42 research outputs found
The matroid secretary problem for minor-closed classes and random matroids
We prove that for every proper minor-closed class of matroids
representable over a prime field, there exists a constant-competitive matroid
secretary algorithm for the matroids in . This result relies on the
extremely powerful matroid minor structure theory being developed by Geelen,
Gerards and Whittle.
We also note that for asymptotically almost all matroids, the matroid
secretary algorithm that selects a random basis, ignoring weights, is
-competitive. In fact, assuming the conjecture that almost all
matroids are paving, there is a -competitive algorithm for almost all
matroids.Comment: 15 pages, 0 figure
Generalized Laminar Matroids
Nested matroids were introduced by Crapo in 1965 and have appeared frequently
in the literature since then. A flat of a matroid is Hamiltonian if it has
a spanning circuit. A matroid is nested if and only if its Hamiltonian
flats form a chain under inclusion; is laminar if and only if, for every
-element independent set , the Hamiltonian flats of containing
form a chain under inclusion. We generalize these notions to define the classes
of -closure-laminar and -laminar matroids. This paper focuses on
structural properties of these classes noting that, while the second class is
always minor-closed, the first is if and only if . The main results
are excluded-minor characterizations for the classes of 2-laminar and
2-closure-laminar matroids.Comment: 12 page
Laminar Matroids
A laminar family is a collection of subsets of a set such
that, for any two intersecting sets, one is contained in the other. For a
capacity function on , let be \{I:|I\cap A|
\leq c(A)\text{ for all A\in\mathscr{A}}\}. Then is the
collection of independent sets of a (laminar) matroid on . We present a
method of compacting laminar presentations, characterize the class of laminar
matroids by their excluded minors, present a way to construct all laminar
matroids using basic operations, and compare the class of laminar matroids to
other well-known classes of matroids.Comment: 17 page
Matroid Secretary for Regular and Decomposable Matroids
In the matroid secretary problem we are given a stream of elements and asked
to choose a set of elements that maximizes the total value of the set, subject
to being an independent set of a matroid given in advance. The difficulty comes
from the assumption that decisions are irrevocable: if we choose to accept an
element when it is presented by the stream then we can never get rid of it, and
if we choose not to accept it then we cannot later add it. Babaioff, Immorlica,
and Kleinberg [SODA 2007] introduced this problem, gave O(1)-competitive
algorithms for certain classes of matroids, and conjectured that every matroid
admits an O(1)-competitive algorithm. However, most matroids that are known to
admit an O(1)-competitive algorithm can be easily represented using graphs
(e.g. graphic and transversal matroids). In particular, there is very little
known about F-representable matroids (the class of matroids that can be
represented as elements of a vector space over a field F), which are one of the
foundational matroid classes. Moreover, most of the known techniques are as
dependent on graph theory as they are on matroid theory. We go beyond graphs by
giving an O(1)-competitive algorithm for regular matroids (the class of
matroids that are representable over every field), and use techniques that are
matroid-theoretic rather than graph-theoretic. We use the regular matroid
decomposition theorem of Seymour to decompose any regular matroid into matroids
which are either graphic, cographic, or isomorphic to R_{10}, and then show how
to combine algorithms for these basic classes into an algorithm for regular
matroids. This allows us to generalize beyond regular matroids to any class of
matroids that admits such a decomposition into classes for which we already
have good algorithms. In particular, we give an O(1)-competitive algorithm for
the class of max-flow min-cut matroids.Comment: 21 page
On Selected Subclasses of Matroids
Matroids were introduced by Whitney to provide an abstract notion of independence.
In this work, after giving a brief survey of matroid theory, we describe structural results for various classes of matroids. A connected matroid is unbreakable if, for each of its flats , the matroid is connected%or, equivalently, if has no two skew circuits. . Pfeil showed that a simple graphic matroid is unbreakable exactly when is either a cycle or a complete graph. We extend this result to describe which graphs are the underlying graphs of unbreakable frame matroids. A laminar family is a collection \A of subsets of a set such that, for any two intersecting sets, one is contained in the other. For a capacity function on \A, let \I be %the set \{I:|I\cap A| \leq c(A)\text{ for all A\in\A}\}. Then \I is the collection of independent sets of a (laminar) matroid on . We characterize the class of laminar matroids by their excluded minors and present a way to construct all laminar matroids using basic operations. %Nested matroids were introduced by Crapo in 1965 and have appeared frequently in the literature since then. A flat of a matroid is Hamiltonian if it has a spanning circuit. A matroid is nested if its Hamiltonian flats form a chain under inclusion; is laminar if, for every -element independent set , the Hamiltonian flats of containing form a chain under inclusion. We generalize these notions to define the classes of -closure-laminar and -laminar matroids. The second class is always minor-closed, and the first is if and only if . We give excluded-minor characterizations of the classes of 2-laminar and 2-closure-laminar matroids
The Outer Limits of Contention Resolution on Matroids and Connections to the Secretary Problem
Contention resolution schemes have proven to be a useful and unifying abstraction for a variety of constrained optimization problems, in both offline and online arrival models. Much of prior work restricts attention to product distributions for the input set of elements, and studies contention resolution for increasingly general packing constraints, both offline and online. In this paper, we instead focus on generalizing the input distribution, restricting attention to matroid constraints in both the offline and online random arrival models. In particular, we study contention resolution when the input set is arbitrarily distributed, and may exhibit positive and/or negative correlations between elements. We characterize the distributions for which offline contention resolution is possible, and establish some of their basic closure properties. Our characterization can be interpreted as a distributional generalization of the matroid covering theorem. For the online random arrival model, we show that contention resolution is intimately tied to the secretary problem via two results. First, we show that a competitive algorithm for the matroid secretary problem implies that online contention resolution is essentially as powerful as offline contention resolution for matroids, so long as the algorithm is given the input distribution. Second, we reduce the matroid secretary problem to the design of an online contention resolution scheme of a particular form
Non-Adaptive Matroid Prophet Inequalities
We consider the problem of matroid prophet inequalities. This problem has been ex-
tensively studied in case of adaptive prices, with [KW12] obtaining a tight 2-competitive
mechanism for all the matroids.
However, the case non-adaptive is far from resolved, although there is a known constant-
competitive mechanism for uniform and graphical matroids (see [Cha+20]).
We improve on constant-competitive mechanism from [Cha+20] for graphical matroids,
present a separate mechanism for cographical matroids, and combine those to obtain
constant-competitive mechanism for all regular matroids