Annamária Kovács

Bayesian Combinatorial Auctions

We study the following Bayesian setting: mitems are sold to nselfish bidders in mindependent second-price auctions. Each bidder has a privatevaluation function that expresses complex preferences over allsubsets of items. Bidders only have beliefsabout the valuation functions of the other bidders, in the form of probability distributions. The objective is to allocate the items to the bidders in a way that provides a good approximation to the optimal social welfare value. We show that if bidders have submodular valuation functions, then every Bayesian Nash equilibrium of the resulting game provides a 2-approximation to the optimal social welfare. Moreover, we show that in the full-information game a pure Nash always exists and can be found in time that is polynomial in both mand n.

Fast monotone 3-approximation algorithm for scheduling related machines

We consider the problem of scheduling n jobs to m machines of different speeds s.t. the makespan is minimized (Q||Cmax). We provide a fast and simple, deterministic monotone 3-approximation algorithm for Q||Cmax Monotonicity is relevant in the context of truthful mechanisms: when each machine speed is only known to the machine itself, we need to motivate that machines declare their true speeds to the scheduling mechanism. As shown by Archer and Tardos, such motivation is possible only if the scheduling algorithm used by the mechanism is monotone. The best previous monotone algorithm that is polynomial in m, was a 5-approximation by Andelman et al. A randomized 2-approximation method, satisfying a weaker definition of truthfulness, is given by Archer. As a core result, we prove the conjecture of Auletta et al., that the greedy algorithm (Lpt) is monotone if machine speeds are all integer powers of 2.

Mechanism design for fractional scheduling on unrelated machines

In this paper, we consider the mechanism design version of the fractional variant of the scheduling problem on unrelated machines. We give a lower bound of 2-1/n for any fractional truthful mechanism, while we propose a truthful mechanism that achieves approximation of 1 + (n - 1)/2, for n machines. We also focus on an interesting family of allocation algorithms, the task-independent algorithms. We give a lower bound of 1 + (n - 1)/2, that holds for every (not only monotone) allocation algorithm of this class. Under this consideration, our truthful independent mechanism is the best that we can hope from this family of algorithms.

Tight Bounds for the Price of Anarchy of Simultaneous First-Price Auctions

We study the price of anarchy (PoA) of simultaneous first-price auctions (FPAs) for buyers with submodular and subadditive valuations. The current best upper bounds for the Bayesian price of anarchy (BPoA) of these auctions are e/(e − 1) [Syrgkanis and Tardos 2013] and 2 [Feldman et al. 2013], respectively. We provide matching lower bounds for both cases even for the case of full information and for mixed Nash equilibria via an explicit construction. We present an alternative proof of the upper bound of e/(e − 1) for FPAs with fractionally subadditive valuations that reveals the worst-case price distribution, which is used as a building block for the matching lower bound construction. We generalize our results to a general class of item bidding auctions that we call bid-dependent auctions (including FPAs and all-pay auctions) where the winner is always the highest bidder and each bidder’s payment depends only on his own bid. Finally, we apply our techniques to discriminatory price multiunit auctions. We complement the results of de Keijzer et al. [2013] for the case of subadditive valuations by providing a matching lower bound of 2. For the case of submodular valuations, we provide a lower bound of 1.109. For the same class of valuations, we were able to reproduce the upper bound of e/(e − 1) using our nonsmooth approach.

A Deterministic Truthful PTAS for Scheduling Related Machines

Scheduling on related machines (

Fast algorithms for two scheduling problems

Q||C_{\max}

A global characterization of envy-free truthful scheduling of two tasks

) is one of the most important problems in the field of algorithmic mechanism design. Each machine is controlled by a selfish agent and her valuation function can be expressed via a single parameter, her speed. Archer and Tardos [Proceedings of the 42nd Annual Symposium on Foundations of Computer Science (FOCS), Las Vegas, NV, 2001, pp. 482--491] showed that, in contrast to other similar problems, a (nonpolynomial) allocation that minimizes the makespan can be truthfully implemented. On the other hand, if we leave out the game-theoretic issues, the complexity of the problem has been completely settled---the problem is strongly NP-hard, while there exists a polynomial-time approximation scheme (PTAS) [D. S. Hochbaum and D. B. Shmoys, SIAM J. Comput., 17 (1988), pp. 539--551, and L. Epstein and J. Sgall, Algorithmica, 39(1) (2004), pp. 43--57]. This problem is the most well studied in single-parameter algorithmic mechanism design. It gives an excellent ground to ...

