26 research outputs found
Polynomial-Time Amoeba Neighborhood Membership and Faster Localized Solving
We derive efficient algorithms for coarse approximation of algebraic
hypersurfaces, useful for estimating the distance between an input polynomial
zero set and a given query point. Our methods work best on sparse polynomials
of high degree (in any number of variables) but are nevertheless completely
general. The underlying ideas, which we take the time to describe in an
elementary way, come from tropical geometry. We thus reduce a hard algebraic
problem to high-precision linear optimization, proving new upper and lower
complexity estimates along the way.Comment: 15 pages, 9 figures. Submitted to a conference proceeding
A low-memory algorithm for finding short product representations in finite groups
We describe a space-efficient algorithm for solving a generalization of the
subset sum problem in a finite group G, using a Pollard-rho approach. Given an
element z and a sequence of elements S, our algorithm attempts to find a
subsequence of S whose product in G is equal to z. For a random sequence S of
length d log_2 n, where n=#G and d >= 2 is a constant, we find that its
expected running time is O(sqrt(n) log n) group operations (we give a rigorous
proof for d > 4), and it only needs to store O(1) group elements. We consider
applications to class groups of imaginary quadratic fields, and to finding
isogenies between elliptic curves over a finite field.Comment: 12 page
Fast local search for the maximum independent set problem
Abstract. Given a graph G = (V, E), the independent set problem is that of finding a maximum-cardinality subset S of V such that no two vertices in S are adjacent. We present a fast local search routine for this problem. Our algorithm can determine in linear time if a maximal solution can be improved by replacing a single vertex with two others. We also show that an incremental version of this method can be useful within more elaborate heuristics. We test our algorithms on instances from the literature as well as on new ones proposed in this paper. 1
Comparing Bacterial Genomes by Searching Their Common Intervals
International audienceComparing bacterial genomes implies the use of a dedicated measure. It relies on comparing circular genomes based on a set of conserved genes. Following this assumption, the common interval appears to be a good candidate. For evidences, we propose herein an approach to compute the common intervals between two circular genomes that takes into account duplications. Its application on a concrete case, comparing E. coli and V. cholerae, is accurate. It indeed emphasizes sets of conserved genes that present high impacts on bacterial functions