215 research outputs found
Circuit and fractional circuit covers of matroids
AbstractLet M be a connected matroid having a ground set E. Lemos and Oxley proved that |E(M)|≤12c(M)c∗(M) where c(M) (resp. c∗(M)) is the circumference (resp. cocircumference) of M. In addition, they conjectured that one can find a collection of at most c∗(M) circuits which cover the elements of M at least twice. In this paper, we verify this conjecture for regular matroids. Moreover, we show that a version of this conjecture is true for fractional circuit covers
Counting matroids in minor-closed classes
A flat cover is a collection of flats identifying the non-bases of a matroid.
We introduce the notion of cover complexity, the minimal size of such a flat
cover, as a measure for the complexity of a matroid, and present bounds on the
number of matroids on elements whose cover complexity is bounded. We apply
cover complexity to show that the class of matroids without an -minor is
asymptotically small in case is one of the sparse paving matroids
, , , , or , thus confirming a few special
cases of a conjecture due to Mayhew, Newman, Welsh, and Whittle. On the other
hand, we show a lower bound on the number of matroids without -minor
which asymptoticaly matches the best known lower bound on the number of all
matroids, due to Knuth.Comment: 13 pages, 3 figure
On the number of matroids
We consider the problem of determining , the number of matroids on
elements. The best known lower bound on is due to Knuth (1974) who showed
that is at least . On the other hand, Piff
(1973) showed that , and it has
been conjectured since that the right answer is perhaps closer to Knuth's
bound.
We show that this is indeed the case, and prove an upper bound on that is within an additive term of Knuth's lower bound. Our proof
is based on using some structural properties of non-bases in a matroid together
with some properties of independent sets in the Johnson graph to give a
compressed representation of matroids.Comment: Final version, 17 page
Idealness of k-wise intersecting families
A clutter is k-wise intersecting if every k members have a common element, yet no element belongs to all members. We conjecture that, for some integer k ≥ 4, every k-wise intersecting clutter is non-ideal. As evidence for our conjecture, we prove it for k = 4 for the class of binary clutters. Two key ingredients for our proof are Jaeger’s 8-flow theorem for graphs, and Seymour’s characterization of the binary matroids with the sums of circuits property. As further evidence for our conjecture, we also note that it follows from an unpublished conjecture of Seymour from 1975. We also discuss connections to the chromatic number of a clutter, projective geometries over the two-element field, uniform cycle covers in graphs, and quarter-integral packings of value two in ideal clutters
Clean clutters and dyadic fractional packings
A vector is dyadic if each of its entries is a dyadic rational number, i.e., an integer multiple of 1 2k for some nonnegative integer k. We prove that every clean clutter with a covering number of at least two has a dyadic fractional packing of value two. This result is best possible for there exist clean clutters with a covering number of three and no dyadic fractional packing of value three. Examples of clean clutters include ideal clutters, binary clutters, and clutters without an intersecting minor. Our proof is constructive and leads naturally to an albeit exponential algorithm. We improve the running time to quasi-polynomial in the rank of the input, and to polynomial in the binary cas
Single Commodity Flow Algorithms for Lifts of Graphic and Cographic Matroids
Consider a binary matroid M given by its matrix representation. We show that if M is a lift of a graphic or a cographic matroid, then in polynomial time we can either solve the single commodity flow problem for M or find an obstruction for which the Max-Flow Min-Cut relation does not hold. The key tool is an algorithmic version of Lehman's Theorem for the set covering polyhedron
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