Lane A. Hemaspaandra

Anyone but him: The complexity of precluding an alternative

Preference aggregation in a multiagent setting is a central issue in both human and computer contexts. In this paper, we study in terms of complexity the vulnerability of preference aggregation to destructive control. That is, we study the ability of an elections chair to, through such mechanisms as voter/candidate addition/suppression/partition, ensure that a particular candidate (equivalently, alternative) does not win. And we study the extent to which election systems can make it impossible, or computationally costly (NP-complete), for the chair to execute such control. Among the systems we study--plurality, Condorcet, and approval voting--we find cases where systems immune or computationally resistant to a chair choosing the winner nonetheless are vulnerable to the chair blocking a victory. Beyond that, we see that among our studied systems no one system offers the best protection against destructive control. Rather, the choice of a preference aggregation system will depend closely on which types of control one wishes to be protected against. We also find concrete cases where the complexity of or susceptibility to control varies dramatically based on the choice among natural tie-handling rules.

How hard is bribery in elections

We study the complexity of influencing elections through bribery: How computationally complex is it for an external actor to determine whether by paying certain voters to change their preferences a specified candidate can be made the elections winner? We study this problem for election systems as varied as scoring protocols and Dodgson voting, and in a variety of settings regarding homogeneous-vs.-nonhomogeneous electorate bribability, bounded-size-vs.-arbitrary-sized candidate sets, weighted-vs.-unweighted voters, and succinct-vs.-nonsuccinct input specification. We obtain both polynomial-time bribery algorithms and proofs of the intractability of bribery, and indeed our results show that the complexity of bribery is extremely sensitive to the setting. For example, we find settings in which bribery is NP-complete but manipulation (by voters) is in P, and we find settings in which bribing weighted voters is NP-complete but bribing voters with individual bribe thresholds is in P. For the broad class of elections (including plurality, Borda, k- approval,and veto) known as scoring protocols, we prove a dichotomy result for bribery of weighted voters: We find a simple-to-evaluate condition that classifies every case as either NP-complete or in P.

Using complexity to protect elections

Computational complexity may truly be the shield against election manipulation.

A Richer Understanding of the Complexity of Election Systems

We provide an overview of some recent progress on the complexity of election systems. The issues studied include the complexity of the winner, manipulation, bribery, and control problems.

Bypassing combinatorial protections: polynomial-time algorithms for single-peaked electorates

For many election systems, bribery (and related) attacks have been shown NP-hard using constructions on combinatorially rich structures such as partitions and covers. It is important to learn how robust these hardness protection results are, in order to find whether they can be relied on in practice. This paper shows that for voters who follow the most central political-science model of electorates—single-peaked preferences—those protections vanish. By using single-peaked preferences to simplify combinatorial covering challenges, we show that NP-hard bribery problems—including those for Kemeny and Llull elections—fall to polynomial time. By using single-peaked preferences to simplify combinatorial partition challenges, we show that NP-hard partition-of-voters problems fall to polynomial time. We furthermore show that for single-peaked electorates, the winner problems for Dodgson and Kemeny elections, though Θp2-complete in the general case, fall to polynomial time. And we completely classify the complexity of weighted coalition manipulation for scoring protocols in single-peaked electorates.

Threshold Computation and Cryptographic Security

Threshold machines are Turing machines whose acceptance is determined by what portion of the machines computation paths are accepting paths. Probabilistic machines are Turing machines whose acceptance is determined by the probability weight of the machines accepting computation paths. In 1975, Simon proved that for unbounded-error polynomial-time machines these two notions yield the same class, PP\@. Perhaps because Simons result seemed to collapse the threshold and probabilistic modes of computation, the relationship between threshold and probabilistic computing for the case of bounded error has remained unexplored. In this paper, we compare the bounded-error probabilistic class BPP with the analogous threshold class,

Computational Aspects of Approval Voting

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SIGACT news complexity theory column 42

, and, more generally, we study the structural properties of

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

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NONDETERMINISTICALLY SELECTIVE SETS

. We prove that