Let M be the polynomial-time TM and p be the polynomial bound such that f(x) = #{y ∈ {0, 1}p(|x|) | M(x, y) = 1}. Define M ′(x,N, y′) for N a length p(|x|) binary string viewed as an integer and |y′| = p(|x|) + 1 by M ′(x,N, 0y) = M(x, y) and M ′(x,N, 1y) = 1 iff y ≥ N . Note that there will be precisely 2p(|x|) − N strings y′ = 1y such that M ′(x,N, y′) accepts. Therefore f(x) > N if and only ...

We consider the problem of computing the Euler characteristic of an abstract simplicial complex given by its vertices and facets. We show that this problem is #P-complete and present two new practical algorithms for computing Euler characteristic. The two new algorithms are derived using combinatorial commutative algebra and we also give a second description of them that requires no algebra. We...

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...

Let T be a set of tasks. Each task has a non-negative processing time and a deadline. The problem of determining whether or not there is a schedule of the tasks in T such that a single machine can finish processing each of them before its deadline is polynomially solvable. We prove that counting the number of schedules satisfying this condition is #P-complete.

We describe fully polynomial randomized approximation schemes for the problems of determining the number of Hamilton paths and cycles in an n-vertex graph with minimum degree (g + e)n, for any fixed e > 0. We show that the exact counting problems are #P-complete. We also describe fully polynomial randomized approximation schemes for counting paths and cycles of all sizes in such graphs.

In this paper, we prove that the numerical-semigroup-gap counting problem is #NP-complete as a main theorem. A numerical semigroup is an additive semigroup over the set of all nonnegative integers. A gap of a numerical semigroup is defined as a positive integer that does not belong to the numerical semigroup. The computation of gaps of numerical semigroups has been actively studied from the 19t...

Shakashaka is a pencil-and-paper puzzle proposed by Guten and popularized by the Japanese publisher Nikoli (like Sudoku). We determine the computational complexity by proving that Shakashaka is NP-complete, and furthermore that counting the number of solutions if #P-complete. Next we formulate Shakashaka as an integer programming (IP) problem, and show that an IP solver can solve every instance...

Jaw muscle silent periods in response to chin taps were recorded from immediate complete denture patients before extraction of a residual anterior dentition, after insertion of the dentures, and three and six months after insertion. After three months' use of the dentures, the mean EMG silent period duration was significantly increased compared to the pre-extraction stage.

Reconstructing phylogenetic trees is a fundamental task in evolutionary biology. Various algorithms exist for this purpose, many of which come under the heading of ‘supertree methods.’ These methods amalgamate a collection P of phylogenetic trees into a single parent tree. In this paper, we show that, in both the rooted and unrooted settings, counting the number of parent trees that preserve al...

Over the years, researchers have studied the complexity of several decision versions of Nash equilibrium in (symmetric) two-player games (bimatrix games). To the best of our knowledge, the last remaining open problem of this sort is the following; it was stated by Papadimitriou in 2007: find a non-symmetric Nash equilibrium (NE) in a symmetric game. We show that this problem is NP-complete and ...