57 research outputs found
Quantized VCG Mechanisms for Polymatroid Environments
Many network resource allocation problems can be viewed as allocating a
divisible resource, where the allocations are constrained to lie in a
polymatroid. We consider market-based mechanisms for such problems. Though the
Vickrey-Clarke-Groves (VCG) mechanism can provide the efficient allocation with
strong incentive properties (namely dominant strategy incentive compatibility),
its well-known high communication requirements can prevent it from being used.
There have been a number of approaches for reducing the communication costs of
VCG by weakening its incentive properties. Here, instead we take a different
approach of reducing communication costs via quantization while maintaining
VCG's dominant strategy incentive properties. The cost for this approach is a
loss in efficiency which we characterize. We first consider quantizing the
resource allocations so that agents need only submit a finite number of bids
instead of full utility function. We subsequently consider quantizing the
agent's bids
An Algorithmic Theory of Dependent Regularizers, Part 1: Submodular Structure
We present an exploration of the rich theoretical connections between several
classes of regularized models, network flows, and recent results in submodular
function theory. This work unifies key aspects of these problems under a common
theory, leading to novel methods for working with several important models of
interest in statistics, machine learning and computer vision.
In Part 1, we review the concepts of network flows and submodular function
optimization theory foundational to our results. We then examine the
connections between network flows and the minimum-norm algorithm from
submodular optimization, extending and improving several current results. This
leads to a concise representation of the structure of a large class of pairwise
regularized models important in machine learning, statistics and computer
vision.
In Part 2, we describe the full regularization path of a class of penalized
regression problems with dependent variables that includes the graph-guided
LASSO and total variation constrained models. This description also motivates a
practical algorithm. This allows us to efficiently find the regularization path
of the discretized version of TV penalized models. Ultimately, our new
algorithms scale up to high-dimensional problems with millions of variables
A 3/2-Approximation for the Metric Many-visits Path TSP
In the Many-visits Path TSP, we are given a set of cities along with
their pairwise distances (or cost) , and moreover each city comes
with an associated positive integer request .
The goal is to find a minimum-cost path, starting at city and ending at
city , that visits each city exactly times.
We present a -approximation algorithm for the metric Many-visits
Path TSP, that runs in time polynomial in and poly-logarithmic in the
requests .
Our algorithm can be seen as a far-reaching generalization of the
-approximation algorithm for Path TSP by Zenklusen (SODA 2019), which
answered a long-standing open problem by providing an efficient algorithm which
matches the approximation guarantee of Christofides' algorithm from 1976 for
metric TSP.
One of the key components of our approach is a polynomial-time algorithm to
compute a connected, degree bounded multigraph of minimum cost.
We tackle this problem by generalizing a fundamental result of Kir\'aly, Lau
and Singh (Combinatorica, 2012) on the Minimum Bounded Degree Matroid Basis
problem, and devise such an algorithm for general polymatroids, even allowing
element multiplicities.
Our result directly yields a -approximation to the metric
Many-visits TSP, as well as a -approximation for the problem of
scheduling classes of jobs with sequence-dependent setup times on a single
machine so as to minimize the makespan.Comment: arXiv admin note: text overlap with arXiv:1911.0989
Degree-bounded generalized polymatroids and approximating the metric many-visits TSP
In the Bounded Degree Matroid Basis Problem, we are given a matroid and a
hypergraph on the same ground set, together with costs for the elements of that
set as well as lower and upper bounds and for
each hyperedge . The objective is to find a minimum-cost basis
such that for
each hyperedge . Kir\'aly et al. (Combinatorica, 2012) provided an
algorithm that finds a basis of cost at most the optimum value which violates
the lower and upper bounds by at most , where is the
maximum degree of the hypergraph. When only lower or only upper bounds are
present for each hyperedge, this additive error is decreased to .
We consider an extension of the matroid basis problem to generalized
polymatroids, or g-polymatroids, and additionally allow element multiplicities.
The Bounded Degree g-polymatroid Element Problem with Multiplicities takes as
input a g-polymatroid instead of a matroid, and besides the lower and
upper bounds, each hyperedge has element multiplicities
. Building on the approach of Kir\'aly et al., we provide an
algorithm for finding a solution of cost at most the optimum value, having the
same additive approximation guarantee.
As an application, we develop a -approximation for the metric
Many-Visits TSP, where the goal is to find a minimum-cost tour that visits each
city a positive number of times. Our approach combines our algorithm
for the Bounded Degree g-polymatroid Element Problem with Multiplicities with
the principle of Christofides' algorithm from 1976 for the (single-visit)
metric TSP, whose approximation guarantee it matches.Comment: 17 page
Matroids and Integrality Gaps for Hypergraphic Steiner Tree Relaxations
Until recently, LP relaxations have played a limited role in the design of
approximation algorithms for the Steiner tree problem. In 2010, Byrka et al.
presented a ln(4)+epsilon approximation based on a hypergraphic LP relaxation,
but surprisingly, their analysis does not provide a matching bound on the
integrality gap.
We take a fresh look at hypergraphic LP relaxations for the Steiner tree
problem - one that heavily exploits methods and results from the theory of
matroids and submodular functions - which leads to stronger integrality gaps,
faster algorithms, and a variety of structural insights of independent
interest. More precisely, we present a deterministic ln(4)+epsilon
approximation that compares against the LP value and therefore proves a
matching ln(4) upper bound on the integrality gap.
Similarly to Byrka et al., we iteratively fix one component and update the LP
solution. However, whereas they solve an LP at every iteration after
contracting a component, we show how feasibility can be maintained by a greedy
procedure on a well-chosen matroid. Apart from avoiding the expensive step of
solving a hypergraphic LP at each iteration, our algorithm can be analyzed
using a simple potential function. This gives an easy means to determine
stronger approximation guarantees and integrality gaps when considering
restricted graph topologies. In particular, this readily leads to a 73/60 bound
on the integrality gap for quasi-bipartite graphs.
For the case of quasi-bipartite graphs, we present a simple algorithm to
transform an optimal solution to the bidirected cut relaxation to an optimal
solution of the hypergraphic relaxation, leading to a fast 73/60 approximation
for quasi-bipartite graphs. Furthermore, we show how the separation problem of
the hypergraphic relaxation can be solved by computing maximum flows, providing
a fast independence oracle for our matroids.Comment: Corrects an issue at the end of Section 3. Various other minor
improvements to the expositio
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