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We show that Graph Isomorphism is in the complexity class SPP, and hence it is in ⊕P (in fact, it is in Mod k P for each k ≥ 2). We derive this result as a corollary of a more general result: we show that a generic problem FIND-GROUP has an FP SPP algorithm. This general result has other consequences: for example, it follows that the hidden subgroup problem(More)
An undirected graph G is said to be d-distinguishable if there is a d-coloring of its vertex set V (G) such that no nontrivial automorphism of G preserves the coloring. The distinguishing number of a graph G is the minimum d for which it is d-distinguishable. In this paper we design efficient algorithms for computing the distinguishing numbers of trees and(More)
In this paper we study the complexity of bounded color multiplicity graph isomorphism BCGI/sub b/: the input is a pair of vertex-colored graphs such that the number of vertices of a given color in an input graph is bounded by b. We show that BCGI/sub b/ is in the #L hierarchy (more precisely, the Mod/sub k/L hierarchy for some constant k depending on b).(More)
\begin{abstract} In this paper we study the computational complexity of computing the <i>noncommutative</i> determinant. We first consider the arithmetic circuit complexity of computing the noncommutative determinant polynomial. Then, more generally, we also examine the complexity of algorithms computing the determinant over noncommutative domains. Our(More)
We show that if an NP-complete set or a coNP-complete set is polynomial-time disjunc-tive truth-table reducible to a sparse set then FP NP jj = FP NP log]. With a similar argument we show also that if SAT is O(log n)-approximable then FP NP jj = FP NP log]. Since FP NP jj = FP NP log] implies that SAT is O(logn)-approximable BFT97], it follows from our(More)
Given a system of linear equations $$Ax=b$$ A x = b over the binary field $$\mathbb {F}_2$$ F 2 and an integer $$t\ge 1$$ t ≥ 1 , we study the following three algorithmic problems: 1. Does $$Ax=b$$ A x = b have a solution of weight at most t? 2. Does $$Ax=b$$ A x = b have a solution of weight exactly t? 3. Does $$Ax=b$$ A x = b have a solution of weight at(More)