A truthful constant approximation for maximizing the minimum load on related machines

The thesis deals with problems from two distint areas of scheduling theory. In the first part we consider the preemptive Sum Multicoloring (pSMC) problem. In an instance of pSMC, pairwise conflicting jobs are represented by a conflict graph, and the time demands of jobs are given by integer weights on the nodes. The goal is to schedule the jobs in such a way that the sum of their finish times is minimized. We give the first polynomial algorithm for pSMC on paths and cycles, running in time O(min(n2, n log p)), where n is the number of nodes and p is the largest time demand. This answers a question raised by Halldórsson et al. [51] about the hardness of this problem. Our result identifies a gap between binary-tree conflict graphs – where the question is NP-hard – and paths. In the second part of the thesis we consider the problem of scheduling n jobs on m machines of different speeds s.t. the makespan is minimized (Q||Cmax). We provide a fast and simple, deterministic monotone 2.8-approximation algorithm for Q||Cmax. Monotonicity is relevant in the context of truthful mechanisms: when each machine speed is only known to the machine itself, we need to motivate that machines ’declare’ their true speeds to the scheduling mechanism. So far the best deterministic truthful mechanism that is polynomial in n and m, was a 5-approximation by Andelman et al. [3]. A randomized 2-approximation method, satisfying a weaker definition of truthfulness, was given by Archer and Tardos [4, 5]. As a core result, we prove the conjecture of Auletta et al. [8], that the greedy list scheduling algorithm Lpt is monotone if machine speeds are all integer powers of two (2-divisible machines). Proving the worst case bound of 2.8 involves studying the approximation ratio of Lpt on 2-divisible machines. As a side result, we obtain a tight bound of ( √ 3 + 1)/2 ≈ 1.3660 for the ’one fast machine’ case, i.e., when m − 1 machine speeds are equal, and there is only one faster machine. In this special case the best previous lower and upper bounds were 4/3 − 2 < Lpt/Opt ≤ 3/2 − 1/(2m), shown in a classic paper by Gonzalez et al. [42]. Moreover, the authors of [42] conjectured the bound 4/3 to be tight. Thus, the results of the thesis answer three open questions in scheduling theory.

Online Paging for Flash Memory Devices

We study envy-free and truthful mechanisms for domains with additive valuations, like the ones that arise in scheduling on unrelated machines. We investigate the allocation functions that are both weakly monotone (truthful) and locally efficient (envy-free), in the case of only two tasks, but many players. We show that the only allocation functions that satisfy both conditions are affine minimizers, with strong restrictions on the parameters of the affine minimizer. As a further result, we provide a common payment function, i.e., a single mechanism that is both truthful and envy-free. For additive combinatorial auctions our approach leads us (only) to a non- affine maximizer similar to the counterexample of Lavi et al. [26]. Thus our result demonstrates the inherent difference between the scheduling and the auctions domain, and inspires new questions related to the classic problem of characterizing truthfulness in additive domains.

Mechanisms with Monitoring for truthful RAM allocation

Abstract Designing truthful mechanisms for scheduling on related machines is a very important problem in single-parameter mechanism design. We consider the covering objective, that is we are interested in maximizing the minimum completion time of a machine. This problem falls into the class of problems where the optimal allocation can be truthfully implemented. A major open issue for this class is whether truthfulness affects the polynomial-time implementation. We provide the first constant factor approximation for deterministic truthful mechanisms. In particular we come up with an approximation guarantee of 2 + e , significantly improving on the previous upper bound of min ( m , ( 2 + e ) s m / s 1 ) .