487 research outputs found
An algorithm for weighted fractional matroid matching
Let M be a matroid on ground set E. A subset l of E is called a `line' when
its rank equals 1 or 2. Given a set L of lines, a `fractional matching' in
(M,L) is a nonnegative vector x indexed by the lines in L, that satisfies a
system of linear constraints, one for each flat of M. Fractional matchings were
introduced by Vande Vate, who showed that the set of fractional matchings is a
half-integer relaxation of the matroid matching polytope.
It was shown by Chang et al. that a maximum size fractional matching can be
found in polynomial time. In this paper we give a polynomial time algorithm to
find for any given weights on the lines in L, a maximum weight fractional
matching.Comment: 15 page
Minimizing Submodular Functions on Diamonds via Generalized Fractional Matroid Matchings
In this paper we show the first polynomial-time algorithm for the problem of minimizing submodular functions on the product of diamonds. This submodular function minimization problem is reduced to the membership problem for an associated polyhedron, which is equivalent to the optimization problem over the polyhedron, based on the ellipsoid method. The latter optimization problem is solved by polynomial number of solutions of subproblems, each being a generalization of the weighted fractional matroid matching problem. We give a combinatorial polynomial-time algorithm for this optimization problem by extending the result by Gijswijt and Pap [D.~Gijswijt and G.~Pap, An algorithm for weighted fractional matroid matching, J.\ Combin.\ Theory, Ser.~B 103 (2013), 509--520]
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
Coverage, Matching, and Beyond: New Results on Budgeted Mechanism Design
We study a type of reverse (procurement) auction problems in the presence of
budget constraints. The general algorithmic problem is to purchase a set of
resources, which come at a cost, so as not to exceed a given budget and at the
same time maximize a given valuation function. This framework captures the
budgeted version of several well known optimization problems, and when the
resources are owned by strategic agents the goal is to design truthful and
budget feasible mechanisms, i.e. elicit the true cost of the resources and
ensure the payments of the mechanism do not exceed the budget. Budget
feasibility introduces more challenges in mechanism design, and we study
instantiations of this problem for certain classes of submodular and XOS
valuation functions. We first obtain mechanisms with an improved approximation
ratio for weighted coverage valuations, a special class of submodular functions
that has already attracted attention in previous works. We then provide a
general scheme for designing randomized and deterministic polynomial time
mechanisms for a class of XOS problems. This class contains problems whose
feasible set forms an independence system (a more general structure than
matroids), and some representative problems include, among others, finding
maximum weighted matchings, maximum weighted matroid members, and maximum
weighted 3D-matchings. For most of these problems, only randomized mechanisms
with very high approximation ratios were known prior to our results
Submodular Stochastic Probing on Matroids
In a stochastic probing problem we are given a universe , where each
element is active independently with probability , and only a
probe of e can tell us whether it is active or not. On this universe we execute
a process that one by one probes elements --- if a probed element is active,
then we have to include it in the solution, which we gradually construct.
Throughout the process we need to obey inner constraints on the set of elements
taken into the solution, and outer constraints on the set of all probed
elements. This abstract model was presented by Gupta and Nagarajan (IPCO '13),
and provides a unified view of a number of problems. Thus far, all the results
falling under this general framework pertain mainly to the case in which we are
maximizing a linear objective function of the successfully probed elements. In
this paper we generalize the stochastic probing problem by considering a
monotone submodular objective function. We give a -approximation algorithm for the case in which we are given
matroids as inner constraints and matroids as outer constraints.
Additionally, we obtain an improved -approximation
algorithm for linear objective functions
Online Contention Resolution Schemes
We introduce a new rounding technique designed for online optimization
problems, which is related to contention resolution schemes, a technique
initially introduced in the context of submodular function maximization. Our
rounding technique, which we call online contention resolution schemes (OCRSs),
is applicable to many online selection problems, including Bayesian online
selection, oblivious posted pricing mechanisms, and stochastic probing models.
It allows for handling a wide set of constraints, and shares many strong
properties of offline contention resolution schemes. In particular, OCRSs for
different constraint families can be combined to obtain an OCRS for their
intersection. Moreover, we can approximately maximize submodular functions in
the online settings we consider.
We, thus, get a broadly applicable framework for several online selection
problems, which improves on previous approaches in terms of the types of
constraints that can be handled, the objective functions that can be dealt
with, and the assumptions on the strength of the adversary. Furthermore, we
resolve two open problems from the literature; namely, we present the first
constant-factor constrained oblivious posted price mechanism for matroid
constraints, and the first constant-factor algorithm for weighted stochastic
probing with deadlines.Comment: 33 pages. To appear in SODA 201
Changing Bases: Multistage Optimization for Matroids and Matchings
This paper is motivated by the fact that many systems need to be maintained
continually while the underlying costs change over time. The challenge is to
continually maintain near-optimal solutions to the underlying optimization
problems, without creating too much churn in the solution itself. We model this
as a multistage combinatorial optimization problem where the input is a
sequence of cost functions (one for each time step); while we can change the
solution from step to step, we incur an additional cost for every such change.
We study the multistage matroid maintenance problem, where we need to maintain
a base of a matroid in each time step under the changing cost functions and
acquisition costs for adding new elements. The online version of this problem
generalizes online paging. E.g., given a graph, we need to maintain a spanning
tree at each step: we pay for the cost of the tree at time
, and also for the number of edges changed at
this step. Our main result is an -approximation, where is
the number of elements/edges and is the rank of the matroid. We also give
an approximation for the offline version of the problem. These
bounds hold when the acquisition costs are non-uniform, in which caseboth these
results are the best possible unless P=NP.
We also study the perfect matching version of the problem, where we must
maintain a perfect matching at each step under changing cost functions and
costs for adding new elements. Surprisingly, the hardness drastically
increases: for any constant , there is no
-approximation to the multistage matching maintenance
problem, even in the offline case
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