By Eiichi Bannai

ISBN-10: 0805304908

ISBN-13: 9780805304909

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**Extra resources for Algebraic Combinatorics I: Association Schemes**

**Example text**

And, if so, returns V . Let us execute it with K = n + 2m. If the answer from A is yes, then the transversal must be of a size equal to n+2m. In fact, any transversal needs at least n vertices in order to cover the n edges corresponding to the variables of φ (one vertex per edge) and 2m vertices to cover the edges of m triangles (two vertices per triangle). As a result, if A answers yes, it will have calculated a transversal of exactly n + 2m vertices. In the light of the previous observation, given such a transversal V , we state that xi = 1 if the extremity xi of the edge (xi , x ¯i ) is taken in V ; if the extremity x ¯i is included, then we state that x ¯i = 1, that is xi = 0.

Nk . The problem therefore consists of deciding whether or not x belongs to the intersection of the tables, that is whether there is a table that does not contain x. The analogy with the hidden coins problem can be seen if we make a copper coin correspond to the tables containing x, and a silver coin to the tables not containing x. As for the hidden coin problem, the instance is harder if almost all the tables contain x, both for the deterministic and the probabilistic algorithms, and a probabilistic algorithm will end more quickly than a deterministic algorithm if only half the tables contain x.

A Guide to the Theory of NP-completeness, W. H. Freeman, San Francisco, 1979. , Introduction to Automata Theory, Languages and Computation, Addison-Wesley, 1979. , Complexity of computer computations, p. 85–103, Plenum Press, New York, 1972. , “A polynomial algorithm for linear programming”, Dokladi Akademiy Nauk SSSR, vol. 244, p. 1093–1096, 1979. , “On the structure of polynomial time reducibility”, J. Assoc. Comput. , vol. 22, p. 155–171, 1975. , Elements of the Theory of Computation, Prentice-Hall, New Jersey, 1981.

### Algebraic Combinatorics I: Association Schemes by Eiichi Bannai

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