Edith Hemaspaandra

Exact analysis of Dodgson elections: Lewis Carroll's 1876 voting system is complete for parallel access to NP

In 1876, Lewis Carroll proposed a voting system in which the winner is the candidate who with the fewest changes in voters preferences becomes a Condorcet winner—a candidate who beats all other candidates in pairwise majority-rule elections. Bartholdi, Tovey, and Trick provided a lower bound—NP-hardness—on the computational complexity of determining the election winner in Carrolls system. We provide a stronger lower bound and an upper bound that matches our lower bound. In particular, determining the winner in Carrolls system is complete for parallel access to NP, that is, it is complete for Theta_2p for which it becomes the most natural complete problem known. It follows that determining the winner in Carrolls elections is not NP-complete unless the polynomial hierarchy collapses.

P-Selective Sets and Reducing Search to Decision vs Self-Reducibility

We distinguish self-reducibility of a languageLwith the question of whether search reduces to decision forL. Results include: (i) If NE?E, then there exists a setLin NP?P such that search reduces to decision forL, search doesnotnonadaptively reduce to decision forLandLis not self-reducible. (ii) If UE?E, then there exists a languageL?UP?P such that search nonadaptively reduces to decision for L, but L is not self-reducible. (iii) If UE?co-UE?E, then there is a disjunctive self-reducible languageL?UP?P for which search doesnotnonadaptively reduce to decision. We prove that if NE?BPE, then there is a languageL?NP?BPP such thatLis randomly self-reducible,notnonadaptively randomly self-reducible, andnotself-reducible. We obtain results concerning trade-offs in multiprover interactive proof systems and results that distinguish checkable languages from those that are nonadaptively checkable. Many of our results are proven by constructing p-selective sets. We obtain a p-selective set that isnot?Ptt-equivalent to any tally language, and we show that if P=PP, then every p-selective set is ?PT-equivalent to a tally language. Similarly, if P=NP, then every cheatable set is ?Pm-equivalent to a tally language. We construct a recursive p-selective tally set that isnotcheatable.

The minimization problem for Boolean formulas

We investigate the computational complexity of the minimization problem for Boolean formulas. Depending on the definition, these problems are trivially in /spl Sigma//sub 2//sup P/ or II/sub 2//sup P/, and these are the best upper bounds known. The only previously known lower bounds are also trivial, and are coNP lower bounds at best, thus leaving quite a large gap between the upper and lower bounds. In this paper, we prove much better lower bounds: hardness for parallel access to NP for those cases in which coNP was the best previously known lower bound, and coNP-hardness for the case in which no lower bound was previously known.

Raising NP lower bounds to parallel NP lower bounds

This issues column surveys recent progress in raising NP-hardness lower bounds to parallel NP lower bounds. Complexity theorists will learn that Lewis Carroll (unbeknownst to himself) was a fellow complexity theorist. So that readers specializing in algorithms dont feel left out, let me mention that they are in even better company. Algorithms for apportioning the US House of Representatives were designed and debated by Adam, Hamilton, Jefferson, Webster, and other famous figures from American history, as recounted in a page-turner by Balinski and Young ([BY82]; see also [BY85] for a shorter take and [HRSZ96] for a more explicitly computational spin). Looking forward, the next issues column will be a survey, by Mitsunori Ogihara, of DNA-based computing.

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. [9] proved that Llull voting (which is here denoted by Copeland1) and a variant (here denoted by Copeland0) 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?, where the parameter ?is a rational number between 0 and 1 that specifies how ties are valued in the pairwise comparisons of candidates in Copeland elections. For each rational ?, 0 < ?< 1, and each previously studied control scenario, we either prove that Copeland?is computationally vulnerable to control in that scenario (i.e., we give a P-time algorithm that determines whether control is possible, and if so, determines exactly how to exert the control) or we prove that Copeland?is computationally resistant to control in that scenario (i.e., we prove that control problem to be NP-hard). In particular, we prove that Copeland0.5, the system commonly referred to as Copeland voting, provides full resistance to constructive control. Among systems with a polynomial-time winner problem, this is the first natural election system proven to have full resistance to constructive control. Looking at rational ?, 0 < ?< 1, we give a broad set of results on bribery and on the fixed-parameter tractability of bounded-case control for Copeland?(previously only Copeland0and Copeland1had been studied), and we introduce and obtain fixed-parameter tractability results even in a new, more flexible model of control (that we dub extended control).

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.

Query Order in the Polynomial Hierarchy

We study query order within the polynomial hierarchy.

A Downward Translation in the Polynomial Hierarchy

P^{cal C : cal D}

Complexity transfer for modal logic

denotes the class of languages computable by a polynomial-time machine that is allowed one query to

Exact Analysis of Dodgson Elections: Lewis Carroll's 1876 Voting System is Complete for Parallel Access to NP

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