Algorithms for Manipulating Sequential Allocation

Sequential allocation is a simple and widely studied mechanism to allocate indivisible items in turns to agents according to a pre-specified picking sequence of agents. At each turn, the current agent in the picking sequence picks its most preferred item among all items having not been allocated yet. This problem is well-known to be not strategyproof, i.e., an agent may get more utility by reporting an untruthful preference ranking of items. It arises the problem: how to find the best response of an agent? It is known that this problem is polynomially solvable for only two agents and NP-complete for arbitrary number of agents. The computational complexity of this problem with three agents was left as an open problem. In this paper, we give a novel algorithm that solves the problem in polynomial time for each fixed number of agents. We also show that an agent can always get at least half of its optimal utility by simply using its truthful preference as the response

Group Activity Selection with Few Agent Types

The Group Activity Selection Problem (GASP) models situations where a group of agents needs to be distributed to a set of activities while taking into account preferences of the agents w.r.t. individual activities and activity sizes. The problem, along with its well-known variants sGASP and gGASP, has previously been studied in the parameterized complexity setting with various parameterizations, such as number of agents, number of activities and solution size. However, the complexity of the problem parameterized by the number of types of agents, a natural parameter proposed already in the first paper that introduced GASP, has so far remained unexplored. In this paper we establish the complexity map for GASP, sGASP and gGASP when the number of types of agents is the parameter. Our positive results, consisting of one fixed-parameter algorithm and one XP algorithm, rely on a combination of novel Subset Sum machinery (which may be of general interest) and identifying certain compression steps which allow us to focus on solutions which are "acyclic". These algorithms are complemented by matching lower bounds, which among others close a gap to a recently obtained tractability result of Gupta, Roy, Saurabh and Zehavi (2017). In this direction, the techniques used to establish W[1]-hardness of sGASP are of particular interest: as an intermediate step, we use Sidon sequences to show the W[1]-hardness of a highly restricted variant of multi-dimensional Subset Sum, which may find applications in other settings as well

Pareto Front Identification with Regret Minimization

We consider Pareto front identification for linear bandits (PFILin) where the goal is to identify a set of arms whose reward vectors are not dominated by any of the others when the mean reward vector is a linear function of the context. PFILin includes the best arm identification problem and multi-objective active learning as special cases. The sample complexity of our proposed algorithm is $\tilde{O}(d/\Delta^2)$, where $d$ is the dimension of contexts and $\Delta$ is a measure of problem complexity. Our sample complexity is optimal up to a logarithmic factor. A novel feature of our algorithm is that it uses the contexts of all actions. In addition to efficiently identifying the Pareto front, our algorithm also guarantees $\tilde{O}(\sqrt{d/t})$ bound for instantaneous Pareto regret when the number of samples is larger than $\Omega(d\log dL)$ for $L$ dimensional vector rewards. By using the contexts of all arms, our proposed algorithm simultaneously provides efficient Pareto front identification and regret minimization. Numerical experiments demonstrate that the proposed algorithm successfully identifies the Pareto front while minimizing the regret.Comment: 25 pages including appendi

Core Stability in Additively Separable Hedonic Games of Low Treewidth

Additively Separable Hedonic Game (ASHG) are coalition-formation games where we are given a graph whose vertices represent $n$ selfish agents and the weight of each edge $uv$ denotes how much agent $u$ gains (or loses) when she is placed in the same coalition as agent $v$. We revisit the computational complexity of the well-known notion of core stability of ASHGs, where the goal is to construct a partition of the agents into coalitions such that no group of agents would prefer to diverge from the given partition and form a new (blocking) coalition. Since both finding a core stable partition and verifying that a given partition is core stable are intractable problems ($\Sigma_2^p$-complete and coNP-complete respectively) we study their complexity from the point of view of structural parameterized complexity, using standard graph-theoretic parameters, such as treewidth