Strategy Proof Mechanisms for Facility Location with Capacity Limits

An important feature of many real world facility location problems are capacity limits on the facilities. We show here how capacity constraints make it harder to design strategy proof mechanisms for facility location, but counter-intuitively can improve the guarantees on how well we can approximate the optimal solution.

Strategy Proof Mechanisms for Facility Location in Euclidean and Manhattan Space

We study the impact on mechanisms for facility location of moving from one dimension to two (or more) dimensions and Euclidean or Manhattan distances. We consider three fundamental axiomatic properties: anonymity which is a basic fairness property, Pareto optimality which is one of the most important efficiency properties, and strategy proofness which ensures agents do not have an incentive to mis-report. We also consider how well such mechanisms can approximate the optimal welfare. Our results are somewhat negative. Moving from one dimension to two (or more) dimensions often makes these axiomatic properties more difficult to achieve. For example, with two facilities in Euclidean space or with just a single facility in Manhattan space, no mechanism is anonymous, Pareto optimal and strategy proof. By contrast, mechanisms on the line exist with all three properties.We also show that approximation ratios may increase when moving to two (or more) dimensions. All our impossibility results are minimal. If we drop one of the three axioms (anonymity, Pareto optimality or strategy proofness) multiple mechanisms satisfy the other two axioms.

Strategy Proof Mechanisms for Facility Location at Limited Locations

Facility location problems often permit facilities to be located at any position. But what if this is not the case in practice? What if facilities can only be located at particular locations like a highway exit or close to a bus stop? We consider here the impact of such constraints on the location of facilities on the performance of strategy proof mechanisms for locating facilities.We study four different performance objectives: the total distance agents must travel to their closest facility, the maximum distance any agent must travel to their closest facility, and the utilitarian and egalitarian welfare.We show that constraining facilities to a limited set of locations makes all four objectives harder to approximate in general.

Adventures in Mathematical Reasoning

"Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness, where the explorers often get lost. Rigour should be a signal to the historian that the maps have been made, and the real explorers have gone elsewhere." W.S. Anglin, the Mathematical Intelligencer, 4 (4), 1982.

On the Complexity of Breaking Symmetry

We can break symmetry by eliminating solutions within a symmetry class that are not least in the lexicographical ordering. This is often referred to as the lex-leader method. Unfortunately, as symmetry groups can be large, the lexleader method is not tractable in general. We prove that using other total orderings besides the usual lexicographical ordering will not reduce the computational complexity of breaking symmetry in general. It follows that breaking symmetry with other orderings like the Gray code ordering or the Snake-Lex ordering is intractable in general.

Fair Division: The Computer Scientist's Perspective

I survey recent progress on a classic and challenging problem in social choice: the fair division of indivisible items. I discuss how a computational perspective has provided interesting insights into and understanding of how to divide items fairly and efficiently. This has involved bringing to bear tools such as those used in knowledge representation, computational complexity, approximation methods, game theory, online analysis and communication complexity

A Pebble in the AI Race

Bhutan is sometimes described as \a pebble between two boulders", a small country caught between the two most populous nations on earth: India and China. This pebble is, however, about to be caught up in a vortex: the transformation of our economic, political and social orders by new technologies like Artificial Intelligence. What can a small nation like Bhutan hope to do in the face of such change? What should the nation do, not just to weather this storm, but to become a better place in which to live?

Partial Queries for Constraint Acquisition

Learning constraint networks is known to require a number of membership queries exponential in the number of variables. In this paper, we learn constraint networks by asking the user partial queries. That is, we ask the user to classify assignments to subsets of the variables as positive or negative. We provide an algorithm, called QUACQ, that, given a negative example, focuses onto a constraint of the target network in a number of queries logarithmic in the size of the example. The whole constraint network can then be learned with a polynomial number of partial queries. We give information theoretic lower bounds for learning some simple classes of constraint networks and show that our generic algorithm is optimal in some cases.

From Matching with Diversity Constraints to Matching with Regional Quotas

In the past few years, several new matching models have been proposed and studied that take into account complex distributional constraints. Relevant lines of work include (1) school choice with diversity constraints where students have (possibly overlapping) types and (2) hospital-doctor matching where various regional quotas are imposed. In this paper, we present a polynomial-time reduction to transform an instance of (1) to an instance of (2) and we show how the feasibility and stability of corresponding matchings are preserved under the reduction. Our reduction provides a formal connection between two important strands of work on matching with distributional constraints. We then apply the reduction in two ways. Firstly, we show that it is NP-complete to check whether a feasible and stable outcome for (1) exists. Due to our reduction, these NP-completeness results carry over to setting (2). In view of this, we help unify some of the results that have been presented in the literature. Secondly, if we have positive results for (2), then we have corresponding results for (1). One key conclusion of our results is that further developments on axiomatic and algorithmic aspects of hospital-doctor matching with regional quotas will result in corresponding results for school choice with diversity constraints.

Greedy Algorithms for Fair Division of Mixed Manna

We consider a multi-agent model for fair division of mixed manna (i.e. items for which agents can have positive, zero or negative utilities), in which agents have additive utilities for bundles of items. For this model, we give several general impossibility results and special possibility results for three common fairness concepts (i.e. EF1, EFX, EFX3) and one popular efficiency concept (i.e. PO). We also study how these interact with common welfare objectives such as the Nash, disutility Nash and egalitarian welfares. For example, we show that maximizing the Nash welfare with mixed manna (or minimizing the disutility Nash welfare) does not ensure an EF1 allocation whereas with goods and the Nash welfare it does. We also prove that an EFX3 allocation may not exist even with identical utilities. By comparison, with tertiary utilities, EFX and PO allocations, or EFX3 and PO allocations always exist. Also, with identical utilities, EFX and PO allocations always exist. For these cases, we give polynomial-time algorithms, returning such allocations and approximating further the Nash, disutility Nash and egalitarian welfares in special cases.