41 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
Integrality Gap of the Hypergraphic Relaxation of Steiner Trees: a short proof of a 1.55 upper bound
Recently Byrka, Grandoni, Rothvoss and Sanita (at STOC 2010) gave a
1.39-approximation for the Steiner tree problem, using a hypergraph-based
linear programming relaxation. They also upper-bounded its integrality gap by
1.55. We describe a shorter proof of the same integrality gap bound, by
applying some of their techniques to a randomized loss-contracting algorithm
On the Equivalence of the Bidirected and Hypergraphic Relaxations for Steiner Tree
The bottleneck of the currently best (ln(4) + epsilon)-approximation algorithm for the NP-hard Steiner tree problem is the solution of its large, so called hypergraphic, linear programming relaxation (HYP). Hypergraphic LPs are NP-hard to solve exactly, and it is a formidable computational task to even approximate them sufficiently well.
We focus on another well-studied but poorly understood LP relaxation of the problem: the bidirected cut relaxation (BCR). This LP is compact, and can therefore be solved efficiently. Its integrality gap is known to be greater than 1.16, and while this is widely conjectured to be close to the real answer, only a (trivial) upper bound of 2 is known.
In this paper, we give an efficient constructive proof that BCR and HYP are polyhedrally equivalent in instances that do not have an (edge-induced) claw on Steiner vertices, i.e., they do not contain a Steiner vertex with 3 Steiner neighbors. This implies faster ln(4)-approximations for these graphs, and is a significant step forward from the previously known equivalence for (so called quasi-bipartite) instances in which Steiner vertices form an independent set. We complement our results by showing that even restricting to instances where Steiner vertices induce one single star, determining whether the two relaxations are equivalent is NP-hard
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
Linear Programming Tools and Approximation Algorithms for Combinatorial Optimization
We study techniques, approximation algorithms, structural properties and lower bounds related to applications of linear programs in combinatorial optimization. The following "Steiner tree problem" is central: given a graph with a distinguished subset of required vertices, and costs for each edge, find a minimum-cost subgraph that connects the required vertices. We also investigate the areas of network design, multicommodity flows, and packing/covering integer programs. All of these problems are NP-complete so it is natural to seek approximation algorithms with the best provable approximation ratio.
Overall, we show some new techniques that enhance the already-substantial corpus of LP-based approximation methods, and we also look for limitations of these techniques.
The first half of the thesis deals with linear programming relaxations for the Steiner tree problem. The crux of our work deals with hypergraphic relaxations obtained via the well-known full component decomposition of Steiner trees; explicitly, in this view the fundamental building blocks are not edges, but hyperedges containing two or more required vertices. We introduce a new hypergraphic LP based on partitions. We show the new LP has the same value as several previously-studied hypergraphic ones; when no Steiner nodes are adjacent, we show that the value of the well-known bidirected cut relaxation is also the same. A new partition uncrossing technique is used to demonstrate these equivalences, and to show that extreme points of the new LP are well-structured. We improve the best known integrality gap on these LPs in some special cases. We show that several approximation algorithms from the literature on Steiner trees can be re-interpreted through linear programs, in particular our hypergraphic relaxation yields a new view of the Robins-Zelikovsky 1.55-approximation algorithm for the Steiner tree problem.
The second half of the thesis deals with a variety of fundamental problems in combinatorial optimization. We show how to apply the iterated LP relaxation framework to the problem of multicommodity integral flow in a tree, to get an approximation ratio that is asymptotically optimal in terms of the minimum capacity. Iterated relaxation gives an infeasible solution, so we need to finesse it back to feasibility without losing too much value. Iterated LP relaxation similarly gives an O(k^2)-approximation algorithm for packing integer programs with at most k occurrences of each variable; new LP rounding techniques give a k-approximation algorithm for covering integer programs with at most k variable per constraint. We study extreme points of the standard LP relaxation for the traveling salesperson problem and show that they can be much more complex than was previously known. The k-edge-connected spanning multi-subgraph problem has the same LP and we prove a lower bound and conjecture an upper bound on the approximability of variants of this problem. Finally, we show that for packing/covering integer programs with a bounded number of constraints, for any epsilon > 0, there is an LP with integrality gap at most 1 + epsilon
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