23 research outputs found
Quantum Algorithms for Identifying Hidden Strings with Applications to Matroid Problems
In this paper, we explore quantum speedups for the problem, inspired by
matroid theory, of identifying a pair of -bit binary strings that are
promised to have the same number of 1s and differ in exactly two bits, by using
the max inner product oracle and the sub-set oracle. More specifically, given
two string satisfying the above constraints, for any
the max inner product oracle returns the max
value between and , and the sub-set oracle
indicates whether the index set of the 1s in is a subset of that in or
. We present a quantum algorithm consuming queries to the max inner
product oracle for identifying the pair , and prove that any
classical algorithm requires queries. Also, we present a
quantum algorithm consuming queries to the subset
oracle, and prove that any classical algorithm requires at least
queries. Therefore, quantum speedups are revealed in the two oracle models.
Furthermore, the above results are applied to the problem in matroid theory of
finding all the bases of a 2-bases matroid, where a matroid is called -bases
if it has bases
A-Tint: A polymake extension for algorithmic tropical intersection theory
In this paper we study algorithmic aspects of tropical intersection theory.
We analyse how divisors and intersection products on tropical cycles can
actually be computed using polyhedral geometry. The main focus of this paper is
the study of moduli spaces, where the underlying combinatorics of the varieties
involved allow a much more efficient way of computing certain tropical cycles.
The algorithms discussed here have been implemented in an extension for
polymake, a software for polyhedral computations.Comment: 32 pages, 5 figures, 4 tables. Second version: Revised version, to be
published in European Journal of Combinatoric
Covering Vectors by Spaces: Regular Matroids
We consider the problem of covering a set of vectors of a given finite dimensional linear space (vector space) by a subspace generated by a set of vectors of minimum size. Specifically, we study the Space Cover problem, where we are given a matrix M and a subset of its columns T; the task is to find a minimum set F of columns of M disjoint with T such that that the linear span of F contains all vectors of T. This is a fundamental problem arising in different domains, such as coding theory, machine learning, and graph algorithms.
We give a parameterized algorithm with running time 2^{O(k)}||M|| ^{O(1)} solving this problem in the case when M is a totally unimodular matrix over rationals, where k is the size of F. In other words, we show that the problem is fixed-parameter tractable parameterized by the rank of the covering subspace. The algorithm is "asymptotically optimal" for the following reasons.
Choice of matrices: Vector matroids corresponding to totally unimodular matrices over rationals are exactly the regular matroids. It is known that for matrices corresponding to a more general class of matroids, namely, binary matroids, the problem becomes W[1]-hard being parameterized by k.
Choice of the parameter: The problem is NP-hard even if |T|=3 on matrix-representations of a subclass of regular matroids, namely cographic matroids. Thus for a stronger parameterization, like by the size of T, the problem becomes intractable.
Running Time: The exponential dependence in the running time of our algorithm cannot be asymptotically improved unless Exponential Time Hypothesis (ETH) fails.
Our algorithm exploits the classical decomposition theorem of Seymour for regular matroids
Polynomial-Delay Enumeration of Large Maximal Common Independent Sets in Two Matroids
Finding a maximum cardinality common independent set in two matroids (also known as Matroid Intersection) is a classical combinatorial optimization problem, which generalizes several well-known problems, such as finding a maximum bipartite matching, a maximum colorful forest, and an arborescence in directed graphs. Enumerating all maximal common independent sets in two (or more) matroids is a classical enumeration problem. In this paper, we address an "intersection" of these problems: Given two matroids and a threshold ?, the goal is to enumerate all maximal common independent sets in the matroids with cardinality at least ?. We show that this problem can be solved in polynomial delay and polynomial space. We also discuss how to enumerate all maximal common independent sets of two matroids in non-increasing order of their cardinalities