Simulated annealing and Boltzmann machines: a stochastic approach to combinatorial optimization and neural computing
NP-completeness of some problems concerning voting games
International Journal of Game Theory
Probabilistic polynomial time is closed under parity reductions
Information Processing Letters
PP is as hard as the polynomial-time hierarchy
SIAM Journal on Computing
The history and status of the P versus NP question
STOC '92 Proceedings of the twenty-fourth annual ACM symposium on Theory of computing
Counting classes are at least as hard as the polynomial-time hierarchy
SIAM Journal on Computing
Closure properties and witness reduction
Journal of Computer and System Sciences
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computational Politics: Electoral Systems
MFCS '00 Proceedings of the 25th International Symposium on Mathematical Foundations of Computer Science
The Complexity of Power-Index Comparison
AAIM '08 Proceedings of the 4th international conference on Algorithmic Aspects in Information and Management
Note: The complexity of power-index comparison
Theoretical Computer Science
Survey: The consequences of eliminating NP solutions
Computer Science Review
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We measure the performance, in the task of apportioning the Congress of the United States, of an algorithm combining a heuristic-driven (simulated annealing) search with an exact-computation dynamic programming evaluation of the apportionments visited in the search. We compare this with the actual algorithm currently used in the United States to apportion Congress, and with a number of other algorithms that have been proposed. We conclude that on every set of census data in this country's history, the heuristic-driven apportionment provably yields far fairer apportionments than those of any of the other algorithm considered, including the algorithm currently used by the United States for Congressional apportionment.