Jörg Rothe

Exact Complexity of the Winner Problem for Young Elections

Abstract. In 1977 Young proposed a voting scheme that extends the Condorcet Principle based on the fewest possible number of voters whose removal yields a Condorcet winner. We prove that both the winner and the ranking problem for Young elections is complete for \p||NP , the class of problems solvable in polynomial time by parallel access to NP. Analogous results for Lewis Carrolls 1876 voting scheme were recently established by Hemaspaandra et al. In contrast, we prove that the winner and ranking problems in Fishburns homogeneous variant of Carrolls voting scheme can be solved efficiently by linear programming.

The Cost of Stability in Coalitional Games

A key question in cooperative game theory is that of coalitional stability, usually captured by the notion of the core --the set of outcomes such that no subgroup of players has an incentive to deviate. However, some coalitional games have empty cores, and any outcome in such a game is unstable. In this paper, we investigate the possibility of stabilizing a coalitional game by using external payments. We consider a scenario where an external party, which is interested in having the players work together, offers a supplemental payment to the grand coalition (or, more generally, a particular coalition structure). This payment is conditional on players not deviating from their coalition(s). The sum of this payment plus the actual gains of the coalition(s) may then be divided among the agents so as to promote stability. We define the cost of stability (CoS) as the minimal external payment that stabilizes the game. We provide general bounds on the cost of stability in several classes of games, and explore its algorithmic properties. To develop a better intuition for the concepts we introduce, we provide a detailed algorithmic study of the cost of stability in weighted voting games, a simple but expressive class of games which can model decision-making in political bodies, and cooperation in multiagent settings. Finally, we extend our model and results to games with coalition structures.

Computational Aspects of Approval Voting

“Yes, we can!” – Barack Obama’s campaign slogan inspired enough of his supporters to go to the polls and give him their “yes” votes that he won the 2008 U.S. presidential election. And this happened notwithstanding the fact that many other voters said “no” when pollsters asked if they viewed Barack Obama as qualified for the office. “Yes” and “no” are perhaps the most basic ways for us, as voters, to express our preferences about candidates, and “yes” and “no” are what approval voting is all about.

Creating Strong, Total, Commutative, Associative One-Way Functions from Any One-Way Function in Complexity Theory

Rabi and Sherman presented novel digital signature and unauthenticated secret-key agreement protocols, developed by themselves and by Rivest and Sherman. These protocols use strong, total, commutative (in the case of multiparty secret-key agreement), associative one-way functions as their key building blocks. Although Rabi and Sherman did prove that associative one-way functions exist if P?NP, they left as an open question whether any natural complexity-theoretic assumption is sufficient to ensure the existence of strong, total, commutative, associative one-way functions. In this paper, we prove that if P?NP then strong, total, commutative, associative one-way functions exist.

Quantum cryptography: A survey

We survey some results in quantum cryptography. After a brief introduction to classical cryptography, we provide the quantum-mechanical background needed to present some fundamental protocols from quantum cryptography. In particular, we review quantum key distribution via the BB84 protocol and its security proof, as well as the related quantum bit commitment protocol and its proof of insecurity.

A survey of approximability and inapproximability results for social welfare optimization in multiagent resource allocation

Multiagent resource allocation provides mechanisms to allocate bundles of resources to agents, where resources are assumed to be indivisible and nonshareable. A central goal is to maximize social welfare of such allocations, which can be measured in terms of the sum of utilities realized by the agents (utilitarian social welfare), in terms of their minimum (egalitarian social welfare), and in terms of their product (Nash product social welfare). Unfortunately, social welfare optimization is a computationally intractable task in many settings. We survey recent approximability and inapproximability results on social welfare optimization in multiagent resource allocation, focusing on the two most central representation forms for utility functions of agents, the bundle form and the k-additive form. In addition, we provide some new (in)approximability results on maximizing egalitarian social welfare and social welfare with respect to the Nash product when restricted to certain special cases.

The shield that never was: societies with single-peaked preferences are more open to manipulation 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.

Challenges to complexity shields that are supposed to protect elections against manipulation and control: a survey

In the context of voting, manipulation and control refer to attempts to influence the outcome of elections by either setting some of the votes strategically (i.e., by reporting untruthful preferences) or by altering the structure of elections via adding, deleting, or partitioning either candidates or voters. Since by the celebrated Gibbard–Satterthwaite theorem (and other results expanding its scope) all reasonable voting systems are manipulable in principle and since many voting systems are in principle susceptible to many control types modeling natural control scenarios, much work has been done to use computational complexity as a shield to protect elections against manipulation and control. However, most of this work has merely yielded NP-hardness results, showing that certain voting systems resist certain types of manipulation or control only in the worst case. Various approaches, including studies of the typical case (where votes are given according to some natural distribution), pose serious challenges to such worst-case complexity results and might allow successful manipulation or control attempts, despite the NP-hardness of the corresponding problems. We survey and discuss some recent results on these challenges to complexity results for manipulation and control, including typical-case analyses and experiments, fixed-parameter tractability, domain restrictions (single-peakedness), and approximability.

How hard is it to bribe the judges? a study of the complexity of bribery in judgment aggregation

Endriss et al. [1,2] initiated the complexity-theoretic study of problems related to judgment aggregation. We extend their results for manipulating two specific judgment aggregation procedures to a whole class of such procedures, and we obtain stronger results by considering not only the classical complexity (NP-hardness) but the parameterized complexity (W[2]-hardness) of these problems with respect to natural parameters. Furthermore, we introduce and study the closely related issue of bribery in judgment aggregation, inspired by work on bribery in voting (see, e.g., [3,4,5]). In manipulation scenarios one of the judges seeks to influence the outcome of the judgment aggregation procedure used by reporting an insincere judgment set. In bribery scenarios, however, an external actor, the briber, seeks to influence the outcome of the judgment aggregation procedure used by bribing some of the judges without exceeding his or her budget. We study three variants of bribery and show W[2]-hardness of the corresponding problems for natural parameters and for one specific judgment aggregation procedure. We also show that in certain special cases one can determine in polynomial time whether there is a successful bribery action.

Recognizing when greed can approximate maximum independent sets is complete for parallel access to NP

Abstract Bodlaender, Thilikos, and Yamazaki (1997) investigate the computational complexity of the problem of whether the Minimum Degree Greedy Algorithm can approximate a maximum independent set of a graph within a constant factor of r , for fixed rational r ⩾ 1. They denote this problem by S r and prove that for each rational r ⩾ 1, S r is coNP-hard. They also provide a P NP upper bound of S r , leaving open the question of whether this gap between the upper and the lower bound of S r can be closed. For the special case of r = 1, they show that S 1 is even DP-hard, again leaving open the question of whether S 1 can be shown to be complete for DP or some larger class such as P NP . In this note, we completely solve all the questions left open by Bodlaender et al. Our main result is that for each rational r ⩾ 1, S r is complete for P ∥ NP , the class of sets solvable via parallel access to NP.