The n-Queens problem is to place n chess queens on an n by n chessboard so that no two queens are on the same row, column or diagonal. The n-Queens Completion problem is a variant, dating to 1850, in which some queens are already placed and the solver is asked to place the rest, if possible. We show that n-Queens Completion is both NP-Complete and #P-Complete. A corollary is that any non-attack...

We prove that the counting problems #1-in-3Sat, #Not-All-Equal 3Sat and #3-Colorability, whose decision counterparts have been the most frequently used in proving NP-hardness of new decision problems, are #P-complete. On one hand, the explicit #P-completeness proof of #1-in-3Sat could be useful to prove complexity results within uniication theory. On the other hand, the fact that #3-Colorabilit...

Aslam presents an algorithm he claims will count the number of perfect matchings in any incomplete bipartite graph with an algorithm in the function-computing version of NC, which is itself a subset of FP. Counting perfect matchings is known to be #P-complete; therefore if Aslam's algorithm is correct, then NP=P. However, we show that Aslam's algorithm does not correctly count the number of per...

We show that computing the dimension of a semi-algebraic set of R n is a NP R-complete problem in the Blum-Shub-Smale model of computation over the reals. Since this problem is easily seen to be NP R-hard, the main ingredient of the proof is a NP R algorithm for computing the dimension. On montre que le calcul de la dimension d'un ensemble semi-alg ebrique de R n est un probl eme NP R-complet d...

A partial order or poset   , P X   on a (finite) base set X determines the set   P  of linear extensions of P . The problem of computing, for a poset P , the cardinality of   P  is #P-complete. A set   1 2 , , , k P P P  of posets on X covers the set of linear orders that is the union of the   i P  . Given linear orders 1 2 , , , m L L L  on X , the Poset Cover problem is to de...

We study the complexity of evaluating positive equality-free sentences of first-order (FO) logic over fixed, finite structures B. This may be seen as a natural generalisation of the non-uniform quantified constraint satisfaction problem QCSP(B). Extending the algebraic methods of a previous paper, we derive a complete complexity classification for these problems as B ranges over structures of d...

A positive temporal template (or a positive temporal constraint language) is a relational structure whose relations can be defined over a dense linear order of rational numbers using a relational symbol ≤, logical conjunction and disjunction. We provide a complexity characterization for quantified constraint satisfaction problems (QCSP) over positive temporal languages. The considered QCSP prob...

Qualitative coalitional games (QCGs) are a variation of conventional coalitional games in which each coalition may choose to cooperate in a number of different ways, with different choices resulting in potentially different sets of goals being achieved; each agent is associated with a set of goals, the intuition being that an agent is “satisfied” if any of its goals are achieved, but is indiffe...