We consider a social choice setting in which agents and alternatives are represented by points in a metric space, and the cost of an agent for an alternative is the distance between the corresponding points in the space. The goal is to choose a single alternative to (approximately) minimize the social cost (cost of all agents) or the maximum cost of any agent, when only limited information about the preferences of the agents is given. Previous work has shown that the best possible distortion one can hope to achieve is $3$ when access to the ordinal preferences of the agents is given, even when the distances between alternatives in the metric space are known. We improve upon this bound of $3$ by designing deterministic mechanisms that exploit a bit of cardinal information. We show that it is possible to achieve distortion $1+\sqrt{2}$ by using the ordinal preferences of the agents, the distances between alternatives, and a threshold approval set per agent that contains all alternatives for whom her cost is within an appropriately chosen factor of her cost for her most-preferred alternative. We show that this bound is the best possible for any deterministic mechanism in general metric spaces, and also provide improved bounds for the fundamental case of a line metric.
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Density power divergence (DPD) [Basu et al. (1998), Biometrika], designed to estimate the underlying distribution of the observations robustly, comprises an integral term of the power of the parametric density models to be estimated. While the explicit form of the integral term can be obtained for some specific densities (such as normal density and exponential density), its computational intractability has prohibited the application of DPD-based estimation to more general parametric densities, over a quarter of a century since the proposal of DPD. This study proposes a stochastic optimization approach to minimize DPD for general parametric density models and explains its adequacy by referring to conventional theories on stochastic optimization. The proposed approach also can be applied to the minimization of another density power-based $\gamma$-divergence with the aid of unnormalized models [Kanamori and Fujisawa (2015), Biometrika].
We consider the problem of incentivising desirable behaviours in multi-agent systems by way of taxation schemes. Our study employs the concurrent games model: in this model, each agent is primarily motivated to seek the satisfaction of a goal, expressed as a Linear Temporal Logic (LTL) formula; secondarily, agents seek to minimise costs, where costs are imposed based on the actions taken by agents in different states of the game. In this setting, we consider an external principal who can influence agents' preferences by imposing taxes (additional costs) on the actions chosen by agents in different states. The principal imposes taxation schemes to motivate agents to choose a course of action that will lead to the satisfaction of their goal, also expressed as an LTL formula. However, taxation schemes are limited in their ability to influence agents' preferences: an agent will always prefer to satisfy its goal rather than otherwise, no matter what the costs. The fundamental question that we study is whether the principal can impose a taxation scheme such that, in the resulting game, the principal's goal is satisfied in at least one or all runs of the game that could arise by agents choosing to follow game-theoretic equilibrium strategies. We consider two different types of taxation schemes: in a static scheme, the same tax is imposed on a state-action profile pair in all circumstances, while in a dynamic scheme, the principal can choose to vary taxes depending on the circumstances. We investigate the main game-theoretic properties of this model as well as the computational complexity of the relevant decision problems.
Social distance games have been extensively studied as a coalition formation model where the utilities of agents in each coalition were captured using a utility function u that took into account distances in a given social network. In this paper, we consider a non-normalized score-based definition of social distance games where the utility function u_v depends on a generic scoring vector v, which may be customized to match the specifics of each individual application scenario. As our main technical contribution, we establish the tractability of computing a welfare-maximizing partitioning of the agents into coalitions on tree-like networks, for every score-based function u_v. We provide more efficient algorithms when dealing with specific choices of u_v or simpler networks, and also extend all of these results to computing coalitions that are Nash stable or individually rational. We view these results as a further strong indication of the usefulness of the proposed score-based utility function: even on very simple networks, the problem of computing a welfare-maximizing partitioning into coalitions remains open for the originally considered canonical function u.
\textit{Pursuit-evasion games} have been intensively studied for several decades due to their numerous applications in artificial intelligence, robot motion planning, database theory, distributed computing, and algorithmic theory. \textsc{Cops and Robber} (\CR) is one of the most well-known pursuit-evasion games played on graphs, where multiple \textit{cops} pursue a single \textit{robber}. The aim is to compute the \textit{cop number} of a graph, $k$, which is the minimum number of cops that ensures the \textit{capture} of the robber. From the viewpoint of parameterized complexity, \CR is W[2]-hard parameterized by $k$~[Fomin et al., TCS, 2010]. Thus, we study structural parameters of the input graph. We begin with the \textit{vertex cover number} ($\mathsf{vcn}$). First, we establish that $k \leq \frac{\mathsf{vcn}}{3}+1$. Second, we prove that \CR parameterized by $\mathsf{vcn}$ is \FPT by designing an exponential kernel. We complement this result by showing that it is unlikely for \CR parameterized by $\mathsf{vcn}$ to admit a polynomial compression. We extend our exponential kernels to the parameters \textit{cluster vertex deletion number} and \textit{deletion to stars number}, and design a linear vertex kernel for \textit{neighborhood diversity}. Additionally, we extend all of our results to several well-studied variations of \CR.
