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We study the problem of fair allocation for indivisible goods. We use the maxmin share paradigm introduced by Budish [16] as a measure for fairness. Kurokawa, Procaccia, and Wang [36] were the first to investigate this fundamental problem in the additive setting. They show that a maxmin guarantee (1-MMS allocation) is not always possible even when the number of agents is limited to 3. While the existence of an approximation solution (e.g. a 1/2-MMS allocation) is quite straightforward, improving the guarantee becomes subtler for larger constants. Kurokawa et al. [36] provide a proof for the existence of a 2/3-MMS allocation and leave the question open for better guarantees. Our main... 

We conduct a study on the problem of fair allocation of indivisible goods when maximin share [1] is used as the measure of fairness. Most of the current studies on this notion are limited to the case that the valuations are additive. In this paper, we go beyond additive valuations and consider the cases that the valuations are submodular, fractionally subadditive, and subadditive. We give constant approximation guarantees for agents with submodular and XOS valuations, and a logarithmic bound for the case of agents with subadditive valuations. Furthermore, we complement our results by providing close upper bounds for each class of valuation functions. Finally, we present algorithms to find... 

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... 

In Spatial Voting Theory, distortion is a measure of how good the winner is. It has been proved that no deterministic voting mechanism can guarantee a distortion better than 3, even for simple metrics such as a line. In this study, we wish to answer the following question: How does the distortion value change if we allow less motivated agents to abstain from the election? We consider an election with two candidates and suggest an abstention model, which is a general form of the abstention model proposed by Kirchgässner. Our results characterize the distortion value and provide a rather complete picture of the model. ©2021 AI Access Foundation  

We study the minimum diameter problem for a set of inexact points. By inexact, we mean that the precise location of the points is not known. Instead, the location of each point is restricted to a continuous region ({Imprecise} model) or a finite set of points ({Indecisive} model). Given a set of inexact points in one of {Imprecise} or {Indecisive} models, we wish to provide a lower-bound on the diameter of the real points. In the first part of the paper, we focus on {Indecisive} model. We present an (formula presented) time approximation algorithm of factor (1+epsilon) for finding minimum diameter of a set of points in d dimensions. This improves the previously proposed algorithms for this... 

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 the share allocated to an agent may affect the utilities of other agents. In this paper, we conduct a study of fair allocation of indivisible goods with positive externalities. Inspired by the models in the context of network diffusion, 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, extended-maximin-share (EMMS).... 

Edit distance is one of the most fundamental problems in combinatorial optimization to measure the similarity between strings. Ulam distance is a special case of edit distance where no character is allowed to appear more than once in a string. Recent developments have been very fruitful for obtaining fast and parallel algorithms for both edit distance and Ulam distance. In this work, we present an almost optimal MPC (massively parallel computation) algorithm for Ulam distance and improve MPC algorithms for edit distance. Our algorithm for Ulam distance is almost optimal in the sense that (1) the approximation factor of our algorithm is $1+epsilon$1+ϵ, (2) the round complexity of our... 

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 the share allocated to an agent may affect the utilities of other agents. In this paper, we conduct a study of fair allocation of indivisible goods with positive externalities. Inspired by the models in the context of network diffusion, 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, extended-maximin-share (EMMS).... 

Edit distance is one of the most fundamental problems in computer science. Tree edit distance is a natural generalization of edit distance to ordered rooted trees. Such a generalization extends the applications of edit distance to areas such as computational biology, structured data analysis (e.g., XML), image analysis, and compiler optimization. Perhaps the most notable application of tree edit distance is in the analysis of RNA molecules in computational biology where the secondary structure of RNA is typically represented as a rooted tree. The best-known solution for tree edit distance runs in cubic time. Recently, Bringmann et al. show that an O(n2.99) algorithm for weighted tree edit... 

In this paper, we consider two new variants of the unit covering problem in color-spanning set model: Given a set of n points in d-dimensional plane colored with m colors, the MinCSBC problem is to select m points of different colors minimizing the minimum number of unit balls needed to cover them. Similarly, the MaxCSBC problem is to choose one point of each color to maximize the minimum number of needed unit balls. We show that MinCSBC is NP-hard and hard to approximate within any constant factor even in one dimension. For d = 1, however, we propose an ln(m)-approximation algorithm and present a constant-factor approximation algorithm for fixed f, where f is the maximum frequency of the... 

We study the problem of fair division of a heterogeneous resource among strategic players. Given a divisible heterogeneous cake, we wish to divide the cake among n players to meet these conditions: (I) every player (weakly) prefers his allocated cake to any other player’s share (such notion is known as envy-freeness), (II) the allocation is dominant strategy-proof (truthful) (III) the number of cuts made on the cake is small. We provide methods for dividing the cake under different assumptions on the valuation functions of the players. First, we suppose that the valuation function of every player is a single interval with a special property, namely ordering property. For this case, we... 

We study the problem of fair division of a heterogeneous resource among strategic players. Given a divisible heterogeneous cake, we wish to divide the cake among n players to meet these conditions: (I) every player (weakly) prefers his allocated cake to any other player’s share (such notion is known as envy-freeness), (II) the allocation is dominant strategy-proof (truthful) (III) the number of cuts made on the cake is small. We provide methods for dividing the cake under different assumptions on the valuation functions of the players. First, we suppose that the valuation function of every player is a single interval with a special property, namely ordering property. For this case, we... 

People make decisions and express their opinions according to their communities. A natural idea for controlling the diffusion of a behavior is to find influential people, and employ them to spread a desired behavior. We investigate an influencing problem when individuals' behaviors are affected by their friends in an opinion formation process. Our goal is to design efficient algorithms for finding opinion leaders such that changing their opinions has a great impact on the overall external behaviors in the society. We study directed social networks and define a set of problems like maximizing the sum of individuals' behaviors or maximizing the number of individuals whose external behaviors... 

The closed neighbourhood NG[e] of an edge e in a graph G is the set consisting of e and of all edges having a common end vertex with e. Let f be a function on the edges of G into the set {-1, 1}. If ∑e∈NG[x] f(e) ≥ 1 for every x ∈ E(G), then f is called a signed edge domination function of G. The minimum value of ∑x∈E(G) f(x), taken over every signed edge domination function f of G, is called signed edge domination number of G and denoted by γ's (G). It has been proved that γ's(G) ≥ n - m for every graph G of order n and size m. In this paper we prove that γ's (G) ≥ 2α'(G)-m 3 for every simple graph G, where α'(G) is the size of a maximum matching of G. We also prove that for a simple graph... 

We study the problem of fair division of a heterogeneous resource among strategic players. Given a divisible heterogeneous cake, we wish to divide the cake among n players in a way that meets the following criteria: (I) every player (weakly) prefers his allocated cake to any other player's share (such notion is known as envy-freeness), (II) the mechanism is strategy-proof (truthful), and (III) the number of cuts made on the cake is minimal. We provide methods, namely expansion process and expansion process with unlocking, for dividing the cake under different assumptions on the valuation functions of the players. Copyright © 2017, Association for the Advancement of Artificial Intelligence... 

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 the share 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. Inspired by the models in the context of network diffusion, 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,...