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Prize-collecting set multicovering with submodular pricing

Daniel Porumbel 1
1 CEDRIC - OC - CEDRIC. Optimisation Combinatoire
CEDRIC - Centre d'études et de recherche en informatique et communications
Abstract : We consider a ground set E and a submodular function f : 2 E → R acting on it. We first propose a Set Multi-Covering problem in which the value (price) of any S ⊆ E is f (S). We show that the Linear Program (LP) of this problem can be directly solved by applying a Submodular Function Minimization (SFM) algorithm on the dual LP. However, the main results of this study concern Prize-Collecting Multi-Covering With Submodular Pricing, i.e., a more general and more difficult Multi-Covering problem in which elements can be left uncovered by paying a penalty. We formulate it as a large-scale LP (with 2 |E| variables representing all subsets of E) that could be tackled by Column Generation (CG), see [18] for a CG algorithm for Set-Covering problems with submodular pricing. However, we do not solve this large-scale LP by CG, but we solve it in polynomial time by exploiting its integrality properties. More exactly, after appropriate restructuring, the dual LP can be transformed into an LP that has been thoroughly studied (as a primal) in the SFM theory. Solving this LP reduces to optimizing a strong map of O(n) submodular functions. For this, we use the Fleischer-Iwata framework [6] that optimizes all these O(n) functions within the same asymptotic running time as solving a single SFM, i.e., in O(n 7 γ + n 8), where n = |E| and γ is the complexity of one submodular evaluation. Besides solving the problem, the proposed approach can be useful to: (1) simultaneously find the best solution of up to O(n 5) functions under strong map relations in O(n 8 γ + n 9) time, (2) perform sensitivity analysis in very short time (even at no extra cost), (3) reveal combinatorial insight into the primal-dual optimal solutions. We present several potential applications along the paper, from production planning to combinatorial auctions.
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Daniel Porumbel. Prize-collecting set multicovering with submodular pricing. International Transactions in Operational Research, Wiley, 2018, 25 (4), pp.1221-1239. ⟨10.1111/itor.12420⟩. ⟨hal-02542673⟩

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