178 research outputs found
Hypergraphic LP Relaxations for Steiner Trees
We investigate hypergraphic LP relaxations for the Steiner tree problem,
primarily the partition LP relaxation introduced by Koenemann et al. [Math.
Programming, 2009]. Specifically, we are interested in proving upper bounds on
the integrality gap of this LP, and studying its relation to other linear
relaxations. Our results are the following. Structural results: We extend the
technique of uncrossing, usually applied to families of sets, to families of
partitions. As a consequence we show that any basic feasible solution to the
partition LP formulation has sparse support. Although the number of variables
could be exponential, the number of positive variables is at most the number of
terminals. Relations with other relaxations: We show the equivalence of the
partition LP relaxation with other known hypergraphic relaxations. We also show
that these hypergraphic relaxations are equivalent to the well studied
bidirected cut relaxation, if the instance is quasibipartite. Integrality gap
upper bounds: We show an upper bound of sqrt(3) ~ 1.729 on the integrality gap
of these hypergraph relaxations in general graphs. In the special case of
uniformly quasibipartite instances, we show an improved upper bound of 73/60 ~
1.216. By our equivalence theorem, the latter result implies an improved upper
bound for the bidirected cut relaxation as well.Comment: Revised full version; a shorter version will appear at IPCO 2010
Variational approximation of functionals defined on 1-dimensional connected sets: the planar case
In this paper we consider variational problems involving 1-dimensional
connected sets in the Euclidean plane, such as the classical Steiner tree
problem and the irrigation (Gilbert-Steiner) problem. We relate them to optimal
partition problems and provide a variational approximation through
Modica-Mortola type energies proving a -convergence result. We also
introduce a suitable convex relaxation and develop the corresponding numerical
implementations. The proposed methods are quite general and the results we
obtain can be extended to -dimensional Euclidean space or to more general
manifold ambients, as shown in the companion paper [11].Comment: 30 pages, 5 figure
Constant distortion embeddings of Symmetric Diversities
Diversities are like metric spaces, except that every finite subset, instead
of just every pair of points, is assigned a value. Just as there is a theory of
minimal distortion embeddings of finite metric spaces into , there is a
similar, yet undeveloped, theory for embedding finite diversities into the
diversity analogue of spaces. In the metric case, it is well known that
an -point metric space can be embedded into with
distortion. For diversities, the optimal distortion is unknown. Here, we
establish the surprising result that symmetric diversities, those in which the
diversity (value) assigned to a set depends only on its cardinality, can be
embedded in with constant distortion.Comment: 14 pages, 3 figure
Dual Growth with Variable Rates: An Improved Integrality Gap for Steiner Tree
A promising approach for obtaining improved approximation algorithms for
Steiner tree is to use the bidirected cut relaxation (BCR). The integrality gap
of this relaxation is at least , and it has long been conjectured that
its true value is very close to this lower bound. However, the best upper bound
for general graphs is still . With the aim of circumventing the asymmetric
nature of BCR, Chakrabarty, Devanur and Vazirani [Math. Program., 130 (2011),
pp. 1--32] introduced the simplex-embedding LP, which is equivalent to it.
Using this, they gave a -approximation algorithm for quasi-bipartite
graphs and showed that the integrality gap of the relaxation is at most
for this class of graphs.
In this paper, we extend the approach provided by these authors and show that
the integrality gap of BCR is at most on quasi-bipartite graphs via a
fast combinatorial algorithm. In doing so, we introduce a general technique, in
particular a potentially widely applicable extension of the primal-dual schema.
Roughly speaking, we apply the schema twice with variable rates of growth for
the duals in the second phase, where the rates depend on the degrees of the
duals computed in the first phase. This technique breaks the disadvantage of
increasing dual variables in a monotone manner and creates a larger total dual
value, thus presumably attaining the true integrality gap.Comment: A completely rewritten version of a previously retracted manuscript,
using the simplex-embedding LP. The idea of growing duals with variable rates
is still there. 23 pages, 7 figure
Combinatorial Optimization
Combinatorial Optimization is a very active field that benefits from bringing together ideas from different areas, e.g., graph theory and combinatorics, matroids and submodularity, connectivity and network flows, approximation algorithms and mathematical programming, discrete and computational geometry, discrete and continuous problems, algebraic and geometric methods, and applications. We continued the long tradition of triannual Oberwolfach workshops, bringing together the best researchers from the above areas, discovering new connections, and establishing new and deepening existing international collaborations
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
Universal Algorithms for Clustering Problems
This paper presents universal algorithms for clustering problems, including
the widely studied -median, -means, and -center objectives. The input
is a metric space containing all potential client locations. The algorithm must
select cluster centers such that they are a good solution for any subset of
clients that actually realize. Specifically, we aim for low regret, defined as
the maximum over all subsets of the difference between the cost of the
algorithm's solution and that of an optimal solution. A universal algorithm's
solution for a clustering problem is said to be an -approximation if for all subsets of clients , it satisfies , where is the cost of the
optimal solution for clients and is the minimum regret achievable by
any solution.
Our main results are universal algorithms for the standard clustering
objectives of -median, -means, and -center that achieve -approximations. These results are obtained via a novel framework for
universal algorithms using linear programming (LP) relaxations. These results
generalize to other -objectives and the setting where some subset of
the clients are fixed. We also give hardness results showing that -approximation is NP-hard if or is at most a certain
constant, even for the widely studied special case of Euclidean metric spaces.
This shows that in some sense, -approximation is the strongest
type of guarantee obtainable for universal clustering.Comment: Appeared in ICALP 2021, Track A. Fixed mismatch between paper title
and arXiv titl
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