198 research outputs found
Algorithms for Enumerating Circuits in Matroids
We present an incremental polynomial-time algorithm for enumerating all circuits of a matroid or, more generally, all minimal spanning sets for a flat. This result implies, in particular, that for a given infeasible system of linear equations, all its maximal feasible subsystems, as well as all minimal infeasible subsystems, can be enumerated in incremental polynomial time. We also show the NP-hardness of several related enumeration problems
A Family of matroid intersection algorithms for the computation of approximated symbolic network functions
In recent years, the technique of simplification during generation has turned out to be very promising for the efficient computation of approximate symbolic network functions for large transistor circuits. In this paper it is shown how symbolic network functions can be simplified during their generation with any well-known symbolic network analysis method. The underlying algorithm for the different techniques is always a matroid intersection algorithm. It is shown that the most efficient technique is the two-graph method. An implementation of the simplification during generation technique with the two-graph method illustrates its benefits for the symbolic analysis of large analog circuits
Computing Algebraic Matroids
An affine variety induces the structure of an algebraic matroid on the set of
coordinates of the ambient space. The matroid has two natural decorations: a
circuit polynomial attached to each circuit, and the degree of the projection
map to each base, called the base degree. Decorated algebraic matroids can be
computed via symbolic computation using Groebner bases, or through linear
algebra in the space of differentials (with decorations calculated using
numerical algebraic geometry). Both algorithms are developed here. Failure of
the second algorithm occurs on a subvariety called the non-matroidal or NM-
locus. Decorated algebraic matroids have widespread relevance anywhere that
coordinates have combinatorial significance. Examples are computed from applied
algebra, in algebraic statistics and chemical reaction network theory, as well
as more theoretical examples from algebraic geometry and matroid theory.Comment: 15 pages; added link to references, note on page 1, and small
formatting fixe
Simplicial and Cellular Trees
Much information about a graph can be obtained by studying its spanning
trees. On the other hand, a graph can be regarded as a 1-dimensional cell
complex, raising the question of developing a theory of trees in higher
dimension. As observed first by Bolker, Kalai and Adin, and more recently by
numerous authors, the fundamental topological properties of a tree --- namely
acyclicity and connectedness --- can be generalized to arbitrary dimension as
the vanishing of certain cellular homology groups. This point of view is
consistent with the matroid-theoretic approach to graphs, and yields
higher-dimensional analogues of classical enumerative results including
Cayley's formula and the matrix-tree theorem. A subtlety of the
higher-dimensional case is that enumeration must account for the possibility of
torsion homology in trees, which is always trivial for graphs. Cellular trees
are the starting point for further high-dimensional extensions of concepts from
algebraic graph theory including the critical group, cut and flow spaces, and
discrete dynamical systems such as the abelian sandpile model.Comment: 39 pages (including 5-page bibliography); 5 figures. Chapter for
forthcoming IMA volume "Recent Trends in Combinatorics
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
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
Determinantal Sieving
We introduce determinantal sieving, a new, remarkably powerful tool in the
toolbox of algebraic FPT algorithms. Given a polynomial on a set of
variables and a linear matroid of
rank , both over a field of characteristic 2, in
evaluations we can sieve for those terms in the monomial expansion of which
are multilinear and whose support is a basis for . Alternatively, using
evaluations of we can sieve for those monomials whose odd support
spans . Applying this framework, we improve on a range of algebraic FPT
algorithms, such as:
1. Solving -Matroid Intersection in time and -Matroid
Parity in time , improving on (Brand and Pratt,
ICALP 2021)
2. -Cycle, Colourful -Path, Colourful -Linkage in undirected
graphs, and the more general Rank -Linkage problem, all in
time, improving on respectively (Fomin et al., SODA 2023)
3. Many instances of the Diverse X paradigm, finding a collection of
solutions to a problem with a minimum mutual distance of in time
, improving solutions for -Distinct Branchings from time
to (Bang-Jensen et al., ESA 2021), and for Diverse
Perfect Matchings from to (Fomin et al.,
STACS 2021)
All matroids are assumed to be represented over a field of characteristic 2.
Over general fields, we achieve similar results at the cost of using
exponential space by working over the exterior algebra. For a class of
arithmetic circuits we call strongly monotone, this is even achieved without
any loss of running time. However, the odd support sieving result appears to be
specific to working over characteristic 2
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