Martin Fürer

Faster integer multiplication

For more than 35 years, the fastest known method for integer multiplication has been the Schönhage-Strassen algorithm running in time O(n log n log log n). Under certain restrictive conditions there is a corresponding Ω(n log n) lower bound. The prevailing conjecture has always been that the complexity of an optimal algorithm is Θ(n log n). We present a major step towards closing the gap from above by presenting an algorithm running in time n log n, 2O(log* n). The main result is for boolean circuits as well as for multitape Turing machines, but it has consequences to other models of computation as well.

Approximating the minimum-degree Steiner tree to within one of optimal

Abstract The problem of constructing a spanning tree for a graph G = (V, E) with n vertices whose maximal degree is the smallest among all spanning trees of G is considered. This problem is easily shown to be NP-hard. In the Steiner version of this problem, along with the input graph, a set of distinguished vertices D ⊆ V is given. A minimum-degree Steiner tree is a tree of minimum degree which spans at least the set D. Iterative polynomial time approximation algorithms for the problems are given. The algorithms compute trees whose maximal degree is at most Δ* + 1, where Δ* is the degree of some optimal tree for the respective problems. Unless P = NP, this is the best bound achievable in polynomial time.

Approximation of k -set cover by semi-local optimization

In this thesis a powerful new approximation technique called semi-local optimization is introduced. It provides very natural heuristics that are distinctly more effective than those based on local optimization. With an appropriate metric, semi-local optimization can still be viewed as a local optimization, but it has the advantage of making global changes to an approximate solution in polynomial time. Semi-local optimization generalizes recent heuristics of Halldorsson for the 3-Set Cover problem, the Color Saving problem, and the k-Set Cover problem. Greatly improved performance ratios of 4/3 for the 3-Set Cover problem and 6/5 for the Color Saving problem in graphs without independent sets of size 4 are obtained and shown to be the best possible with semi-local optimization. Also, based on the result for the 3-Set Cover problem and a restricted greedy phase for big sets, the performance ratio, for the k-Set Cover problem is correspondingly improved to

Algorithms for Counting 2-Sat Solutions and Colorings with Applications

{\cal H}\sb{k}-1/2

The tight deterministic time hierarchy

. This result is also tight with the semi-local optimization technique. For larger values of k, further improvement better than

Improved hardness results for approximating the chromatic number

{\cal H}\sb{k}-1/2

A faster algorithm for finding maximum independent sets in sparse graphs

are also possible when the greedy selection of big sets is replaced by a local optimization selection. In the Color Saving problem, when larger independent sets exist, an improvement of the performance ratio to

An Exponential Time 2-Approximation Algorithm for Bandwidth

{240}\over{193}

Normal forms for trivalent graphs and graphs of bounded valence

for general graphs can be obtained.

An exponential time 2-approximation algorithm for bandwidth

An algorithm is presented for exactly counting the number of maximum weight satisfying assignments of a 2- Cnf formula. The worst case running time of O( 1.246n) for formulas with nvariables improves on the previous bound of O( 1.256n) by Dahllof, Jonsson, and Wahlstrom. The algorithm uses only polynomial space. As a direct consequence we get an O(1.246n) time algorithm for counting maximum weighted independent sets in a graph.