Single-Source Stochastic Routing
We introduce and study the following model for routing uncertain demands through a network. We are given a capacitated multicommodity flow network with a single source and multiple sinks, and demands that have known values but unknown sizes. We assume that the sizes of demands are governed by independent distributions, and that we know only the means of these distributions and an upper bound on the maximum-possible size. Demands are irrevocably routed one-by-one, and the size of a demand is unveiled only after it is routed.
A routing policy is a function that selects an unrouted demand and a path for it, as a function of the residual capacity in the network. Our objective is to maximize the expected value of the demands successfully routed by our routing policy. We distinguish between safe routing policies, which never violate capacity constraints, and unsafe policies, which can attempt to route a demand on any path with strictly positive residual capacity.
We design safe routing policies that obtain expected value close to that of an optimal unsafe policy in planar graphs. Unlike most previous work on similar stochastic optimization problems, our routing policies are fundamentally adaptive. Our policies iteratively solve a sequence of linear programs to guide the selection of both demands and routes.
KeywordsPlanar Graph Path Decomposition Stochastic Optimization Problem Planar Embedding Edge Capacity
Unable to display preview. Download preview PDF.
- 2.Chawla, S., Roughgarden, T.: Single-source stochastic routing, http://www.cs.cmu.edu/~shuchi/papers/stoch-routing.ps
- 4.Dean, B., Goemans, M., Vondrak, J.: Adaptivity and approximation for stochastic packing problems. In: SODA 2005, pp. 395–404 (2005)Google Scholar
- 5.Dean, B., Goemans, M., Vondrak, J.: The benefit of adaptivity: Approximating the stochastic knapsack problem. In: FOCS 2004, pp. 208–217 (2004)Google Scholar
- 6.Dean, B.: Approximation Algorithms for Stochastic Scheduling Problems. Ph.D thesis, Massachusetts Institute of Technology, Massachusetts (2005)Google Scholar
- 9.Gupta, A., Pal, M., Ravi, R., Sinha, A.: Boosted sampling: Approximation algorithms for stochastic optimization. In: STOC 2004, pp. 417–426 (2004)Google Scholar
- 11.Gupta, A., Ravi, R., Sinha, A.: An edge in time saves nine: Lp rounding approximation algorithms for stochastic network design. In: FOCS 2004, pp. 218–227 (2004)Google Scholar
- 12.Immorlica, N., Karger, D., Minkoff, M., Mirrokni, V.: On the costs and benefits of procrastination: Approximation algorithms for stochastic combinatorial optimization problems. In: SODA 2004, pp. 684–693 (2004)Google Scholar
- 13.Kleinberg, J.: Single-source unsplittable flow. In: FOCS 1996, pp. 68–77 (1996)Google Scholar
- 15.Ravi, R., Sinha, A.: Hedging uncertainty: approximation algorithms for stochastic optimization problems. Mathematical Programming (2005)Google Scholar
- 16.Shmoys, D., Swamy, C.: Sampling-based approximation algorithms for multi-stage stochastic optimization. In: FOCS 2005 (2005)Google Scholar
- 17.Shmoys, D., Swamy, C.: Stochastic optimization is (almost) as easy as deterministic optimization. In: FOCS 2004, pp. 228–237 (2004)Google Scholar
- 18.Stochastic programming community homepage, http://stoprog.org/