Holger Spakowski

The complexity of Kemeny elections

Kemeny proposed a voting scheme which is distinguished by the fact that it is the unique voting scheme that is neutral, consistent, and Condorcet. Bartholdi, Tovey, and Trick showed that determining the winner in Kemenys system is NP-hard. We provide a stronger lower bound and an upper bound matching the lower bound, namely, we show that determining the winner in Kemenys system is complete for P||NP, the class of sets solvable via parallel access to NP.

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.

Theta2p-Completeness: A Classical Approach for New Results

In this paper we present an approach for proving Θ2p- completeness. There are several papers in which different problems of logic, of combinatorics, and of approximation are stated to be complete for parallel access to NP, i.e. Θ2p-complete. There is a special acceptance concept for nondeterministic Turing machines which allows a characterization of Θ2p as a polynomial-time bounded class. This characterization is the starting point of this paper. It makes a master reduction from that type of Turing machines to suitable boolean formula problems possible. From the reductions we deduce a couple of conditions that are sufficient for proving Θ2p-hardness. These new conditions are applicable in a canonical way. Thus we are able to do the following: (i) we can prove the Θ2p-completeness for different combinatorial problems (e.g. max-card-clique compare) as well as for optimization problems (e.g. the Kemeny voting scheme), (ii) we can simplify known proofs for Θ2p-completeness (e.g. for the Dodgson voting scheme), and (iii) we can transfer this technique for proving Δ2p-completeness (e.g. TSPcompare).

Generalized juntas and NP-hard sets

We show that many NP-hard sets have heuristic polynomial-time algorithms with high probability weight of correctness with respect to generalizations of Procaccia and Rosenscheins junta distributions.

An improved exact algorithm for the domatic number problem

The 3-domatic number problem asks whether a given graph can be partitioned into three dominating sets. We prove that this problem can be solved by a deterministic algorithm in time 2.695n (up to polynomial factors). This result improves the previous bound of 2.8805 n, which is due to Fomin et al. (2005). To prove our result, we combine an algorithm by Fomin et al. (2005) with Yamamotos algorithm for the satisfiability problem. In addition, we show that the 3-domatic number problem can be solved for graphs G with bounded maximum degree Delta (G) by a randomized algorithm, whose running time is better than the previous bound due to Riege and Rothe (2005) whenever Delta(G) > 5. Our new randomized algorithm employs Schonings approach to constraint satisfaction problems by U. Schoning (1999)

An Improved Exact Algorithm for the Domatic Number Problem

The 3-domatic number problem asks whether a given graph can be partitioned into three dominating sets. We prove that this problem can be solved by a deterministic algorithm in time 2.695n (up to polynomial factors). This result improves the previous bound of 2.8805 n, which is due to Fomin et al. (2005). To prove our result, we combine an algorithm by Fomin et al. (2005) with Yamamotos algorithm for the satisfiability problem. In addition, we show that the 3-domatic number problem can be solved for graphs G with bounded maximum degree Delta (G) by a randomized algorithm, whose running time is better than the previous bound due to Riege and Rothe (2005) whenever Delta(G) > 5. Our new randomized algorithm employs Schonings approach to constraint satisfaction problems by U. Schoning (1999)

Exact Complexity of Exact-Four-Colorability and of the Winner Problem for Young Elections

We classify two problems: Exact-Four-Colorability and the winner problem for Young elections. Regarding the former problem, Wagner raised the question of whether it is DP-complete to determine if the chromatic number of a given graph is exactly four. We prove a general result that in particular solves Wagner’s question in the affirmative.

Quantum and classical complexity classes: separations, collapses, and closure properties

We study the complexity of quantum complexity classes such as EQP, BQP, and NQP (quantum analogs of P, BPP, and NP, respectively) using classical complexity classes such as ZPP, WPP, and C=P. The contributions of this paper are threefold. First, via oracle constructions, we show that no relativizable proof technique can improve the best known classical upper bound for BQP (BQP⊆AWPP [Journal of Computer and System Sciences 59(2) (1999) 240]) to BQP⊆WPP and the best known classical lower bound for EQP (P⊆EQP) to ZPP⊆EQP. Second, we prove that there are oracles A and B such that, relative to A, coRP is immune to NQP and relative to B, BQP is immune to pC=P. Extending a result of de Graaf and Valiant [Technical Report quant-ph/0211179, Quantum Physics (2002)], we construct a relativized world where EQP is immune to MODpkP. Third, motivated by the fact that counting classes (e.g., LWPP, AWPP, etc.) are the best known classical upper bounds on quantum complexity classes, we study properties of these counting classes. We prove that WPP is closed under polynomial-time truth-table reductions, while we construct an oracle relative to which WPP is not closed under polynomial-time Turing reductions. The latter result implies that proving the equality of the similar appearing classes LWPP and WPP would require nonrelativizable proof techniques. We also prove that both AWPP and APP are closed under ≤TUP reductions. We use closure properties of WPP and AWPP to prove interesting consequences, in terms of the complexity of the polynomial-hierarchy, of the following hypotheses: NQP⊆BQP and EQP=NQP.

Frequency of correctness versus average polynomial time

We prove that every distributional problem solvable in polynomial time on the average with respect to the uniform distribution has a frequently self-knowingly correct polynomial-time algorithm.

Degree Bounds on Polynomials and Relativization Theory

We demonstrate the applicability of the polynomial degree bound technique to notions such as the nonexistence of Turing-hard sets in some relativized world, (non)uniform gap-definability, and relativized separations. This way, we settle certain open questions of Hemaspaandra, Ramachandran & Zimand [HRZ95 ] and Fenner, Fortnow & Kurtz [FFK94], extend results of Hemaspaandra, Jain & Vereshchagin [HJV93] and construct oracles achieving desired results.