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journal contribution

posted on 2000-09-01, 00:00 authored by Anupam GuptaAnupam Gupta, Martin Pál, Ramamoorthi RaviRamamoorthi Ravi, Amitabh SinhaSeveral combinatorial optimization problems choose elements to minimize the total cost of constructing
a feasible solution that satisfies requirements of clients. For example, in the STEINER TREE problem,
edges must be chosen to connect terminals (clients); in VERTEX COVER, vertices must be chosen to
cover edges (clients); in FACILITY LOCATION, facilities must be chosen and demand vertices (clients)
connected to these chosen facilities.
We consider a stochastic version of such a problem where the solution is constructed in two stages: Before
the actual requirements materialize, we can choose elements in a first stage. The actual requirements
are then revealed, drawn from a pre-specified probability distribution π; thereupon, some more elements
may be chosen to obtain a feasible solution for the actual requirements. However, in this second (recourse)
stage, choosing an element is costlier by a factor of σ > 1. The goal is to minimize the first stage
cost plus the expected second stage cost.
We give a general yet simple technique to adapt approximation algorithms for several deterministic
problems to their stochastic versions via the following method.
• First stage: Draw σ independent sets of clients from the distribution π and apply the approximation
algorithm to construct a feasible solution for the union of these sets.
• Second stage: Since the actual requirements have now been revealed, augment the first-stage solution
to be feasible for these requirements.
We use this framework to derive constant factor approximations for stochastic versions of VERTEX
COVER, STEINER TREE and UNCAPACITATED FACILITY LOCATION for arbitrary distributions π in
one fell swoop. For special (product) distributions, we obtain additional and improved results. Our
techniques adapt and use the notion of strict cost-shares introduced in [5].