We consider additive Schwarz methods for boundary value problems involving the $p$-Laplacian. Although the existing theoretical estimates indicate a sublinear convergence rate for these methods, empirical evidence from numerical experiments demonstrates a linear convergence rate. In this paper, we narrow the gap between these theoretical and empirical results by presenting a novel convergence analysis. Firstly, we present an abstract convergence theory of additive Schwarz methods written in terms of a quasi-norm. This quasi-norm exhibits behavior similar to the Bregman distance of the convex energy functional associated to the problem. Secondly, we provide a quasi-norm version of the Poincar'{e}--Friedrichs inequality, which is essential for deriving a quasi-norm stable decomposition for a two-level domain decomposition setting. By utilizing these two key elements, we establish a new bound for the linear convergence rate of the methods.
In this work we study systems consisting of a group of moving particles. In such systems, often some important parameters are unknown and have to be estimated from observed data. Such parameter estimation problems can often be solved via a Bayesian inference framework. However in many practical problems, only data at the aggregate level is available and as a result the likelihood function is not available, which poses challenge for Bayesian methods. In particular, we consider the situation where the distributions of the particles are observed. We propose a Wasserstein distance based sequential Monte Carlo sampler to solve the problem: the Wasserstein distance is used to measure the similarity between the observed and the simulated particle distributions and the sequential Monte Carlo samplers is used to deal with the sequentially available observations. Two real-world examples are provided to demonstrate the performance of the proposed method.
We deal with linear programming problems involving absolute values in their formulations, so that they are no more expressible as standard linear programs. The presence of absolute values causes the problems to be nonconvex and nonsmooth, so hard to solve. In this paper, we study fundamental properties on the topology and the geometric shape of the solution set, and also conditions for convexity, connectedness, boundedness and integrality of the vertices. Further, we address various complexity issues, showing that many basic questions are NP-hard to solve. We show that the feasible set is a (nonconvex) polyhedral set and, more importantly, every nonconvex polyhedral set can be described by means of absolute value constraints. We also provide a necessary and sufficient condition when a KKT point of a nonconvex quadratic programming reformulation solves the original problem.
We consider the multilinear polytope which arises naturally in binary polynomial optimization. Del Pia and Di Gregorio introduced the class of odd $\beta$-cycle inequalities valid for this polytope, showed that these generally have Chv{\'a}tal rank 2 with respect to the standard relaxation and that, together with flower inequalities, they yield a perfect formulation for cycle hypergraph instances. Moreover, they describe a separation algorithm in case the instance is a cycle hypergraph. We introduce a weaker version, called simple odd $\beta$-cycle inequalities, for which we establish a strongly polynomial-time separation algorithm for arbitrary instances. These inequalities still have Chv{\'a}tal rank 2 in general and still suffice to describe the multilinear polytope for cycle hypergraphs. Finally, we report about computational results of our prototype implementation. The simple odd $\beta$-cycle inequalities sometimes help to close more of the integrality gap in the experiments; however, the preliminary implementation has substantial computational cost, suggesting room for improvement in the separation algorithm.
Positive linear programs (LPs) model many graph and operations research problems. One can solve for a $(1+\epsilon)$-approximation for positive LPs, for any selected $\epsilon$, in polylogarithmic depth and near-linear work via variations of the multiplicative weight update (MWU) method. Despite extensive theoretical work on these algorithms through the decades, their empirical performance is not well understood. In this work, we implement and test an efficient parallel algorithm for solving positive LP relaxations, and apply it to graph problems such as densest subgraph, bipartite matching, vertex cover and dominating set. We accelerate the algorithm via a new step size search heuristic. Our implementation uses sparse linear algebra optimization techniques such as fusion of vector operations and use of sparse format. Furthermore, we devise an implicit representation for graph incidence constraints. We demonstrate the parallel scalability with the use of threading OpenMP and MPI on the Stampede2 supercomputer. We compare this implementation with exact libraries and specialized libraries for the above problems in order to evaluate MWU's practical standing for both accuracy and performance among other methods. Our results show this implementation is faster than general purpose LP solvers (IBM CPLEX, Gurobi) in all of our experiments, and in some instances, outperforms state-of-the-art specialized parallel graph algorithms.
Image segmentation is an important component of many image understanding systems. It aims to group pixels in a spatially and perceptually coherent manner. Typically, these algorithms have a collection of parameters that control the degree of over-segmentation produced. It still remains a challenge to properly select such parameters for human-like perceptual grouping. In this work, we exploit the diversity of segments produced by different choices of parameters. We scan the segmentation parameter space and generate a collection of image segmentation hypotheses (from highly over-segmented to under-segmented). These are fed into a cost minimization framework that produces the final segmentation by selecting segments that: (1) better describe the natural contours of the image, and (2) are more stable and persistent among all the segmentation hypotheses. We compare our algorithm's performance with state-of-the-art algorithms, showing that we can achieve improved results. We also show that our framework is robust to the choice of segmentation kernel that produces the initial set of hypotheses.
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