Nonuniform Bribery

We study the concept of bribery in the situation where voters are willing to change their votes as we ask them, but where their prices depend on the nature of the change we request. Our model is an extension of the one of Faliszewski et al. [FHH06], where each voter has a single price for any change we may ask for. We show polynomial-time algorithms for our version of bribery for a broad range of voting protocols, including plurality, veto, approval, and utility based voting. In addition to our polynomial-time algorithms we provide NP-completeness results for a couple of our nonuniform bribery problems for weighted voters, and a couple of approximation algorithms for NP-complete bribery problems defined in [FHH06] (in particular, an FPTAS for plurality-weighted-$bribery problem)

Open Questions in the Theory of Semifeasible Computation

The study of semifeasible algorithms was initiated by Selman's work a quarter of century ago [Sel79,Sel81,Sel82]. Informally put, this research stream studies the power of those sets L for which there is a deterministic (or in some cases, the function may belong to one of various nondeterministic function classes) polynomial-time function f such that when at least one of x and y belongs to L, then f(x,y) \in L \cap \{x,y\}. The intuition here is that it is saying: ``Regarding membership in L, if you put a gun to my head and forced me to bet on one of x or y as belonging to L, my money would be on f(x,y).'' In this article, we present a number of open problems from the theory of semifeasible algorithms. For each we present its background and review what partial results, if any, are known

Separating the Notions of Self- and Autoreducibility

Recently Gla{\ss}er et al. have shown that for many classes $C$ including PSPACE and NP it holds that all of its nontrivial many-one complete languages are autoreducible. This immediately raises the question of whether all many-one complete languages are Turing self-reducible for such classes $C$. This paper considers a simpler version of this question---whether all PSPACE-complete NP-complete languages are length-decreasing self-reducible. We show that if all PSPACE-complete languages are length-decreasing self-reducible then PSPACE = P and that if all NP-complete languages are length-decreasing self-reducible then NP = P. The same type of result holds for many other natural complexity classes. In particular, we show that (1) not all NL-complete sets are logspace length-decreasing self-reducible, (2) unconditionally not all PSPACE-complete languages are logspace length-decreasing self-reducible, and (3) unconditionally not all EXP-complete languages are polynomial-time length-decreasing self-reducible

Manipulation of elections : algorithms and infeasibility results

Thesis (Ph. D.)--University of Rochester. Dept. of Computer Science, 2009.Voting and elections are at the core of democratic societies. People vote to elect leaders, decide policies, and organize their lives, but elections also have natural applications in computer science. For example, agents in multiagent systems often need to work together to complete some task, but each agent may have its own set of beliefs, preferences, and goals. Voting provides agents with a natural way to reach decisions that take all their preferences into account. With elections playing such an important role both in real-life political settings and in computer science, it is natural to ask about their resistance to misuse. Two particular types of election misuse are manipulation and bribery. In manipulation, a group of voters chooses to misrepresent its preferences in order to obtain a more desirable outcome, and in bribery an outside agent, the briber, asks (possibly at a cost) a group of voters to change its votes, to obtain some outcome desirable for the briber. Classical results from political science show that, for any reasonable election system, there are scenarios where at least some voters have an incentive to attempt manipulation. In this thesis we seek to protect elections from manipulators and bribers by making their computational task of finding good manipulations/bribes prohibitively expensive. When this is not possible, we seek to better understand (and even improve) the algorithmic attacks that manipulators and bribers can employ. In doing so, we develop new models of manipulation and bribery, and provide new approaches to studying the computational complexity of bribery and manipulation in elections

The Consequences of Eliminating NP Solutions

Given a function based on the computation of an NP machine, can one in general eliminate some solutions? That is, can one in general decrease the ambiguity? This simple question remains, even after extensive study by many researchers over many years, mostly unanswered. However, complexity-theoretic consequences and enabling conditions are known. In this tutorial-style article we look at some of those, focusing on the most natural framings: reducing the number of solutions of NP functions, refining the solutions of NP functions, and subtracting from or otherwise shrinking #P functions. We will see how small advice strings are important here, but we also will see how increasing advice size to achieve robustness is central to the proof of a key ambiguity-reduction result for NP functions

