Abstract. In 1995, Beauquier, Nivat, Rémila, and Robson showed that tiling of general regions with two rectangles is NP-complete, except for a few trivial special cases. In a different direction, in 2005, Rémila showed that for simply connected regions by two rectangles, the tileability can be solved in quadratic time (in the area). We prove that there is a finite set of at most 10 rectangles f...

We determine the computational power of preparing Projected Entangled Pair States (PEPS), as well as the complexity of classically simulating them, and generally the complexity of contracting tensor networks. While creating PEPS allows to solve PP problems, the latter two tasks are both proven to be #P-complete. We further show how PEPS can be used to approximate ground states of gapped Hamilto...

In 1979, Valiant proved that computing the permanent of a 01-matrix is #PComplete. In this paper we present another proof for the same result. Our proof uses \black box" methodology, which facilitates its presentation. We also prove that deciding whether the permanent is divisible by a small prime is #P-Hard. We conclude by proving that a polynomially bounded function can not be #P-Complete und...

In this paper we prove that calculating the average covering tree value recently proposed as a single-valued solution of graph games is #P-complete.

#2SAT is a classical #P-complete problem. We present here, a novel algorithm for given a 2-CF Σ, to build a minimum spanning tree for its constraint graph GΣ assuming dynamic weights on the edges of the input graph.

Let G denote a graph and let K be a subset of vertices that are a set of target vertices of G. The Kterminal reliability of G is defined as the probability that all target vertices in K are connected, considering the possible failures of non-target vertices of G. The problem of computing K-terminal reliability is known to be #P-complete for polygon-circle graphs, and can be solved in polynomial...

We study the complexity of counting the number of solutions to a system of equations over a fixed finite semigroup. We show that this problem is always either in FP or #P-complete and describe the borderline precisely. We use these results to convey some intuition about the conjectured dichotomy for the complexity of counting the number of solutions in constraint satisfaction problems.

A linear extension of a poset P is a permutation of the elements of the set that respects the partial order. Let L(P ) denote the number of linear extensions. It is a #P complete problem to determine L(P ) exactly for an arbitrary poset, and so randomized approximation algorithms that draw randomly from the set of linear extensions are used. In this work, the set of linear extensions is embedde...

A binary matrix satisfies the consecutive ones property (C1P) if its columns can be permuted such that the 1s in each row of the resulting matrix are consecutive. Equivalently, a family of sets F = {Q1, . . . , Qm}, where Qi ⊆ R for some universe R, satisfies the C1P if the symbols in R can be permuted such that the elements of each set Qi ∈ F occur consecutively, as a contiguous segment of the...

Minesweeper is a popular single player game. It has been shown that the Minesweeper consistency problem is NPcomplete and the Minesweeper counting problem is #P-complete. We present a polynomial algorithm for solving these problems for minesweeper graphs with bounded treewidth.