Branch-and-Prune Search Strategies for Numerical Constraint Solving

When solving numerical constraints such as nonlinear equations and inequalities, solvers often exploit pruning techniques, which remove redundant value combinations from the domains of variables, at pruning steps. To find the complete solution set, most of these solvers alternate the pruning steps with branching steps, which split each problem into subproblems. This forms the so-called branch-and-prune framework, well known among the approaches for solving numerical constraints. The basic branch-and-prune search strategy that uses domain bisections in place of the branching steps is called the bisection search. In general, the bisection search works well in case (i) the solutions are isolated, but it can be improved further in case (ii) there are continuums of solutions (this often occurs when inequalities are involved). In this paper, we propose a new branch-and-prune search strategy along with several variants, which not only allow yielding better branching decisions in the latter case, but also work as well as the bisection search does in the former case. These new search algorithms enable us to employ various pruning techniques in the construction of inner and outer approximations of the solution set. Our experiments show that these algorithms speed up the solving process often by one order of magnitude or more when solving problems with continuums of solutions, while keeping the same performance as the bisection search when the solutions are isolated.Comment: 43 pages, 11 figure

Size versus truthfulness in the house allocation problem

We study the House Allocation problem (also known as the Assignment problem), i.e., the problem of allocating a set of objects among a set of agents, where each agent has ordinal preferences (possibly involving ties) over a subset of the objects. We focus on truthful mechanisms without monetary transfers for finding large Pareto optimal matchings. It is straightforward to show that no deterministic truthful mechanism can approximate a maximum cardinality Pareto optimal matching with ratio better than 2. We thus consider randomized mechanisms. We give a natural and explicit extension of the classical Random Serial Dictatorship Mechanism (RSDM) specifically for the House Allocation problem where preference lists can include ties. We thus obtain a universally truthful randomized mechanism for finding a Pareto optimal matching and show that it achieves an approximation ratio of eovere-1. The same bound holds even when agents have priorities (weights) and our goal is to find a maximum weight (as opposed to maximum cardinality) Pareto optimal matching. On the other hand we give a lower bound of 18 over 13 on the approximation ratio of any universally truthful Pareto optimal mechanism in settings with strict preferences. In the case that the mechanism must additionally be non-bossy, an improved lower bound of eovere-1 holds. This lower bound is tight given that RSDM for strict preference lists is non-bossy. We moreover interpret our problem in terms of the classical secretary problem and prove that our mechanism provides the best randomized strategy of the administrator who interviews the applicants

Size versus truthfulness in the House Allocation problem

We study the House Allocation problem (also known as the Assignment problem), i.e., the problem of allocating a set of objects among a set of agents, where each agent has ordinal preferences (possibly involving ties) over a subset of the objects. We focus on truthful mechanisms without monetary transfers for finding large Pareto optimal matchings. It is straightforward to show that no deterministic truthful mechanism can approximate a maximum cardinality Pareto optimal matching with ratio better than 2. We thus consider randomised mechanisms. We give a natural and explicit extension of the classical Random Serial Dictatorship Mechanism (RSDM) specifically for the House Allocation problem where preference lists can include ties. We thus obtain a universally truthful randomised mechanism for finding a Pareto optimal matching and show that it achieves an approximation ratio of $\frac{e}{e-1}$. The same bound holds even when agents have priorities (weights) and our goal is to find a maximum weight (as opposed to maximum cardinality) Pareto optimal matching. On the other hand we give a lower bound of $\frac{18}{13}$ on the approximation ratio of any universally truthful Pareto optimal mechanism in settings with strict preferences. In the case that the mechanism must additionally be non-bossy with an additional technical assumption, we show by utilising a result of Bade that an improved lower bound of $\frac{e}{e-1}$ holds. This lower bound is tight since RSDM for strict preference lists is non-bossy. We moreover interpret our problem in terms of the classical secretary problem and prove that our mechanism provides the best randomised strategy of the administrator who interviews the applicants.Comment: To appear in Algorithmica (preliminary version appeared in the Proceedings of EC 2014

Multi-Winner Voting with Approval Preferences

Approval-based committee (ABC) rules are voting rules that output a fixed-size subset of candidates, a so-called committee. ABC rules select committees based on dichotomous preferences, i.e., a voter either approves or disapproves a candidate. This simple type of preferences makes ABC rules widely suitable for practical use. In this book, we summarize the current understanding of ABC rules from the viewpoint of computational social choice. The main focus is on axiomatic analysis, algorithmic results, and relevant applications.Comment: This is a draft of the upcoming book "Multi-Winner Voting with Approval Preferences

Using Choice Experiments for Non-Market Valuation

This paper provides the latest research developments in the method of choice experiments applied to valuation of non-market goods. Choice experiments, along with the, by now, well-known contingent valuation method, are very important tools for valuing non-market goods and the results are used in both cost-benefit analyses and litigations related to damage assessments. The paper should provide the reader with both the means to carry out a choice experiment and to conduct a detailed critical analysis of its performance in order to give informed advice about the results. A discussion of the underlying economic model of choice experiments is incorporated, as well as a presentation of econometric models consistent with economic theory. Furthermore, a detailed discussion on the development of a choice experiment is provided, which in particular focuses on the design of the experiment and tests of validity. Finally, a discussion on different ways to calculate welfare effects is presented.Choice experiments; non-market goods; stated preference methods; valuation

Multi-Winner Voting with Approval Preferences

From fundamental concepts and results to recent advances in computational social choice, this open access book provides a thorough and in-depth look at multi-winner voting based on approval preferences. The main focus is on axiomatic analysis, algorithmic results and several applications that are relevant in artificial intelligence, computer science and elections of any kind. What is the best way to select a set of candidates for a shortlist, for an executive committee, or for product recommendations? Multi-winner voting is the process of selecting a fixed-size set of candidates based on the preferences expressed by the voters. A wide variety of decision processes in settings ranging from politics (parliamentary elections) to the design of modern computer applications (collaborative filtering, dynamic Q&A platforms, diversity in search results, etc.) share the problem of identifying a representative subset of alternatives. The study of multi-winner voting provides the principled analysis of this task. Approval-based committee voting rules (in short: ABC rules) are multi-winner voting rules particularly suitable for practical use. Their usability is founded on the straightforward form in which the voters can express preferences: voters simply have to differentiate between approved and disapproved candidates. Proposals for ABC rules are numerous, some dating back to the late 19th century while others have been introduced only very recently. This book explains and discusses these rules, highlighting their individual strengths and weaknesses. With the help of this book, the reader will be able to choose a suitable ABC voting rule in a principled fashion, participate in, and be up to date with the ongoing research on this topic

General equilibrium programming

Equilibrium Theory;Algorithm

Allocation in Practice

How do we allocate scarcere sources? How do we fairly allocate costs? These are two pressing challenges facing society today. I discuss two recent projects at NICTA concerning resource and cost allocation. In the first, we have been working with FoodBank Local, a social startup working in collaboration with food bank charities around the world to optimise the logistics of collecting and distributing donated food. Before we can distribute this food, we must decide how to allocate it to different charities and food kitchens. This gives rise to a fair division problem with several new dimensions, rarely considered in the literature. In the second, we have been looking at cost allocation within the distribution network of a large multinational company. This also has several new dimensions rarely considered in the literature.Comment: To appear in Proc. of 37th edition of the German Conference on Artificial Intelligence (KI 2014), Springer LNC