Query Evaluation in Election Databases

Standard voting rules usually assume that the preferences of voters are provided in the form of complete rankings over a fixed set of alternatives. This assumption does not hold in applications like recommendation systems where the set of alternatives is extremely large and only partial preferences can be elicited. In this paper, we study the problem of strategic manipulation of voting rules that aggregate voter preferences provided in the form of pairwise comparisons between alternatives. Our contributions are twofold: first, we show that any onto pairwise voting rule is manipulable in principle. Next, we analyze how the computational complexity of manipulation of such rules varies with the structure of the graph induced by the pairs of alternatives that the manipulator is allowed to vote over and the type of the preference relation. Building on natural connections between the pairwise manipulation and sports elimination problems (including a mixed-elimination variant that we introduce in this paper), we show that manipulating pairwise voting rules can be computationally hard even in the single-manipulator setting, a setting where most standard voting rules are known to be easy to manipulate.

Classical results in voting theory show that strategic manipulation by voters is inevitable if a voting rule simultaneously satisfies certain desirable properties. Motivated by this, we study the relevant question of how often a voting rule is manipulable. It is well known that elections with a large number of voters are rarely manipulable under impartial culture (IC) assumption. However, the manipulability of voting rules when the number of candidates is large has hardly been addressed in the literature and our article focuses on this problem. First, we propose two properties: (i) asymptotic strategy-proofness and (ii) asymptotic collusion-proofness, with respect to new voters, which makes the two notions more relevant from the perspective of computational problem of manipulation. In addition to IC, we explore a new culture of society where all score vectors of the candidates are equally likely. This new notion has its motivation in computational social choice and we call it impartial scores culture (ISC) assumption. We study asymptotic strategy-proofness and asymptotic collusion-proofness for plurality, veto and [Formula: see text]-approval, and Borda voting rules under IC as well as ISC assumptions. Specifically, we prove bounds for the fraction of manipulable profiles when the number of candidates is large. Our results show that the size of the coalition and the tie-breaking rule play a crucial role in determining whether or not a voting rule satisfies the above two properties.

We introduce a general problem about bribery in voting systems. In the $\mathcal{R}$-Multi-Bribery problem, the goal is to bribe a set of voters at minimum cost such that a desired candidate wins the perturbed election under the voting rule R. Voters assign prices for withdrawing their vote, for swapping the positions of two consecutive candidates in their preference order, and for perturbing their approval count to favour candidates. As our main result, we show that $\mathcal{R}$-Multi-Bribery is fixed-parameter tractable parameterized by the number of candidates for many natural voting rules $\mathcal{R}$, including Kemeny rule, all scoring protocols, maximin rule, Bucklin rule, fallback rule, SP-AV, and any C1 rule. In particular, our result resolves the parameterized complexity of $\mathcal{R}$-Swap Bribery for all those voting rules, thereby solving a long-standing open problem and "Challenge #2" of the "Nine Research Challenges in Computational Social Choice" by Bredereck et al. Further, our algorithm runs in single-exponential time for arbitrary cost; it thus improves the earlier double-exponential time algorithm by Dorn and Schlotter that is restricted to the uniform-cost case for all scoring protocols, the maximin rule, and Bucklin rule.

The complexity of query evaluation in indefinite temporal constraint databases

Lu and Boutilier proposed a novel approach based on "minimax regret" to use classical score based voting rules in the setting where preferences can be any partial (instead of complete) orders over the set of alternatives. We show here that such an approach is vulnerable to a new kind of manipulation which was not present in the classical (where preferences are complete orders) world of voting. We call this attack "manipulative elicitation." More specifically, it may be possible to (partially) elicit the preferences of the agents in a way that makes some distinguished alternative win the election who may not be a winner if we elicit every preference completely. More alarmingly, we show that the related computational task is polynomial time solvable for a large class of voting rules which includes all scoring rules, maximin, Copeland α for every α ∈ [0,1], simplified Bucklin voting rules, etc. We then show that introducing a parameter per pair of alternatives which specifies the minimum number of partial preferences where this pair of alternatives must be comparable makes the related computational task of manipulative elicitation NP-complete for all common voting rules including a class of scoring rules which includes the plurality, k-approval, k-veto, veto, and Borda voting rules, maximin, Copeland α for every α ∈ [0,1], and simplified Bucklin voting rules. Hence, in this work, we discover a fundamental vulnerability in using minimax regret based approach in partial preferential setting and propose a novel way to tackle it.

A voting rule maps voter preferences into outcomes, and is called sophisticated if there exists a voting tree whose sophisticated outcomes coincide with the voting rule for every voter preference. As yet, no complete characterization of such rules is available. In this paper, we provide an important step toward this characterization by completely solving the problem when there are two possible sets of voter preferences.

We investigate the query evaluation problem for fixed queries over fully dynamic databases where tuples can be inserted or deleted. The task is to design a dynamic data structure that can immediately report the new result of a fixed query after every database update. We consider unions of conjunctive queries (UCQs) and focus on the query evaluation tasks testing (decide whether an input tuple belongs to the query result), enumeration (enumerate, without repetition, all tuples in the query result), and counting (output the number of tuples in the query result). We identify three increasingly restrictive classes of UCQs which we call t-hierarchical, q-hierarchical, and exhaustively q-hierarchical UCQs. Our main results provide the following dichotomies: If the query's homomorphic core is t-hierarchical (q-hierarchical, exhaustively q-hierarchical), then the testing (enumeration, counting) problem can be solved with constant update time and constant testing time (delay, counting time). Otherwise, it cannot be solved with sublinear update time and sublinear testing time (delay, counting time), unless the OV-conjecture and/or the OMv-conjecture fails. We also study the complexity of query evaluation in the dynamic setting in the presence of integrity constraints, and we obtain similar dichotomy results for the special case of small domain constraints (i.e., constraints which state that all values in a particular column of a relation belong to a fixed domain of constant size).

