We show that P2T-the problem of deciding whether the edge set of a simple graph can be partitioned into two trees or not-is NP-complete. It is a well known that deciding whether the edge set of a graph can be partitioned into k spanning trees or not is in P [1]. Recently András Frank asked what we know about partitioning the edge set of a graph into k (not necessarily spanning) trees. One can e...

We consider the problem of higher-order matching restricted to the set of linear λ-terms (i.e., λ-terms where each abstraction λx.M is such that there is exactly one free occurrence of x in M). We prove that this problem is decidable by showing that it belongs to NP. Then we prove that this problem is in fact NP-complete. Finally, we discuss some heuristics for a practical algorithm.

Matroids and oriented matroids are fundamental objects in combinatorial geometry. While matroids model the behavior of vector configurations over general fields, oriented matroids model the behavior of vector configurations over ordered fields. For every oriented matroid there is a corresponding underlying oriented matroid. This paper addresses the question how difficult it is to algorithmicall...

The mathematical formalism of quantum resonance combined with tensor product decomposability of unitary evolutions is mapped onto a class of NP-complete combinatorial problems. It has been demonstrated that nature has polynomial resources for solving NP-complete problems and that may help to develop a new strategy for artificial intelligence, as well as to re-evaluate the role of natural select...

We will show that the question of whether or not a given Spiral Galaxies puzzle has a solution is NP-complete. To do so, we construct Spiral Galaxies puzzles which correspond to arbitrary Boolean circuits. A circuit will be satisfied, (that is, have a set of inputs which give the desired outputs) if and only if the corresponding puzzle has a solution. Since Satisfiability is the canonical NP-co...

Algorithms for learning Bayesian networks from data have two components: a scoring metric and a search procedure. The scoring metric computes a score re ecting the goodness-oft of the structure to the data. The search procedure tries to identify network structures with high scores. Heckerman et al. (1995) introduce a Bayesian metric, called the BDe metric, that computes the relative posterior p...

Given a graph G and a bipartition of its vertices, the edge-ratio is the minimum for both classes so defined of their number of internal edges divided by their number of cut edges. We prove that maximizing edgeratio is NP-complete.

Bounded Second-Order Unification is the problem of deciding, for a given second-order equation t ? = u and a positive integer m, whether there exists a unifier σ such that, for every second-order variable F , the terms instantiated for F have at most m occurrences of every bound variable. It is already known that Bounded Second-Order Unification is decidable and NP-hard, whereas general Second-...

We prove that all of Karp’s 21 original NP-complete problems have a version that is hard to approximate. These versions are obtained from the original problems by adding essentially the same simple constraint. We further show that these problems are absurdly hard to approximate. In fact, no polynomial-time algorithm can even approximate log(k) of the magnitude of these problems to within any co...

In a Magnets puzzle, one must pack magnets in a box subject to polarity and numeric constraints. We show that deciding solvability of Magnets instances is NP-complete.