341 research outputs found
On the unique representability of spikes over prime fields
For an integer , a rank- matroid is called an -spike if it
consists of three-point lines through a common point such that, for all
, the union of every set of of these lines has
rank . Spikes are very special and important in matroid theory. In 2003 Wu
found the exact numbers of -spikes over fields with 2, 3, 4, 5, 7 elements,
and the asymptotic values for larger finite fields. In this paper, we prove
that, for each prime number , a ) representable -spike is only
representable on fields with characteristic provided that .
Moreover, is uniquely representable over .Comment: 8 page
Subdeterminant Maximization via Nonconvex Relaxations and Anti-concentration
Several fundamental problems that arise in optimization and computer science
can be cast as follows: Given vectors and a
constraint family , find a set that
maximizes the squared volume of the simplex spanned by the vectors in . A
motivating example is the data-summarization problem in machine learning where
one is given a collection of vectors that represent data such as documents or
images. The volume of a set of vectors is used as a measure of their diversity,
and partition or matroid constraints over are imposed in order to ensure
resource or fairness constraints. Recently, Nikolov and Singh presented a
convex program and showed how it can be used to estimate the value of the most
diverse set when corresponds to a partition matroid. This result was
recently extended to regular matroids in works of Straszak and Vishnoi, and
Anari and Oveis Gharan. The question of whether these estimation algorithms can
be converted into the more useful approximation algorithms -- that also output
a set -- remained open.
The main contribution of this paper is to give the first approximation
algorithms for both partition and regular matroids. We present novel
formulations for the subdeterminant maximization problem for these matroids;
this reduces them to the problem of finding a point that maximizes the absolute
value of a nonconvex function over a Cartesian product of probability
simplices. The technical core of our results is a new anti-concentration
inequality for dependent random variables that allows us to relate the optimal
value of these nonconvex functions to their value at a random point. Unlike
prior work on the constrained subdeterminant maximization problem, our proofs
do not rely on real-stability or convexity and could be of independent interest
both in algorithms and complexity.Comment: in FOCS 201
Fairness in Streaming Submodular Maximization over a Matroid Constraint
Streaming submodular maximization is a natural model for the task of
selecting a representative subset from a large-scale dataset. If datapoints
have sensitive attributes such as gender or race, it becomes important to
enforce fairness to avoid bias and discrimination. This has spurred significant
interest in developing fair machine learning algorithms. Recently, such
algorithms have been developed for monotone submodular maximization under a
cardinality constraint.
In this paper, we study the natural generalization of this problem to a
matroid constraint. We give streaming algorithms as well as impossibility
results that provide trade-offs between efficiency, quality and fairness. We
validate our findings empirically on a range of well-known real-world
applications: exemplar-based clustering, movie recommendation, and maximum
coverage in social networks.Comment: Accepted to ICML 2
An algebraic framework for the greedy algorithm with applications to the core and Weber set of cooperative games
An algebraic model generalizing submodular polytopes is presented, where modular functions on partially ordered sets take over the role of vectors in . This model unifies various generalizations of combinatorial models in which the greedy algorithm and the Monge algorithm are successful and generalizations of the notions of core and Weber set in cooperative game theory. As a further application, we show that an earlier model of ours as well as the algorithmic model of Queyranne, Spieksma and Tardella for the Monge algorithm can be treated within the framework of usual matroid theory (on unordered ground-sets), which permits also the efficient algorithmic solution of the intersection problem within this model. \u
Matroid Online Bipartite Matching and Vertex Cover
The Adwords and Online Bipartite Matching problems have enjoyed a renewed
attention over the past decade due to their connection to Internet advertising.
Our community has contributed, among other things, new models (notably
stochastic) and extensions to the classical formulations to address the issues
that arise from practical needs. In this paper, we propose a new generalization
based on matroids and show that many of the previous results extend to this
more general setting. Because of the rich structures and expressive power of
matroids, our new setting is potentially of interest both in theory and in
practice.
In the classical version of the problem, the offline side of a bipartite
graph is known initially while vertices from the online side arrive one at a
time along with their incident edges. The objective is to maintain a decent
approximate matching from which no edge can be removed. Our generalization,
called Matroid Online Bipartite Matching, additionally requires that the set of
matched offline vertices be independent in a given matroid. In particular, the
case of partition matroids corresponds to the natural scenario where each
advertiser manages multiple ads with a fixed total budget.
Our algorithms attain the same performance as the classical version of the
problems considered, which are often provably the best possible. We present
-competitive algorithms for Matroid Online Bipartite Matching under the
small bid assumption, as well as a -competitive algorithm for Matroid
Online Bipartite Matching in the random arrival model. A key technical
ingredient of our results is a carefully designed primal-dual waterfilling
procedure that accommodates for matroid constraints. This is inspired by the
extension of our recent charging scheme for Online Bipartite Vertex Cover.Comment: 19 pages, to appear in EC'1
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
Exchange distance of basis pairs in split matroids
The basis exchange axiom has been a driving force in the development of
matroid theory. However, the axiom gives only a local characterization of the
relation of bases, which is a major stumbling block to further progress, and
providing a global understanding of the structure of matroid bases is a
fundamental goal in matroid optimization.
While studying the structure of symmetric exchanges, Gabow proposed the
problem that any pair of bases admits a sequence of symmetric exchanges. A
different extension of the exchange axiom was proposed by White, who
investigated the equivalence of compatible basis sequences. Farber studied the
structure of basis pairs, and conjectured that the basis pair graph of any
matroid is connected. These conjectures suggest that the family of bases of a
matroid possesses much stronger structural properties than we are aware of.
In the present paper, we study the distance of basis pairs of a matroid in
terms of symmetric exchanges. In particular, we give an upper bound on the
minimum number of exchanges needed to transform a basis pair into another for
split matroids, a class that was motivated by the study of matroid polytopes
from a tropical geometry point of view. As a corollary, we verify the above
mentioned long-standing conjectures for this large class. Being a subclass of
split matroids, our result settles the conjectures for paving matroids as well.Comment: 17 page
Reconfiguration of basis pairs in regular matroids
In recent years, combinatorial reconfiguration problems have attracted great
attention due to their connection to various topics such as optimization,
counting, enumeration, or sampling. One of the most intriguing open questions
concerns the exchange distance of two matroid basis sequences, a problem that
appears in several areas of computer science and mathematics. In 1980, White
proposed a conjecture for the characterization of two basis sequences being
reachable from each other by symmetric exchanges, which received a significant
interest also in algebra due to its connection to toric ideals and Gr\"obner
bases. In this work, we verify White's conjecture for basis sequences of length
two in regular matroids, a problem that was formulated as a separate question
by Farber, Richter, and Shan and Andres, Hochst\"attler, and Merkel. Most of
previous work on White's conjecture has not considered the question from an
algorithmic perspective. We study the problem from an optimization point of
view: our proof implies a polynomial algorithm for determining a sequence of
symmetric exchanges that transforms a basis pair into another, thus providing
the first polynomial upper bound on the exchange distance of basis pairs in
regular matroids. As a byproduct, we verify a conjecture of Gabow from 1976 on
the serial symmetric exchange property of matroids for the regular case.Comment: 28 pages, 6 figure
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