13 research outputs found
Non-Smooth, H\"older-Smooth, and Robust Submodular Maximization
We study the problem of maximizing a continuous DR-submodular function that
is not necessarily smooth. We prove that the continuous greedy algorithm
achieves an [(1-1/e)\OPT-\epsilon] guarantee when the function is monotone
and H\"older-smooth, meaning that it admits a H\"older-continuous gradient. For
functions that are non-differentiable or non-smooth, we propose a variant of
the mirror-prox algorithm that attains an [(1/2)\OPT-\epsilon] guarantee. We
apply our algorithmic frameworks to robust submodular maximization and
distributionally robust submodular maximization under Wasserstein ambiguity. In
particular, the mirror-prox method applies to robust submodular maximization to
obtain a single feasible solution whose value is at least (1/2)\OPT-\epsilon.
For distributionally robust maximization under Wasserstein ambiguity, we deduce
and work over a submodular-convex maximin reformulation whose objective
function is H\"older-smooth, for which we may apply both the continuous greedy
and the mirror-prox algorithms
Online Resource Allocation in Episodic Markov Decision Processes
This paper studies a long-term resource allocation problem over multiple
periods where each period requires a multi-stage decision-making process. We
formulate the problem as an online allocation problem in an episodic
finite-horizon constrained Markov decision process with an unknown
non-stationary transition function and stochastic non-stationary reward and
resource consumption functions. We propose the observe-then-decide regime and
improve the existing decide-then-observe regime, while the two settings differ
in how the observations and feedback about the reward and resource consumption
functions are given to the decision-maker. We develop an online dual mirror
descent algorithm that achieves near-optimal regret bounds for both settings.
For the observe-then-decide regime, we prove that the expected regret against
the dynamic clairvoyant optimal policy is bounded by where is the budget parameter,
is the length of the horizon, and are the numbers of states and
actions, and is the number of episodes. For the decide-then-observe regime,
we show that the regret against the static optimal policy that has access to
the mean reward and mean resource consumption functions is bounded by with high probability. We test the numerical
efficiency of our method for a variant of the resource-constrained inventory
management problem
Characterizing matroids whose bases form graphic delta-matroids
We introduce delta-graphic matroids, which are matroids whose bases form
graphic delta-matroids. The class of delta-graphic matroids contains graphic
matroids as well as cographic matroids and is a proper subclass of the class of
regular matroids. We give a structural characterization of the class of
delta-graphic matroids. We also show that every forbidden minor for the class
of delta-graphic matroids has at most elements.Comment: 40 pages, 4 figures; revise
A chain theorem for sequentially -rank-connected graphs with respect to vertex-minors
Tutte (1961) proved the chain theorem for -connected graphs with respect
to minors, which states that every -connected graph has a -connected
minor with one vertex fewer than , unless is a wheel graph. Bouchet
(1987) proved an analog for prime graphs with respect to vertex-minors. We
present a chain theorem for higher connectivity with respect to vertex-minors,
showing that every sequentially -rank-connected graph has a sequentially
-rank-connected vertex-minor with one vertex fewer than , unless
.Comment: 21 page
Projection-Free Online Convex Optimization with Stochastic Constraints
This paper develops projection-free algorithms for online convex optimization
with stochastic constraints. We design an online primal-dual projection-free
framework that can take any projection-free algorithms developed for online
convex optimization with no long-term constraint. With this general template,
we deduce sublinear regret and constraint violation bounds for various
settings. Moreover, for the case where the loss and constraint functions are
smooth, we develop a primal-dual conditional gradient method that achieves
regret and constraint violations. Furthermore, for
the setting where the loss and constraint functions are stochastic and strong
duality holds for the associated offline stochastic optimization problem, we
prove that the constraint violation can be reduced to have the same asymptotic
growth as the regret
-graphic delta-matroids and their applications
For an abelian group , a -labelled graph is a graph whose
vertices are labelled by elements of . We prove that a certain
collection of edge sets of a -labelled graph forms a delta-matroid,
which we call a -graphic delta-matroid, and provide a polynomial-time
algorithm to solve the separation problem, which allows us to apply the
symmetric greedy algorithm of Bouchet to find a maximum weight feasible set in
such a delta-matroid. We present two algorithmic applications on graphs;
Maximum Weight Packing of Trees of Order Not Divisible by and Maximum
Weight -Tree Packing. We also discuss various properties of -graphic
delta-matroids.Comment: 16 pages, 2 figure
?-Graphic Delta-Matroids and Their Applications
For an abelian group ?, a ?-labelled graph is a graph whose vertices are labelled by elements of ?. We prove that a certain collection of edge sets of a ?-labelled graph forms a delta-matroid, which we call a ?-graphic delta-matroid, and provide a polynomial-time algorithm to solve the separation problem, which allows us to apply the symmetric greedy algorithm of Bouchet to find a maximum weight feasible set in such a delta-matroid. We present two algorithmic applications on graphs; Maximum Weight Packing of Trees of Order Not Divisible by k and Maximum Weight S-Tree Packing. We also discuss various properties of ?-graphic delta-matroids
Characterizing matroids whose bases form graphic delta-matroids
We introduce delta-graphic matroids, which are matroids whose bases form graphic delta-matroids. The class of delta-graphic matroids contains graphic matroids as well as cographic matroids and is a proper subclass of the class of regular matroids. We give a structural characterization of the class of delta-graphic matroids. We also show that every forbidden minor for the class of delta-graphic matroids has at most 48 elements.11Nscopu
Intertwining connectivities for vertex-minors and pivot-minors
We show that for pairs and of disjoint subsets of vertices of
a graph , if is sufficiently large, then there exists a vertex in
such that there are two ways to reduce by a
vertex-minor operation that removes while preserving the connectivity
between and and the connectivity between and . Our theorem
implies an analogous theorem of Chen and Whittle (2014) for matroids restricted
to binary matroids.Comment: 10 page