Coordinating selfish players in scheduling games

We investigate coordination mechanisms that schedule n jobs on m unrelated machines. The objective is to minimize the makespan. It was raised as an open question whether it is possible to design a coordination mechanism that has constant price of anarchy using preemption. We give a negative answer. Next we introduce multi-job players that control a set of jobs, with the aim of minimizing the sum of the completion times of theirs jobs. In this setting, previous mechanisms designed for players with single jobs are inadequate, e.g., having large price of anarchy, or not guaranteeing pure Nash equilibria. To meet this challenge, we design three mechanisms that induce pure Nash equilibria while guaranteeing relatively small price of anarchy. Then we consider multi-job players where each player\u27s objective is to minimize the weighted sum of completion time of her jobs, while the social cost is the sum of players\u27 costs. We first prove that if machines order jobs according to Smith-rule, then the coordination ratio is at most 4, moreover this is best possible among non-preemptive policies. Then we design a preemptive policy, em externality that has coordination ratio 2.618, and complement this result by proving that this ratio is best possible even if we allow for randomization or full information. An interesting consequence of our results is that an $varepsilon-$local optima of $R|,|sum w_iC_i$ for the jump neighborhood can be found in polynomial time and is within a factor of 2.618 of the optimal solution.Wir betrachten Koordinationsmechanismen um n Jobs auf m Maschinen mit individuellen Bearbeitungszeiten zu verteilen. Ziel dabei ist es den Makespan zu minimieren. Es war eine offene Frage, ob es möglich ist einen preämptiven Koordinationsmechanismus zu entwickeln, der einen konstanten Price of Anarchy hat. Wir beantworten diese Frage im negativen Sinne. Als nächstes führen wir Multi-Job-Spieler ein, die eine Menge von Jobs kontrollieren können, mit dem Ziel die Summe der Fertigstellungszeiten ihrer Jobs zu minimieren. In diesem Szenario sind bekannte Mechanismen, die für Ein-Job-Spieler entworfen worden sind, nicht gut genug, und haben beispielsweise einen hohen Price of Anarchy oder können kein reines Nash Gleichgewicht garantieren. Wir entwickeln drei Mechanismen die jeweils ein reines Nash Gleichgewicht besitzen, und einen relativ kleinen Price of Anarchy haben. Zusätzlich betrachten wir Multi-Job-Spieler, mit dem Ziel jeweils die gewichtete Summe der Fertigstellungszeiten ihrer Jobs zu minimieren, während die Gesamtkosten die Summe der Kosten der Spieler sind. Wir zeigen zuerst, dass das Koordinationsverhältnis höchstens $4$ ist, wenn die Maschinen die Jobs nach der Smith-Regel sortieren, was bei nicht-preämptiven Verfahren optimal ist. Danach entwickeln wir ein preämptives Verfahren, Externality, welches ein Koordinationsverhältnis von 2.618 hat, und ergänzen dieses Ergebniss indem wir beweisen, dass dieses Verhältnis optimal ist, auch für den Fall, dass wir Randomisierung oder volle Information erlauben. Eine interessante Folge unserer Ergebnisse ist, dass ein $varepsilon$-lokales Optimum von $R|,|sum w_iC_i$ für die Jump-Neighborhood in Polynomialzeit gefunden werden kann, und innerhalb eines Faktors von 2.618 von der optimalen Lösung ist

On Guillotine Cutting Sequences

Imagine a wooden plate with a set of non-overlapping geometric objects painted on it. How many of them can a carpenter cut out using a panel saw making guillotine cuts, i.e., only moving forward through the material along a straight line until it is split into two pieces? Already fifteen years ago, Pach and Tardos investigated whether one can always cut out a constant fraction if all objects are axis-parallel rectangles. However, even for the case of axis-parallel squares this question is still open. In this paper, we answer the latter affirmatively. Our result is constructive and holds even in a more general setting where the squares have weights and the goal is to save as much weight as possible. We further show that when solving the more general question for rectangles affirmatively with only axis-parallel cuts, this would yield a combinatorial O(1)-approximation algorithm for the Maximum Independent Set of Rectangles problem, and would thus solve a long-standing open problem. In practical applications, like the mentioned carpentry and many other settings, we can usually place the items freely that we want to cut out, which gives rise to the two-dimensional guillotine knapsack problem: Given a collection of axis-parallel rectangles without presumed coordinates, our goal is to place as many of them as possible in a square-shaped knapsack respecting the constraint that the placed objects can be separated by a sequence of guillotine cuts. Our main result for this problem is a quasi-PTAS, assuming the input data to be quasi-polynomially bounded integers. This factor matches the best known (quasi-polynomial time) result for (non-guillotine) two-dimensional knapsack

Near-Optimal Asymmetric Binary Matrix Partitions

We study the asymmetric binary matrix partition problem that was recently introduced by Alon et al. (Proceedings of the 9th Conference on Web and Internet Economics (WINE), pp 1–14, 2013). Instances of the problem consist of an n× m binary matrix A and a probability distribution over its columns. A partition schemeB= (B1, … , Bn) consists of a partition Bifor each row i of A. The partition Biacts as a smoothing operator on row i that distributes the expected value of each partition subset proportionally to all its entries. Given a scheme B that induces a smooth matrix AB, the partition value is the expected maximum column entry of AB. The objective is to find a partition scheme such that the resulting partition value is maximized. We present a 9/10-approximation algorithm for the case where the probability distribution is uniform and a (1 - 1 / e) -approximation algorithm for non-uniform distributions, significantly improving results of Alon et al. Although our first algorithm is combinatorial (and very simple), the analysis is based on linear programming and duality arguments. In our second result we exploit a nice relation of the problem to submodular welfare maximization