Computational Complexity of Fundamental Problems in Social Choice Theory

Computational social choice (comsoc) theory is currently an important area of research in computer science and more specifically in AI. The field started with the pioneering work of Bartholdi et al. in 1989 where they explored the the possibility of using computational intractability as a barrier against manipulation. Following that, a vast amount of research explored computational complexity of various problems in the context of social choice theory. We, in this thesis, study some of the fundamental problems in this domain. Manipulation of voting rules is a well known phenomena is social choice theory. Till date, researchers have studied a plenty of ways to make manipulation either impossible or computationally intractable. Yet, there are not a single satisfactory solution to prevent manipulation. In such a scenario where prevention fails even after considerable research effort of more than four decades, a natural research direction is to explore detection of manipulation. This is precisely the goal of one of our works [2] in this thesis. Another very well studied problem in comsoc is the possible winner problem. There exist quite a large literature studying computational aspects of this problem for various commonly used voting rules. Researchers also studied parameterized complexity of this problem and a fixed parameter tractability result with parameter being the number of candidates follows very easily by reducing it to an integer program. However, the kernelization complexity of this problem is surprisingly a big open problem in comsoc. In one of our works [3], we resolve this open question for many commonly used voting rules. Arguably the most fundamental problem in comsoc is winner determination-given a set of votes and a voting rule, who wins the election? In exit polls and many other applications, people often tries to predict the winner of an upcoming election by sampling a few votes and running the election on those sampled votes. We study the sample complexity of winner prediction for many common voting rules and show upper and lower bounds on the sample complexity [1]. Moreover, the upper and lower bounds match for most of the voting rules.