

Abstract

In the demand routing and slotting problem on unit demands (unit-DRSP), we are given a set of unit demands on an n-node ring. Each demand, which is a (source, destination) pair, must be routed clockwise or counterclockwise and assigned a slot so that no two routes that overlap occupy the same slot. The objective is to minimize the total number of slots used. It is well known that unit-DRSP is NP-complete. The best deterministic approximation algorithm guarantees a solution that is 2 × OPT. A demand of unit-DRSP can be viewed as a chord on the ring. Let w denote the size of the largest set of demand chords that mutually cross in the interior of the ring. We present a simple approximation algorithm that uses at most $(2 - 1/\lceil w/2 \rceil)\times OPT$ slots in an n-node network; this is the first deterministic approximation algorithm that beats the factor of 2 for all values of OPT and therefore for all instances of the input. If randomization is allowed, an algorithm by Kumar produces, with high probability, a solution that uses asymptotically $(1.5 + \frac{1}{2e} +o(1)) \times OPT$ slots. However, when OPT is not large enough, the factor can exceed 2. In this paper, we show how combining our algorithm with Kumar's yields a randomized approximation algorithm that has, with high probability, a constant factor of $2 - 1/\theta(\log n)$. While asymptotically it is not better than Kumar's, the approximation factor holds for all values of OPT.