Sparse NP-complete problems over the reals with addition

The PCP theorem for NP over the reals

In this paper we show that the PCP theorem holds as well in the real number computational model introduced by Blum, Shub, and Smale. More precisely, the real number counterpart NPR of the classical Turing model class NP can be characterized as NPR = PCPR(O(logn), O(1)). Our proof structurally follows the one by Dinur for classical NP. However, a lot of minor and major changes are necessary due ...

New NP-Complete Problems Associated with Lattices

In this paper, we introduce a new decision problem associated with lattices, named the Exact Length Vector Problem (ELVP), and prove the NP-completeness of ELVP in the ∞ norm. Moreover, we define two variants of ELVP. The one is a binary variant of ELVP, named the Binary Exact Length Vector Problem (BELVP), and is shown to be NPcomplete in any p norm (1 ≤ p ≤ ∞). The other is a nonnegative vari...

Complete Problems for Monotone NP

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...

NP-Complete Stable Matching Problems

This paper concerns the complexity analysis of the roommate problem and intern assignment problem with couples. These are two special cases of the matching problems known as stable matching. The roommate problem is that of assigning a set of people to rooms of exactly two occupants in accordance with the preferences of the members of the set. The intern assignment problem with couples is that o...

Lecture 16: NP-complete Problems