The concept of distance rationalizability of voting rules has been explored in recent years by several authors. Roughly speaking, we first choose a consensus set of elections (defined via preferences of voters over candidates) for which the result is specified a priori (intuitively, these are elections on which all voters can easily agree on the result). We also choose a measure of distance between elections. The result of an election outside the consensus set is defined to be the result of the closest consensual election under the distance measure. Most previous work has dealt with a definition in terms of preference profiles. However, most voting rules in common use are anonymous and homogeneous. In this case there is a much more succinct representation (using the voting simplex) of the inputs to the rule. This representation has been widely used in the voting literature, but rarely in the context of distance rationalizability. We show exactly how to connect distance rationalizability on profiles for anonymous and homogeneous rules to geometry in the simplex. We develop the connection for the important special case of votewise distances, recently introduced and studied by Elkind, Faliszewski and Slinko in several papers. This yields a direct interpretation in terms of well-developed mathematical concepts not seen before in the voting literature, namely Kantorovich (also called Wasserstein) distances and the geometry of Minkowski spaces. As an application of this approach, we prove some positive and some negative results about the decisiveness of distance rationalizable anonymous and homogeneous rules. The positive results connect with the recent theory of hyperplane rules, while the negative ones deal with distances that are not metrics, controversial notions of consensus, and the fact that the $\ell^1$-norm is not strictly convex.

In computational social choice, shift bribery is the procedure of paying voters to shift the briber's preferred candidate forward in their preferences so as to make this candidate an election winner; the more general swap bribery procedure also allows one to pay voters to swap other candidates in their preferences. The complexity of swap and shift bribery is well-understood for many voting rules; typically, finding a minimum-cost bribery is computationally hard. In this paper we initiate the study of swap and shift bribery in the setting where voters' preferences are known to be single-peaked or single-crossing. We obtain polynomial-time algorithms for several variants of these problems for classic voting rules, such as Plurality, Borda and Condorcet-consistent rules.

On specifying database updates

Predicting winner and estimating margin of victory in elections using sampling

In this paper, we study the metric distortion of deterministic social choice rules that choose a winning candidate from a set of candidates based on voter preferences. Voters and candidates are located in an underlying metric space. A voter has cost equal to her distance to the winning candidate. Ordinal social choice rules only have access to the ordinal preferences of the voters that are assumed to be consistent with the metric distances. Our goal is to design an ordinal social choice rule with minimum distortion, which is the worst-case ratio, over all consistent metrics, between the social cost of the rule and that of the optimal omniscient rule with knowledge of the underlying metric space. The distortion of the best deterministic social choice rule was known to be between $3$ and $5$. It had been conjectured that any rule that only looks at the weighted tournament graph on the candidates cannot have distortion better than $5$. In our paper, we disprove it by presenting a weighted tournament rule with distortion of $4.236$. We design this rule by generalizing the classic notion of uncovered sets, and further show that this class of rules cannot have distortion better than $4.236$. We then propose a new voting rule, via an alternative generalization of uncovered sets. We show that if a candidate satisfying the criterion of this voting rule exists, then choosing such a candidate yields a distortion bound of $3$, matching the lower bound. We present a combinatorial conjecture that implies distortion of $3$, and verify it for small numbers of candidates and voters by computer experiments. Using our framework, we also show that selecting any candidate guarantees distortion of at most $3$ when the weighted tournament graph is cyclically symmetric.

Optimal defense against election control by deleting voter groups

We study the complexity of estimating the probability of an outcome in an election over probabilistic votes. The focus is on voting rules expressed as positional scoring rules, and two models of probabilistic voters: the uniform distribution over the completions of a partial voting profile (consisting of a partial ordering of the candidates by each voter), and the Repeated Insertion Model (RIM) over the candidates, including the special case of the Mallows distribution. Past research has established that, while exact inference of the probability of winning is computationally hard (#P-hard), an additive polynomial-time approximation (additive FPRAS) is attained by sampling and averaging. There is often, though, a need for multiplicative approximation guarantees that are crucial for important measures such as conditional probabilities. Unfortunately, a multiplicative approximation of the probability of winning cannot be efficient (under conventional complexity assumptions) since it is already NP-complete to determine whether this probability is nonzero. Contrastingly, we devise multiplicative polynomial-time approximations (multiplicative FPRAS) for the probability of the complement event, namely, losing the election.

Comparing and ranking information is an important topic in social and information sciences, and in particular on the web. Its objective is to measure the difference of the preferences of voters on a set of candidates and to compute a consensus ranking. Commonly, each voter provides a total order of all candidates. Recently, this approach was generalized to bucket orders, which allow ties. In this work we further generalize and consider total, bucket, interval and partial orders. The disagreement between two orders is measured by the nearest neighbor Spearman footrule distance, which has not been studied so far. For two bucket orders and for a total and an interval order the nearest neighbor Spearman footrule distance is shown to be computable in linear time, whereas for a total and a partial order the computation is NP-hard, 4-approximable and fixed-parameter tractable. Moreover, in contrast to the well-known efficient solution of the rank aggregation problem for total orders, we prove the NP-completeness for bucket orders and establish a 4-approximation.