We study a natural generalization of the maximum weight many-to-one matching problem. We are given an undirected bipartite graph G = (A∪ P, E) with weights on the edges in E, and with lower and upper quotas on the vertices in P.
An approximation algorithm for an optimization problem runs in polynomial time for all instances and is guaranteed to deliver solutions with bounded optimality gap. We derive such algorithms for different variants of capacitated location routing, an important generalization of vehicle routing where the cost of opening the depots from which vehicles operate is taken into account. Our results originate from combining algorithms and lower bounds for different relaxations of the original problem; along with location routing we also obtain approximation algorithms for multidepot capacitated vehicle routing by this framework. Moreover, we extend our results to further generalizations of both problems, including a prize-collecting variant, a group version, and a variant where cross-docking is allowed. We finally present a computational study of our approximation algorithm for capacitated location routing on benchmark instances and large-scale randomly generated instances. Our study reveals that the quality of the computed solutions is much closer to optimality than the provable approximation factor.
A natural extension of the makespan minimization problem on unrelated machines is to allow jobs to be partially processed by different machines while incurring an arbitrary setup time. In this paper we present increasingly stronger LP-relaxations for this problem and their implications on the approximability of the problem. First we show that the straightforward LP, extending the approach for the original problem, has an integrality gap of 3 and yields an approximation algorithm of the same factor. By applying a lift-and-project procedure, we are able to improve both the integrality gap and the implied approximation factor to 1 + φ, where φ is the golden ratio. Since this bound remains tight for the seemingly stronger machine configuration LP, we propose a new job configuration LP that is based on an infinite continuum of fractional assignments of each job to the machines. We prove that this LP has a finite representation and can be solved in polynomial time up to any accuracy. Interestingly, we show that our problem cannot be approximated within a factor better than e e−1 ≈ 1.582 (unless P = N P), which is larger than the inapproximability bound of 1.5 for the original problem.Mathematics Subject Classification Primary 90B35 · 68W25; Secondary 68Q25 · 90C10
Robust network flows are a concept for dealing with uncertainty and unforeseen failures in the network infrastructure. One of the most basic models is the Maximum Robust Flow problem: Given a network and an integer k, the task is to find a path flow of maximum robust value, i.e., the guaranteed value of surviving flow after removal of any k arcs in the network. The complexity of this problem appeared to have been settled a decade ago: Aneja et al. [1] showed that the problem can be solved efficiently when k = 1, while an article by Du and Chandrasekaran [2] established that the problem is NP -hard for any constant value of k larger than 1.We point to a flaw in the proof of the latter result, leaving the complexity for constant k open once again. For the case that k is not bounded by a constant, we present a new hardness proof, establishing NP -hardness even for instances where the number of paths is polynomial in the size of the network. We further show that computing optimal integral solutions is already NP -hard for k = 2 (whereas for k = 1, an efficient algorithm is known) and give a positive result for the case that capacities are in {1, 2}.
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