Several new graph theoretic problems, which arise naturally from existing coloring algorithms, are de ned. The complementary High Degree Vertex Removal Problem and Low Degree Vertex Removal Problem are both shown to be NP-complete. The Low Degree Subgraph Problem is de ned and shown to be NP-complete, whereas, the \complementary" problem for high degree subgraphs was previously shown to be P-co...

In 1984 Levin put forward a suggestion for a theory of average case complexity. In this theory a problem, called a distributional problem, is defined as a pair consisting of a decision problem and a probability distribution over the instances. Introducing adequate notions of simple distributions and average case preserving reductions, Levin developed a theory analogous to the theory of NP-compl...

Given a simple graph G, a harmonious coloring of G is a proper vertex coloring such that each pair of colors appears together on at most one edge. The harmonious chromatic number is the least integer k for which G admits a harmonious coloring with k colors. Extending previous NP-completeness results of the harmonious coloring problem on subclasses of chordal and co-chordal graphs, we prove that...

For all k≥1, we show that deciding whether a graph is k-planar NP-complete, extending the well-known fact 1-planarity NP-complete. Furthermore, gap version of this decision problem In particular, given with local crossing number either at most k≥1 or least 2k, it NP-complete to decide k 2k. This algorithmic lower bound proves non-existence (2−ϵ)-approximation algorithm for any fixed k≥1. additi...

We show that the problem of deciding whether a digraph has a Hamiltonian path between two specified vertices and the problem of deciding whether a given graph has a cubic subgraph are complete for monotone NP via monotone projection translations. We also show that the problem of deciding whether a uniquely partially orderable (resp. comparability) graph has a cubic subgraph is complete for NP v...

We outline a conceptual framework for teaching the basic notions and results of complexity theory. Our focus is on using deenitions and on organizing the presentation in a way that reeects the fundamental nature of the material. We do not attempt to provide a self-contained presentation of the material itself, but rather outline our suggestions regarding how this material should be presented in...

Ordinary approach to quantum algorithm is based on quantum Turing machine or quantum circuits. It is known that this approach is not powerful enough to solve NP-complete problems. In this paper we study a new approach to quantum algorithm which is a combination of the ordinary quantum algorithm with a chaotic dynamical system. We consider the satisfiability problem as an example of NP-complete ...

Linear logic, introduced by Girard, is a refinement of classical logic with a natural, intrinsic accounting of resources. We show that unlike most other propositional (quantifier-free) logics, full propositional linear logic is undecidable. Further, we prove that without the modal s torage operator, which indicates unboundedness of resources, the decision problem becomes PSPACE-Complete. We als...

Proof search in linear logic is known to be diicult: the provability of propositional linear logic formulas is undecidable. Even without the modalities, multiplicative-additive fragment of propositional linear logic, mall, is known to be pspace-complete, and the pure multiplicative fragment, mll, is known to be np-complete. However, this still leaves open the possibility that there might be pro...

Of course, the even more general problem where the pattern is a graph is already well-known to be NP-complete, as it contains the well-known NP-complete Subgraph Isomorphism problem as the special case k = 1. We thus see here that restricting to tree patterns does not lower the worst-case complexity. The NP-completeness of Disjoint Occurrences of T3 follows immediately from the NP-completeness ...