The Shield that Never Was: Societies with Single-Peaked Preferences are More Open to Manipulations and Control

Much work has been devoted, during the past twenty years, to using complexity to protect elections from manipulation and control. Many results have been obtained showing NP-hardness shields, and recently there has been much focus on whether such worst-case hardness protections can be bypassed by frequently correct heuristics or by approximations. This paper takes a very different approach: We argue that when electorates follow the canonical political science model of societal preferences the complexity shield never existed in the first place. In particular, we show that for electorates having single-peaked preferences, many existing NP-hardness results on manipulation and control evaporate

Llull and Copeland Voting Computationally Resist Bribery and Control

Control and bribery are settings in which an external agent seeks to influence the outcome of an election. Constructive control of elections refers to attempts by an agent to, via such actions as addition/deletion/partition of candidates or voters, ensure that a given candidate wins [BTT92]. Destructive control refers to attempts by an agent to, via the same actions, preclude a given candidate's victory [HHR07a]. An election system in which an agent can affect the result and in which recognizing the inputs on which the agent can succeed is NP-hard (polynomial-time solvable) is said to be resistant (vulnerable) to the given type of control. Aside from election systems with an NP-hard winner problem, the only systems previously known to be resistant to all the standard control types are highly artificial election systems created by hybridization [HHR07b]. We study a parameterized version of Copeland voting, denoted by Copeland^alpha, where the parameter alpha is a rational number between 0 and 1 that specifies how ties are valued in the pairwise comparisons of candidates. In every previously studied constructive or destructive control scenario, we determine which of resistance or vulnerability holds for Copeland^alpha for each rational alpha, 0 <= alpha <= 1. In particular, we prove that Copeland^0.5, the system commonly referred to as ``Copeland voting,'' provides full resistance to constructive control. Among the systems with a polynomial-time winner problem, this is the first natural election system proven to have full resistance to constructive control. In addition, we prove that both Copeland^0 and Copeland^1(interestingly, the latter is an election system developed by the thirteenth-century mystic Ramon Llull) are resistant to all the standard types of constructive control other than one variant of addition of candidates. Moreover, we show that for each rational alpha, 0 <= alpha <= 1, Copeland^alpha voting is fully resistant to bribery attacks, and we establish fixed-parameter tractability of bounded-case control for Copeland^alpha. We also study Copeland^alpha elections under more flexible models such as microbribery and extended control, we integrate the potential irrationality of voter preferences into many of our results, and we prove our results in both the unique-winner and the nonunique-winner model. Our vulnerability results for microbribery are proven via a technique involving min-cost network flow

Copeland Voting Fully Resists Constructive Control

Control and bribery are settings in which an external agent seeks to influence the outcome of an election. Faliszewski et al. [FHHR07] proved that Llull voting (which is here denoted by Copeland^1) and a variant (here denoted by Copeland^0) of Copeland voting are computationally resistant to many, yet not all, types of constructive control and that they also provide broad resistance to bribery. We study a parameterized version of Copeland voting, denoted by Copeland^alpha where the parameter alpha is a rational number between 0 and 1 that specifies how ties are valued in the pairwise comparisons of candidates in Copeland elections. We establish resistance or vulnerability results, in every previously studied control scenario, for Copeland^alpha, for each rational alpha, 0 < alpha < 1. In particular, we prove that Copeland^0.5, the system commonly referred to as ``Copeland voting,'' provides full resistance to constructive control. Among the systems with a polynomial-time winner problem, this is the first natural election system proven to have full resistance to constructive control. Results on bribery and fixed-parameter tractability of bounded-case control proven for Copeland^0 and Copeland^1 in [FHHR07] are extended to Copeland^alpha for each rational alpha, 0 < alpha < 1; we also give results in more flexible models such as microbribery and extended control