Palash Dey

Kernelization complexity of possible winner and coalitional manipulation problems in voting

In the Possible Winner problem in computational social choice theory, we are given a set of partial preferences and the question is whether a distinguished candidate could be made winner by extending the partial preferences to linear preferences. Previous work has provided, for many common voting rules, fixed parameter tractable algorithms for the Possible Winner problem, with number of candidates as the parameter. However, the corresponding kernelization question is still open and in fact, has been mentioned as a key research challenge [10]. In this paper, we settle this open question for many common voting rules. We show that the Possible Winner problem for maximin, Copeland, Bucklin, ranked pairs, and a class of scoring rules that include the Borda voting rule do not admit a polynomial kernel with the number of candidates as the parameter. We show however that the Coalitional Manipulation problem which is an important special case of the Possible Winner problem does admit a polynomial kernel for maximin, Copeland, ranked pairs, and a class of scoring rules that includes the Borda voting rule, when the number of manipulators is polynomial in the number of candidates. A significant conclusion of our work is that the Possible Winner problem is harder than the Coalitional Manipulation problem since the Coalitional Manipulation problem admits a polynomial kernel whereas the Possible Winner problem does not admit a polynomial kernel.

An Optimal Algorithm for l1-Heavy Hitters in Insertion Streams and Related Problems

We give the first optimal bounds for returning the l1-heavy hitters in a data stream of insertions, together with their approximate frequencies, closing a long line of work on this problem. For a stream of m items in {1, 2, ..., n} and parameters 0 < ε < φ ≤ 1, let fi denote the frequency of item i, i.e., the number of times item i occurs in the stream. With arbitrarily large constant probability, our algorithm returns all items i for which fi ≥ φ m, returns no items j for which fj ≤ (φ -ε)m, and returns approximations ~fi with |~fi - fi| ≤ ε m for each item i that it returns. Our algorithm uses O(ε-1 logφ-1 + φ-1 log n + log log m) bits of space, processes each stream update in O(1) worst-case time, and can report its output in time linear in the output size. We also prove a lower bound, which implies that our algorithm is optimal up to a constant factor in its space complexity. A modification of our algorithm can be used to estimate the maximum frequency up to an additive ε m error in the above amount of space, resolving Question 3 in the IITK 2006 Workshop on Algorithms for Data Streams for the case of l1-heavy hitters. We also introduce several variants of the heavy hitters and maximum frequency problems, inspired by rank aggregation and voting schemes, and show how our techniques can be applied in such settings. Unlike the traditional heavy hitters problem, some of these variants look at comparisons between items rather than numerical values to determine the frequency of an item.

Frugal bribery in voting

Bribery in elections is an important problem in computational social choice theory. We introduce and study two important special cases of the bribery problem, namely, FRUGAL-BRIBERY and FRUGAL-

On the Exact Amount of Missing Information that Makes Finding Possible Winners Hard.

BRIBERY where the briber is frugal in nature. By this, we mean that the briber is only able to influence voters who benefit from the suggestion of the briber. More formally, a voter is vulnerable if the outcome of the election improves according to her own preference when she accepts the suggestion of the briber. In the FRUGAL-BRIBERY problem, the goal is to make a certain candidate win the election by changing only the vulnerable votes. In the FRUGAL-

Asymptotic Collusion-proofness of Voting Rules: The Case of Large Number of Candidates

BRIBERY problem, the vulnerable votes have prices and the goal is to make a certain candidate win the election by changing only the vulnerable votes, subject to a budget constraint. We show that both the FRUGAL-BRIBERY and the FRUGAL-

UNO Gets Easier for a Single Player

BRIBERY problems are intractable for many commonly used voting rules for weighted as well as unweighted elections. These intractability results demonstrate that bribery is a hard computational problem, in the sense that several special cases of this problem continue to be computationally intractable. This strengthens the view that bribery, although a possible attack on an election in principle, may be infeasible in practice.

An Optimal Algorithm for ℓ1-Heavy Hitters in Insertion Streams and Related Problems

We consider election scenarios with incomplete information, a situation that arises often in practice. There are several models of incomplete information and accordingly, different notions of outcomes of such elections. In one well-studied model of incompleteness, the votes are given by partial orders over the candidates. In this context we can frame the problem of finding a possible winner, which involves determining whether a given candidate wins in at least one completion of a given set of partial votes for a specific voting rule. The possible winner problem is well-known to be NP-complete in general, and it is in fact known to be NP-complete for several voting rules where the number of undetermined pairs in every vote is bounded only by some constant. In this paper, we address the question of determining precisely the smallest number of undetermined pairs for which the possible winner problem remains NP-complete. In particular, we find the exact values of

Sample Complexity for Winner Prediction in Elections

t

Complexity of manipulation with partial information in voting

for which the possible winner problem transitions to being NP-complete from being in P, where

Detecting Possible Manipulators in Elections

t