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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)
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)
We study the complexity of some computational problems on finite black-box rings whose elements are encoded as strings of a given length and the ring operations are performed by a black-box oracle. We give a polynomial-time quantum algorithm to compute a basis representation for a given black-box ring. Using this result we obtain polynomial-time quantum(More)
In this paper we study the complexity of sets that reduce to sparse sets (and tally sets), and the complexity of the simplest sparse sets to which such sets reduce. We show even with respect to very exible reductions that NP cannot have sparse hard sets unless P = NP; an immediate consequence of our results is: If any NP-complete set conjunctively reduces(More)