Browsing by Author "Verschae Tannenbaum Jose Claudio"

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    A water filling primal dual algorithm for approximating non linear covering problems (2020)
    (Schloss Dagstuhl, Leibniz-Zentrum für Informatik, GmbH, 2020) Fielbaum, Andrés; Morales Bravo Camilo Ignacio; Verschae Tannenbaum Jose Claudio
    Obtaining strong linear relaxations of capacitated covering problems constitute a significant technical challenge even for simple settings. For one of the most basic cases, the Knapsack-Cover (Min-Knapsack) problem, the relaxation based on knapsack-cover inequalities has an integrality gap of 2. These inequalities are exploited in more general problems, many of which admit primal-dual approximation algorithms. Inspired by problems from power and transport systems, we introduce a general setting in which items can be taken fractionally to cover a given demand. The cost incurred by an item is given by an arbitrary non-decreasing function of the chosen fraction. We generalize the knapsack-cover inequalities to this setting an use them to obtain a (2+ε)-approximate primal-dual algorithm. Our procedure has a natural interpretation as a bucket-filling algorithm which effectively overcomes the difficulties implied by having different slopes in the cost functions. More precisely, when some superior segment of an item presents a low slope, it helps to increase the priority of inferior segments. We also present a rounding algorithm with an approximation guarantee of 2.We generalize our algorithm to the Unsplittable Flow-Cover problem on a line, also for the setting of fractional items with non-linear costs. For this problem we obtain a (4+ε)-approximation algorithm in polynomial time, almost matching the 4-approximation algorithm known for the classical setting.
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    Equilibrium Dynamics in Market Games with Exchangeable and Divisible Resources
    (2024) Correa, José; Harks, Tobias; Schedel, Anja; Verschae Tannenbaum Jose Claudio
    We study a market game with n ≥ 2 players competing over m ≥ 1 divisible resources of different finite capacities. Resources are traded via the proportional sharing mechanism, where players are price-anticipating, meaning that they can influence the prices with their bids. Additionally, each player has an initial endowment of the resources which are sold at market prices. Although the players’ total profit functions may be discontinuous in the bids, we prove existence and uniqueness of pure Nash equilibria of the resulting market game. Then, we study a discrete dynamic arising from repeatedly taking the (unique) equilibrium resource allocation as initial endowments for the next market game. We prove that the total utility value of the dynamic converges to either an optimal allocation value (maximizing total utility over the allocation space) or to a restricted optimal allocation value, where the restriction is defined by fixing some tight resources which are exclusively allocated to a single player. As a corollary, it follows that for strictly concave utility functions, the aggregated allocation vector of the dynamic converges to the unique (possibly restricted) optimal aggregated allocation, and for linear utility functions, we even get convergence of the dynamic to a (possibly restricted) optimal solution in the (non-aggregated) original allocation space.