61 research outputs found

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    An EF2X Allocation Protocol for Restricted Additive Valuations

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    We study the problem of fairly allocating a set of mm indivisible goods to aset of nn agents. Envy-freeness up to any good (EFX) criteria -- whichrequires that no agent prefers the bundle of another agent after removal of anysingle good -- is known to be a remarkable analogous of envy-freeness when theresource is a set of indivisible goods. In this paper, we investigate EFXnotion for the restricted additive valuations, that is, every good has somenon-negative value, and every agent is interested in only some of the goods. We introduce a natural relaxation of EFX called EFkX which requires that noagent envies another agent after removal of any kk goods. Our maincontribution is an algorithm that finds a complete (i.e., no good is discarded)EF2X allocation for the restricted additive valuations. In our algorithm wedevise new concepts, namely "configuration" and "envy-elimination" that mightbe of independent interest. We also use our new tools to find an EFX allocation for restricted additivevaluations that discards at most ⌊n/2⌋−1\lfloor n/2 \rfloor -1 goods. This improvesthe state of the art for the restricted additive valuations by a factor of 22.<br

    Integrality Gap of Time-Indexed Linear Programming Relaxation for Coflow Scheduling

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    Coflow is a set of related parallel data flows in a network. The goal of the coflow scheduling is to process all the demands of the given coflows while minimizing the weighted completion time. It is known that the coflow scheduling problem admits several polynomial-time 5-approximation algorithms that compute solutions by rounding linear programming (LP) relaxations of the problem. In this paper, we investigate the time-indexed LP relaxation for coflow scheduling. We show that the integrality gap of the time-indexed LP relaxation is at most 4. We also show that yet another polynomial-time 5-approximation algorithm can be obtained by rounding the solutions to the time-indexed LP relaxation

    The Submodular Santa Claus Problem in the Restricted Assignment Case

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    The submodular Santa Claus problem was introduced in a seminal work by Goemans, Harvey, Iwata, and Mirrokni (SODA\u2709) as an application of their structural result. In the mentioned problem n unsplittable resources have to be assigned to m players, each with a monotone submodular utility function f_i. The goal is to maximize min_i f_i(S_i) where S?,...,S_m is a partition of the resources. The result by Goemans et al. implies a polynomial time O(n^{1/2 +?})-approximation algorithm. Since then progress on this problem was limited to the linear case, that is, all f_i are linear functions. In particular, a line of research has shown that there is a polynomial time constant approximation algorithm for linear valuation functions in the restricted assignment case. This is the special case where each player is given a set of desired resources ?_i and the individual valuation functions are defined as f_i(S) = f(S ? ?_i) for a global linear function f. This can also be interpreted as maximizing min_i f(S_i) with additional assignment restrictions, i.e., resources can only be assigned to certain players. In this paper we make comparable progress for the submodular variant: If f is a monotone submodular function, we can in polynomial time compute an O(log log(n))-approximate solution

    Additive Approximation Schemes for Load Balancing Problems

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    We formalize the concept of additive approximation schemes and apply it to load balancing problems on identical machines. Additive approximation schemes compute a solution with an absolute error in the objective of at most ? h for some suitable parameter h and any given ? > 0. We consider the problem of assigning jobs to identical machines with respect to common load balancing objectives like makespan minimization, the Santa Claus problem (on identical machines), and the envy-minimizing Santa Claus problem. For these settings we present additive approximation schemes for h = p_{max}, the maximum processing time of the jobs. Our technical contribution is two-fold. First, we introduce a new relaxation based on integrally assigning slots to machines and fractionally assigning jobs to the slots. We refer to this relaxation as the slot-MILP. While it has a linear number of integral variables, we identify structural properties of (near-)optimal solutions, which allow us to compute those in polynomial time. The second technical contribution is a local-search algorithm which rounds any given solution to the slot-MILP, introducing an additive error on the machine loads of at most ?? p_{max}
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