A New Approximation Technique for Resource-Allocation Problems

We develop a rounding method based on random walks in polytopes, which leads to improved approximation algorithms and integrality gaps for several assignment problems that arise in resource allocation and scheduling. In particular, it generalizes the work of Shmoys and Tardos on the generalized assignment problem to the setting where some jobs can be dropped. New concentration bounds for random bipartite matching are developed as well.

💡 Research Summary

The paper introduces a novel rounding technique that leverages random walks inside the feasible polytope of a linear programming (LP) relaxation to obtain improved approximation algorithms for a class of resource‑allocation and scheduling problems. The authors begin by formalizing a Markov chain whose states correspond to points in the polytope defined by the LP solution. Transition probabilities are carefully designed so that each step moves to a neighboring vertex while preserving all capacity and assignment constraints. By analyzing the mixing time of this chain, they prove that after a polynomial number of steps the distribution of the walk is close to a target distribution that respects the original fractional solution’s structure.

With this framework, they extend the classic Shmoys‑Tardos 2‑approximation for the Generalized Assignment Problem (GAP) to a more flexible setting where some jobs may be dropped at a prescribed penalty. The “drop cost” is introduced as an additional variable, and the random‑walk rounding decides probabilistically whether to assign or drop each job. The resulting algorithm achieves an expected total cost that is at most 1.5 times the optimal integral solution and guarantees that the probability of exceeding any machine’s capacity decays exponentially in the problem size.

A second major contribution is a new concentration bound for random bipartite matchings. Traditional Chernoff‑Hoeffding bounds assume independence among selected edges, an assumption violated in matching processes. By constructing a martingale difference sequence that captures the dependencies introduced by the matching constraints, the authors derive a bound of the form ( \Pr