Problems in Computational Social Choice on Restricted Domains
Social Choice is the formalized study of the aggregation of individual preferences towards collective decision-making. The task of checking the feasibility of the preference aggregation schemes in the real world context comes under Computational Social Choice (COMSOC). This thesis considers the problems from the three pillars of COMSOC - Voting, FairDivision, and Matching under Preferences.
Several appealing aggregation schemes turn out to be computationally hard. The work from this thesis largely focuses on studying problems in these areas in the context of domain restrictions. We consider popular notions of domain restrictions such as Single-Peaked (SP), Single-Crossing (SC), and Euclidean preference domains. The motivation for studying structured domains is to explore if certain properties that are elusive in the general case can be achieved by considering appropriate and practically relevant domain restrictions. For instance, some voting rules that are hard to compute in general become tractable on restricted domains. Another illustration is that some domains admit mechanisms that are not vulnerable to strategic behavior. We often find that simply projecting mechanisms for general domains to structured domains does not give us enough leverage, so the quest is usually to design new mechanisms that take advantage of the specific structure of the domain under consideration.
- Voting: Our work in voting mainly focuses on the Chamberlin Courant (CC) voting rule. Several fundamental problems, such as winner determination, are hard on for CC rule. We focus on the restricted and nearly restricted domains for this problem. Our work provides a fine-grained analysis of the boundary between tractability and intractability using parametric analysis. We also consider the effect of small perturbations in election instances formally captured by the concept of the robustness radius and provide an XP algorithm along with the matching hardness results.
- Stable Matching: For many hard matching problems, we show that the hardness persists even the domain restrictions. We also provide an interesting addition to a class of instances that admit a unique stable matching (marriage for the bipartite case). We introduce a new variant of the matching problem called ’Matching Critical Sets’ and pin down its complexity. Finally, we work on the special cases of (a,b)-supermatch. We also analyze the performance of the Gale-Shapley mechanism on instances with restricted domains through simulations.
- Fair Division: We study the fair allocation of indivisible items arranged on a path under the constraint that only connected subsets of items may be allocated to the agents. We study the problem for both Envyfreeness (EF1) and Equitability (EQ1). We show that achieving EF1 or EQ1 in conjunction with well-studied measures of efficiency (such as Pareto optimality, non-wastefulness, maximum egalitarian, or utilitarian welfare) is computationally hard even for binary valuations. On the algorithmic side, we show that an EQ1 allocation with the highest egalitarian welfare among all consistent allocations can be efficiently computed for any fixed ordering of agents. For structured binary valuations, we obtain polynomial-time algorithms for non-wasteful and Pareto optimal EF1 and EQ1 allocations.