209 research outputs found
Persistence and stability properties of powers of ideals
We introduce the concept of strong persistence and show that it implies
persistence regarding the associated prime ideals of the powers of an ideal. We
also show that strong persistence is equivalent to a condition on power of
ideals studied by Ratliff. Furthermore, we give an upper bound for the depth of
powers of monomial ideals in terms of their linear relation graph, and apply
this to show that the index of depth stability and the index of stability for
the associated prime ideals of polymatroidal ideals is bounded by their
analytic spread.Comment: 15 pages, 1 figur
Polyhedra with the Integer Caratheodory Property
A polyhedron P has the Integer Caratheodory Property if the following holds.
For any positive integer k and any integer vector w in kP, there exist affinely
independent integer vectors x_1,...,x_t in P and positive integers n_1,...,n_t
such that n_1+...+n_t=k and w=n_1x_1+...+n_tx_t. In this paper we prove that if
P is a (poly)matroid base polytope or if P is defined by a TU matrix, then P
and projections of P satisfy the integer Caratheodory property.Comment: 12 page
Polymatroid Prophet Inequalities
Consider a gambler and a prophet who observe a sequence of independent,
non-negative numbers. The gambler sees the numbers one-by-one whereas the
prophet sees the entire sequence at once. The goal of both is to decide on
fractions of each number they want to keep so as to maximize the weighted
fractional sum of the numbers chosen.
The classic result of Krengel and Sucheston (1977-78) asserts that if both
the gambler and the prophet can pick one number, then the gambler can do at
least half as well as the prophet. Recently, Kleinberg and Weinberg (2012) have
generalized this result to settings where the numbers that can be chosen are
subject to a matroid constraint.
In this note we go one step further and show that the bound carries over to
settings where the fractions that can be chosen are subject to a polymatroid
constraint. This bound is tight as it is already tight for the simple setting
where the gambler and the prophet can pick only one number. An interesting
application of our result is in mechanism design, where it leads to improved
results for various problems
A Polymatroid Approach to Generalized Weights of Rank Metric Codes
We consider the notion of a -polymatroid, due to Shiromoto, and the
more general notion of -demi-polymatroid, and show how generalized
weights can be defined for them. Further, we establish a duality for these
weights analogous to Wei duality for generalized Hamming weights of linear
codes. The corresponding results of Ravagnani for Delsarte rank metric codes,
and Martinez-Penas and Matsumoto for relative generalized rank weights are
derived as a consequence.Comment: 22 pages; with minor revisions in the previous versio
k-Trails: Recognition, Complexity, and Approximations
The notion of degree-constrained spanning hierarchies, also called k-trails,
was recently introduced in the context of network routing problems. They
describe graphs that are homomorphic images of connected graphs of degree at
most k. First results highlight several interesting advantages of k-trails
compared to previous routing approaches. However, so far, only little is known
regarding computational aspects of k-trails.
In this work we aim to fill this gap by presenting how k-trails can be
analyzed using techniques from algorithmic matroid theory. Exploiting this
connection, we resolve several open questions about k-trails. In particular, we
show that one can recognize efficiently whether a graph is a k-trail.
Furthermore, we show that deciding whether a graph contains a k-trail is
NP-complete; however, every graph that contains a k-trail is a (k+1)-trail.
Moreover, further leveraging the connection to matroids, we consider the
problem of finding a minimum weight k-trail contained in a graph G. We show
that one can efficiently find a (2k-1)-trail contained in G whose weight is no
more than the cheapest k-trail contained in G, even when allowing negative
weights.
The above results settle several open questions raised by Molnar, Newman, and
Sebo
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