We study fair allocation of indivisible goods to agents with unequal entitlements. Fair allocation has been the subject of many studies in both divisible and indivisible settings. Our emphasis is on the case where the goods are indivisible and agents have unequal entitlements. This problem is a generalization of the work by Procaccia and Wang [20] wherein the agents are assumed to be symmetric with respect to their entitlements. Although Procaccia and Wang show an almost fair (constant approximation) allocation exists in their setting, our main result is in sharp contrast to their observation. We show that, in some cases with n agents, no allocation can guarantee better than 1/n approximation of a fair allocation when the entitlements are not necessarily equal. Furthermore, we devise a simple algorithm that ensures a 1/n approximation guarantee.Our second result is for a restricted version of the problem where the valuation of every agent for each good is bounded by the total value he wishes to receive in a fair allocation. Although this assumption might seem w.l.o.g, we show it enables us to find a 1/2 approximation fair allocation via a greedy algorithm. Finally, we run some experiments on real-world data and show that, in practice, a fair allocation is likely to exist. We also support our experiments by showing positive results for two stochastic variants of the problem, namely stochastic agents and stochastic items.

Longest common subsequence (LCS) is one of the most fundamental problems in combinatorial optimization. Apart from theoretical importance, LCS has enormous applications in bioinformatics, revision control systems, and data comparison programs 1 . Although a simple dynamic program computes LCS in quadratic time, it has been recently proven that the problem admits a conditional lower bound and may not be solved in truly subquadratic time [2]. In addition to this, LCS is notoriously hard with respect to approximation algorithms. Apart from a trivial sampling technique that obtains a n x approximation solution in time O(n 2−2x ) nothing else is known for LCS. This is in sharp contrast to its dual problem edit distance for which several linear time solutions are obtained in the past two decades [4,5,9,10,16].In this work, we present the first nontrivial algorithm for approximating LCS in linear time. Our main result is a linear time algorithm for the longest common subsequence which has an approximation factor of O(n 0.497956 ). This beats the √ n barrier for approximating LCS in linear time. * A portion of this work was completed while some of the authors were visiting Simons Institute for Theory of Computing. †

We study the problem of fair allocation for indivisible goods. We use the maximin share paradigm introduced by Budish [Budish E (2011) The combinatorial assignment problem: Approximate competitive equilibrium from equal incomes. J. Political Econom. 119(6):1061–1103.] as a measure of fairness. Kurokawa et al. [Kurokawa D, Procaccia AD, Wang J (2018) Fair enough: Guaranteeing approximate maximin shares. J. ACM 65(2):8.] were the first to investigate this fundamental problem in the additive setting. They showed that in delicately constructed examples, not everyone can obtain a utility of at least her maximin value. They mitigated this impossibility result with a beautiful observation: no matter how the utility functions are made, we always can allocate the items to the agents to guarantee each agent’s utility is at least 2/3 of her maximin value. They left open whether this bound can be improved. Our main contribution answers this question in the affirmative. We improve their approximation result to a 3/4 factor guarantee.

One of the important yet insufficiently studied subjects in fair allocation is the externality effect among agents. For a resource allocation problem, externalities imply that a bundle allocated to an agent may affect the utilities of other agents.In this paper, we conduct a study of fair allocation of indivisible goods when the externalities are not negligible. We present a simple and natural model, namely network externalities, to capture the externalities. To evaluate fairness in the network externalities model, we generalize the idea behind the notion of maximin-share (MMS) to achieve a new criterion, namely, extendedmaximin-share (EMMS). Next, we consider two problems concerning our model.First, we discuss the computational aspects of finding the value of EMMS for every agent. For this, we introduce a generalized form of partitioning problem that includes many famous partitioning problems such as maximin, minimax, and leximin partitioning problems. We show that a 1/2-approximation algorithm exists for this partitioning problem.Next, we investigate on finding approximately optimal EMMS allocations. That is, allocations that guarantee every agent a utility of at least a fraction of his extended-maximin-share. We show that under a natural assumption that the agents are α-self-reliant, an α/2-EMMS allocation always exists. The combination of this with the former result yields a polynomial-time α/4-EMMS allocation algorithm. 1 arXiv:1805.06191v1 [cs.GT] 16 May 2018 Our Results and TechniquesIn this paper, we take one step toward understanding the impact of externalities in allocation of indivisible items. We start by proposing a general model to capture the externalities in a fair allocation problem under additive assumptions. Although we present some of our results with regard to this general model, the main focus of the paper is on a more restricted model, namely network externalities, where the influences imposed by the agents can be represented by a weighted directed graph. This model is inspired by the well-studied linear-threshold model in the context of network diffusion.We suggest the extended-maximin-share notion (EMMS) to adapt maximin-share to the environment with externalities. Similar to maximin-share, our extension is motivated by the maximin strategy in cut-and-choose games. We discuss two aspects of our notion.

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