In the field of computational optimization, it is often the case that we are given an instance of an NP problem and asked to enumerate the first few ”best” solutions to the instance. Motivated by the recent research performed in these fields, we propose in this paper a new framework to measure the effective enumerability of NP optimization problems. More specifically, given an instance of an NP...

We prove that multilinear (tensor) analogues of many efficiently computable problems in numerical linear algebra are NP-hard. Our list here includes: determining the feasibility of a system of bilinear equations, deciding whether a 3-tensor possesses a given eigenvalue, singular value, or spectral norm; approximating an eigenvalue, eigenvector, singular vector, or the spectral norm; and determi...

combinatorial optimization is an active field of research in Neural Networks. Since the first attempts to solve the travelling salesman problem with Hopfield nets several progresses have been made. I will present some Neural Network approximate solutions for NP-complete problems that have a sound mathematical foundation and that, beside their theoretical interest, are also numerically encouragi...

String problems arise in various applications ranging from text mining to biological sequence analysis. Many string problems are NP-hard. This motivates the search for (fixed-parameter) tractable special cases of these problems. We survey parameterized and multivariate algorithmics results for NP-hard string problems and identify challenges for future research.

Informally, a language L has speedup if, for any Turing machine (TM) for L, there exists one that is better. Blum [2] showed that there are computable languages that have almost-everywhere speedup. These languages were unnatural in that they were constructed for the sole purpose of having such speedup. We identify a condition apparently only slightly stronger than P 6= NP which implies that acc...

The parameterized computational complexity of a collection of well-known problems including: Bandwidth, Precedence constrained k-processor scheduling, Longest Common Subsequence, DNA physical mapping (or Intervalizing colored graphs), Perfect phylogeny (or Triangulating colored graphs), Colored cutwidth, and Feasible register assignment is explored. It is shown that these problems are hard for ...

Using modular arithmetic we introduce a simple bound which applies to a wide range of bin-packing like problems. This bound must hold if a problem is to be soluble. When it fails, search can therefore be pruned. We show the value of such pruning in a greedy backtracking algorithm for number partitioning problems. As the bound can be used when solving inexact as well as exact partitioning proble...

In this chapter, we discuss approximation algorithms for optimization problems. An optimization problem consists in finding the best (cheapest, heaviest, etc.) element in a large set P, called the feasible region and usually specified implicitly, where the quality of elements of the set are evaluated using a function f(x), the objective function, usually something fairly simple. The element tha...

Can NP-complete problems be solved efficiently in the physical universe? I survey proposals including soap bubbles, protein folding, quantum computing, quantum advice, quantum adiabatic algorithms, quantum-mechanical nonlinearities, hidden variables, relativistic time dilation, analog computing, Malament-Hogarth spacetimes, quantum gravity, closed timelike curves, and “anthropic computing.” The...

Consider SAT, the prototypical example of an NP-complete problem. An instance of this problem consists of a Boolean function f (x1, . . . ,xn) = c1 ∧ . . .∧ cm; the SAT problem asks you to determine whether there exists a satisfying assignment—that is, an input (a1, . . . ,an) such that f (a1, . . . ,an) = 1. UNIQUE-SAT is a variant of SAT that poses the same problem with the restriction that f...