On an Optimal Deterministic Algorithm for SAT
Proceedings of the 12th International Workshop on Computer Science Logic
A personal view of average-case complexity
SCT '95 Proceedings of the 10th Annual Structure in Complexity Theory Conference (SCT'95)
Foundations and Trends® in Theoretical Computer Science
Canonical labelling of graphs in linear average time
SFCS '79 Proceedings of the 20th Annual Symposium on Foundations of Computer Science
On optimal algorithms and optimal proof systems
STACS'99 Proceedings of the 16th annual conference on Theoretical aspects of computer science
Optimal acceptors and optimal proof systems
TAMC'10 Proceedings of the 7th annual conference on Theory and Applications of Models of Computation
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An acceptor for a language L is an algorithm that accepts every input in L and does not stop on every input not in L. An acceptor Opt for a language L is called optimal if for every acceptor A for this language there exists polynomial p"A such that for every x@?L, the running time time"O"p"t(x) of Opt on x is bounded by p"A(time"A(x)+|x|) for every x@?L. (Note that the comparison of the running time is done pointwise, i.e., for every word of the language.) The existence of optimal acceptors is an important open question equivalent to the existence of p-optimal proof systems for many important languages (Krajicek and Pudlak, 1989; Sadowski, 1999; Messner, 1999 [9,13,11]). Yet no nontrivial examples of languages in NP@?co-NP having optimal acceptors are known. In this short note we construct a randomized acceptor for graph nonisomorphism that is optimal up to permutations of the vertices of the input graphs, i.e., its running time on a pair of graphs (G"1,G"2) is at most polynomially larger than the maximum (in fact, even the median) of the running time of any other acceptor taken over all permuted pairs (@p"1(G"1),@p"2(G"2)). One possible motivation is the (pointwise) optimality in the class of acceptors based on graph invariants where the time needed to compute an invariant does not depend much on the representation of the input pair of nonisomorphic graphs. In fact, our acceptor remains optimal even if the running time is compared to the average-case running time over all permuted pairs. We introduce a natural notion of average-case optimality (not depending on the language of graph nonisomorphism) and show that our acceptor remains average-case optimal for any probability distribution on the inputs that respects permutations of vertices.