11 research outputs found
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
On rooted -connectivity problems in quasi-bipartite digraphs
We consider the directed Rooted Subset -Edge-Connectivity problem: given a
set of terminals in a digraph with edge costs and
an integer , find a min-cost subgraph of that contains edge disjoint
-paths for all . The case when every edge of positive cost has
head in admits a polynomial time algorithm due to Frank, and the case when
all positive cost edges are incident to is equivalent to the -Multicover
problem. Recently, [Chan et al. APPROX20] obtained ratio for
quasi-bipartite instances, when every edge in has an end in . We give
a simple proof for the same ratio for a more general problem of covering an
arbitrary -intersecting supermodular set function by a minimum cost edge
set, and for the case when only every positive cost edge has an end in
Approximation Algorithms for Steiner Tree Problems Based on Universal Solution Frameworks
This paper summarizes the work on implementing few solutions for the Steiner
Tree problem which we undertook in the PAAL project. The main focus of the
project is the development of generic implementations of approximation
algorithms together with universal solution frameworks. In particular, we have
implemented Zelikovsky 11/6-approximation using local search framework, and
1.39-approximation by Byrka et al. using iterative rounding framework. These
two algorithms are experimentally compared with greedy 2-approximation, with
exact but exponential time Dreyfus-Wagner algorithm, as well as with results
given by a state-of-the-art local search techniques by Uchoa and Werneck. The
results of this paper are twofold. On one hand, we demonstrate that high level
algorithmic concepts can be designed and efficiently used in C++. On the other
hand, we show that the above algorithms with good theoretical guarantees, give
decent results in practice, but are inferior to state-of-the-art heuristical
approaches
Covering problems in edge- and node-weighted graphs
This paper discusses the graph covering problem in which a set of edges in an
edge- and node-weighted graph is chosen to satisfy some covering constraints
while minimizing the sum of the weights. In this problem, because of the large
integrality gap of a natural linear programming (LP) relaxation, LP rounding
algorithms based on the relaxation yield poor performance. Here we propose a
stronger LP relaxation for the graph covering problem. The proposed relaxation
is applied to designing primal-dual algorithms for two fundamental graph
covering problems: the prize-collecting edge dominating set problem and the
multicut problem in trees. Our algorithms are an exact polynomial-time
algorithm for the former problem, and a 2-approximation algorithm for the
latter problem, respectively. These results match the currently known best
results for purely edge-weighted graphs.Comment: To appear in SWAT 201
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
Linear-Delay Enumeration for Minimal Steiner Problems
Kimelfeld and Sagiv [Kimelfeld and Sagiv, PODS 2006], [Kimelfeld and Sagiv,
Inf. Syst. 2008] pointed out the problem of enumerating -fragments is of
great importance in a keyword search on data graphs. In a graph-theoretic term,
the problem corresponds to enumerating minimal Steiner trees in (directed)
graphs. In this paper, we propose a linear-delay and polynomial-space algorithm
for enumerating all minimal Steiner trees, improving on a previous result in
[Kimelfeld and Sagiv, Inf. Syst. 2008]. Our enumeration algorithm can be
extended to other Steiner problems, such as minimal Steiner forests, minimal
terminal Steiner trees, and minimal directed Steiner trees. As another variant
of the minimal Steiner tree enumeration problem, we study the problem of
enumerating minimal induced Steiner subgraphs. We propose a polynomial-delay
and exponential-space enumeration algorithm of minimal induced Steiner
subgraphs on claw-free graphs. Contrary to these tractable results, we show
that the problem of enumerating minimal group Steiner trees is at least as hard
as the minimal transversal enumeration problem on hypergraphs
Bridging the Gap Between Tree and Connectivity Augmentation: Unified and Stronger Approaches
We consider the Connectivity Augmentation Problem (CAP), a classical problem
in the area of Survivable Network Design. It is about increasing the
edge-connectivity of a graph by one unit in the cheapest possible way. More
precisely, given a -edge-connected graph and a set of extra edges,
the task is to find a minimum cardinality subset of extra edges whose addition
to makes the graph -edge-connected. If is odd, the problem is
known to reduce to the Tree Augmentation Problem (TAP) -- i.e., is a
spanning tree -- for which significant progress has been achieved recently,
leading to approximation factors below (the currently best factor is
). However, advances on TAP did not carry over to CAP so far. Indeed,
only very recently, Byrka, Grandoni, and Ameli (STOC 2020) managed to obtain
the first approximation factor below for CAP by presenting a
-approximation algorithm based on a method that is disjoint from recent
advances for TAP.
We first bridge the gap between TAP and CAP, by presenting techniques that
allow for leveraging insights and methods from TAP to approach CAP. We then
introduce a new way to get approximation factors below , based on a new
analysis technique. Through these ingredients, we obtain a
-approximation algorithm for CAP, and therefore also TAP. This leads to
the currently best approximation result for both problems in a unified way, by
significantly improving on the above-mentioned -approximation for CAP and
also the previously best approximation factor of for TAP by Grandoni,
Kalaitzis, and Zenklusen (STOC 2018). Additionally, a feature we inherit from
recent TAP advances is that our approach can deal with the weighted setting
when the ratio between the largest to smallest cost on extra links is bounded,
in which case we obtain approximation factors below
Timing-Constrained Global Routing with Buffered Steiner Trees
This dissertation deals with the combination of two key problems that arise in the physical design of computer chips: global routing and buffering. The task of buffering is the insertion of buffers and inverters into the chip's netlist to speed-up signal delays and to improve electrical properties of the chip. Insertion of buffers and inverters goes alongside with construction of Steiner trees that connect logical sources with possibly many logical sinks and have buffers and inverters as parts of these connections. Classical global routing focuses on packing Steiner trees within the limited routing space. Buffering and global routing have been solved separately in the past. In this thesis we overcome the limitations of the classical approaches by considering the buffering problem as a global, multi-objective problem. We study its theoretical aspects and propose algorithms which we implement in the tool BonnRouteBuffer for timing-constrained global routing with buffered Steiner trees. At its core, we propose a new theoretically founded framework to model timing constraints inherently within global routing. As most important sub-task we have to compute a buffered Steiner tree for a single net minimizing the sum of prices for delays, routing congestion, placement congestion, power consumption, and net length. For this sub-task we present a fully polynomial time approximation scheme to compute an almost-cheapest Steiner tree with a given routing topology and prove that an exact algorithm cannot exist unless P=NP. For topology computation we present a bicriteria approximation algorithm that bounds both the geometric length and the worst slack of the topology. To improve the practical results we present many heuristic modifications, speed-up- and post-optimization techniques for buffered Steiner trees. We conduct experiments on challenging real-world test cases provided by our cooperation partner IBM to demonstrate the quality of our tool. Our new algorithm could produce better solutions with respect to both timing and routability. After post-processing with gate sizing and Vt-assignment, we can even reduce the power consumption on most instances. Overall, our results show that our tool BonnRouteBuffer for timing-constrained global routing is superior to industrial state-of-the-art tools