Results

**1 - 2**of**2**### On the Complexity of Optimal Grammar-Based Compression

, 2004

"... The task of grammar-based compression is to find a small context-free grammar generating exactly one given string. We investigate the relationship between grammar-based compression of strings over unbounded and bounded alphabets. Specifically, we show how to transform a grammar for a string over an ..."

Abstract
- Add to MetaCart

The task of grammar-based compression is to find a small context-free grammar generating exactly one given string. We investigate the relationship between grammar-based compression of strings over unbounded and bounded alphabets. Specifically, we show how to transform a grammar for a string over an unbounded alphabet into a grammar for a block coding of that string over a fixed bounded alphabet and vice versa. From these constructions, we obtain asymptotically tight relationships between the minimum grammar sizes for strings and their block codings. Finally, we exploit an improved bound of our construction for overlap-free block codings to show that a polynomial time algorithm for approximating the minimum grammar for binary strings within a factor of c yields a polynomial time algorithm for approximating the minimum grammar for strings over arbitrary alphabets within a factor of 24c + ε (for arbitrary ε> 0). Since the latter problem is known to be NP-hard to approximate within a factor of 8569/8568, we provide a first step towards solving the long standing open question whether minimum grammar-based compression of binary strings is NP-complete.

### unknown title

"... Given a set of monomials, the Minimum AND-Circuit problem asks for a circuit that computes these monomials using AND-gates of fan-in two and being of minimum size. We prove that the problem is not polynomial time approximable within a factor of less than 1.0051 unless P = NP, even if the monomials a ..."

Abstract
- Add to MetaCart

Given a set of monomials, the Minimum AND-Circuit problem asks for a circuit that computes these monomials using AND-gates of fan-in two and being of minimum size. We prove that the problem is not polynomial time approximable within a factor of less than 1.0051 unless P = NP, even if the monomials are restricted to be of degree at most three. For the latter case, we devise several efficient approximation algorithms, yielding an approximation ratio of 1.278. For the general problem, we achieve an approximation ratio of d − 3/2, where d is the degree of the largest monomial. In addition, we prove that the problem is fixed parameter tractable with the number of monomials as parameter. Finally, we reveal connections between the Minimum AND-Circuit problem and several problems from different areas. 1