82 research outputs found
Pareto-Optimal Allocation of Indivisible Goods with Connectivity Constraints
We study the problem of allocating indivisible items to agents with additive
valuations, under the additional constraint that bundles must be connected in
an underlying item graph. Previous work has considered the existence and
complexity of fair allocations. We study the problem of finding an allocation
that is Pareto-optimal. While it is easy to find an efficient allocation when
the underlying graph is a path or a star, the problem is NP-hard for many other
graph topologies, even for trees of bounded pathwidth or of maximum degree 3.
We show that on a path, there are instances where no Pareto-optimal allocation
satisfies envy-freeness up to one good, and that it is NP-hard to decide
whether such an allocation exists, even for binary valuations. We also show
that, for a path, it is NP-hard to find a Pareto-optimal allocation that
satisfies maximin share, but show that a moving-knife algorithm can find such
an allocation when agents have binary valuations that have a non-nested
interval structure.Comment: 21 pages, full version of paper at AAAI-201
Hedonic Games with Graph-restricted Communication
We study hedonic coalition formation games in which cooperation among the
players is restricted by a graph structure: a subset of players can form a
coalition if and only if they are connected in the given graph. We investigate
the complexity of finding stable outcomes in such games, for several notions of
stability. In particular, we provide an efficient algorithm that finds an
individually stable partition for an arbitrary hedonic game on an acyclic
graph. We also introduce a new stability concept -in-neighbor stability- which
is tailored for our setting. We show that the problem of finding an in-neighbor
stable outcome admits a polynomial-time algorithm if the underlying graph is a
path, but is NP-hard for arbitrary trees even for additively separable hedonic
games; for symmetric additively separable games we obtain a PLS-hardness
result
Cooperative Games with Bounded Dependency Degree
Cooperative games provide a framework to study cooperation among
self-interested agents. They offer a number of solution concepts describing how
the outcome of the cooperation should be shared among the players.
Unfortunately, computational problems associated with many of these solution
concepts tend to be intractable---NP-hard or worse. In this paper, we
incorporate complexity measures recently proposed by Feige and Izsak (2013),
called dependency degree and supermodular degree, into the complexity analysis
of cooperative games. We show that many computational problems for cooperative
games become tractable for games whose dependency degree or supermodular degree
are bounded. In particular, we prove that simple games admit efficient
algorithms for various solution concepts when the supermodular degree is small;
further, we show that computing the Shapley value is always in FPT with respect
to the dependency degree. Finally, we note that, while determining the
dependency among players is computationally hard, there are efficient
algorithms for special classes of games.Comment: 10 pages, full version of accepted AAAI-18 pape
On Parameterized Complexity of Group Activity Selection Problems on Social Networks
In Group Activity Selection Problem (GASP), players form coalitions to
participate in activities and have preferences over pairs of the form
(activity, group size). Recently, Igarashi et al. have initiated the study of
group activity selection problems on social networks (gGASP): a group of
players can engage in the same activity if the members of the group form a
connected subset of the underlying communication structure. Igarashi et al.
have primarily focused on Nash stable outcomes, and showed that many associated
algorithmic questions are computationally hard even for very simple networks.
In this paper we study the parameterized complexity of gGASP with respect to
the number of activities as well as with respect to the number of players, for
several solution concepts such as Nash stability, individual stability and core
stability. The first parameter we consider in the number of activities. For
this parameter, we propose an FPT algorithm for Nash stability for the case
where the social network is acyclic and obtain a W[1]-hardness result for
cliques (i.e., for classic GASP); similar results hold for individual
stability. In contrast, finding a core stable outcome is hard even if the
number of activities is bounded by a small constant, both for classic GASP and
when the social network is a star. Another parameter we study is the number of
players. While all solution concepts we consider become polynomial-time
computable when this parameter is bounded by a constant, we prove W[1]-hardness
results for cliques (i.e., for classic GASP).Comment: 9 pages, long version of accepted AAMAS-17 pape
Forming Probably Stable Communities with Limited Interactions
A community needs to be partitioned into disjoint groups; each community
member has an underlying preference over the groups that they would want to be
a member of. We are interested in finding a stable community structure: one
where no subset of members wants to deviate from the current structure. We
model this setting as a hedonic game, where players are connected by an
underlying interaction network, and can only consider joining groups that are
connected subgraphs of the underlying graph. We analyze the relation between
network structure, and one's capability to infer statistically stable (also
known as PAC stable) player partitions from data. We show that when the
interaction network is a forest, one can efficiently infer PAC stable coalition
structures. Furthermore, when the underlying interaction graph is not a forest,
efficient PAC stabilizability is no longer achievable. Thus, our results
completely characterize when one can leverage the underlying graph structure in
order to compute PAC stable outcomes for hedonic games. Finally, given an
unknown underlying interaction network, we show that it is NP-hard to decide
whether there exists a forest consistent with data samples from the network.Comment: 11 pages, full version of accepted AAAI-19 pape
Envy-free division of multi-layered cakes
We study the problem of dividing a multi-layered cake among heterogeneous
agents under non-overlapping constraints. This problem, recently proposed by
Hosseini et al. (2020), captures several natural scenarios such as the
allocation of multiple facilities over time where each agent can utilize at
most one facility simultaneously, and the allocation of tasks over time where
each agent can perform at most one task simultaneously. We establish the
existence of an envy-free multi-division that is both non-overlapping and
contiguous within each layered cake when the number of agents is a prime
power and the number of layers is at most , thus providing a positive
partial answer to a recent open question. To achieve this, we employ a new
approach based on a general fixed point theorem, originally proven by Volovikov
(1996), and recently applied by Joji\'{c}, Panina, and {\v{Z}}ivaljevi\'{c}
(2020) to the envy-free division problem of a cake. We further show that for a
two-layered cake division among three agents with monotone preferences, an
-approximate envy-free solution that is both non-overlapping and
contiguous can be computed in logarithmic time of .Comment: 21 page
Keeping the Harmony Between Neighbors: Local Fairness in Graph Fair Division
We study the problem of allocating indivisible resources under the
connectivity constraints of a graph . This model, initially introduced by
Bouveret et al. (published in IJCAI, 2017), effectively encompasses a diverse
array of scenarios characterized by spatial or temporal limitations, including
the division of land plots and the allocation of time plots. In this paper, we
introduce a novel fairness concept that integrates local comparisons within the
social network formed by a connected allocation of the item graph. Our
particular focus is to achieve pairwise-maximin fair share (PMMS) among the
"neighbors" within this network. For any underlying graph structure, we show
that a connected allocation that maximizes Nash welfare guarantees a
-PMMS fairness. Moreover, for two agents, we establish that a
-PMMS allocation can be efficiently computed. Additionally, we
demonstrate that for three agents and the items aligned on a path, a PMMS
allocation is always attainable and can be computed in polynomial time. Lastly,
when agents have identical additive utilities, we present a
pseudo-polynomial-time algorithm for a -PMMS allocation, irrespective of
the underlying graph . Furthermore, we provide a polynomial-time algorithm
for obtaining a PMMS allocation when is a tree.Comment: Full version of paper accepted for presentation at AAMAS 202
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