13 research outputs found
Speeding-up Dynamic Programming with Representative Sets - An Experimental Evaluation of Algorithms for Steiner Tree on Tree Decompositions
Dynamic programming on tree decompositions is a frequently used approach to
solve otherwise intractable problems on instances of small treewidth. In recent
work by Bodlaender et al., it was shown that for many connectivity problems,
there exist algorithms that use time, linear in the number of vertices, and
single exponential in the width of the tree decomposition that is used. The
central idea is that it suffices to compute representative sets, and these can
be computed efficiently with help of Gaussian elimination.
In this paper, we give an experimental evaluation of this technique for the
Steiner Tree problem. A comparison of the classic dynamic programming algorithm
and the improved dynamic programming algorithm that employs the table reduction
shows that the new approach gives significant improvements on the running time
of the algorithm and the size of the tables computed by the dynamic programming
algorithm, and thus that the rank based approach from Bodlaender et al. does
not only give significant theoretical improvements but also is a viable
approach in a practical setting, and showcases the potential of exploiting the
idea of representative sets for speeding up dynamic programming algorithms
Technical Communications of ICLP
Abstract Dynamic programming (DP) on tree decompositions is a well studied approach for solving hard problems efficiently. State-of-the-art implementations usually rely on tables for storing information, and algorithms specify how the tuples are manipulated during traversal of the decomposition. However, a major bottleneck of such table-based algorithms is relatively high memory consumption. The goal of the doctoral thesis herein discussed is to mitigate performance and memory shortcomings of such algorithms. The idea is to replace tables with an efficient data structure that no longer requires to enumerate intermediate results explicitly during the computation. To this end, Binary Decision Diagrams (BDDs) and related concepts are studied with respect to their applicability in this setting. Besides native support for efficient storage, from a conceptual point of view BDDs give rise to an alternative approach of how DP algorithms are specified. Instead of tuple-based manipulation operations, the algorithms are specified on a logical level, where sets of models can be conjointly updated. The goal of the thesis is to provide a general tool-set for problems that can be solved efficiently via DP on tree decompositions
Computing a Minimum-Cost -hop Steiner Tree in Tree-Like Metrics
We consider the problem of computing a Steiner tree of minimum cost under a
-hop constraint which requires the depth of the tree to be at most . Our
main result is an exact algorithm for metrics induced by graphs of bounded
treewidth that runs in time . For the special case of a path, we give
a simple algorithm that solves the problem in polynomial time, even if is
part of the input. The main result can be used to obtain, in quasi-polynomial
time, a near-optimal solution that violates the -hop constraint by at most
one hop for more general metrics induced by graphs of bounded highway
dimension
Secluded Connectivity Problems
Consider a setting where possibly sensitive information sent over a path in a
network is visible to every {neighbor} of the path, i.e., every neighbor of
some node on the path, thus including the nodes on the path itself. The
exposure of a path can be measured as the number of nodes adjacent to it,
denoted by . A path is said to be secluded if its exposure is small. A
similar measure can be applied to other connected subgraphs, such as Steiner
trees connecting a given set of terminals. Such subgraphs may be relevant due
to considerations of privacy, security or revenue maximization. This paper
considers problems related to minimum exposure connectivity structures such as
paths and Steiner trees. It is shown that on unweighted undirected -node
graphs, the problem of finding the minimum exposure path connecting a given
pair of vertices is strongly inapproximable, i.e., hard to approximate within a
factor of for any (under an
appropriate complexity assumption), but is approximable with ratio
, where is the maximum degree in the graph. One of
our main results concerns the class of bounded-degree graphs, which is shown to
exhibit the following interesting dichotomy. On the one hand, the minimum
exposure path problem is NP-hard on node-weighted or directed bounded-degree
graphs (even when the maximum degree is 4). On the other hand, we present a
polynomial algorithm (based on a nontrivial dynamic program) for the problem on
unweighted undirected bounded-degree graphs. Likewise, the problem is shown to
be polynomial also for the class of (weighted or unweighted) bounded-treewidth
graphs
Ant Colony Optimization for Multi-objective Digital Convergent Product Network
Convergent product is an assembly shape concept integrating functions and sub-functions to form a final product. To conceptualize the convergent product problem, a web-based network is considered in which a collection of base functions and sub-functions configure the nodes and each arc in the network is considered to be a link between two nodes. The aim is to find an optimal tree of functionalities in the network adding value to the product in the web environment. First, an algorithm is proposed to assign the links among bases and sub-functions. Then, numerical values as benefits and costs are determined for arcs and nodes, respectively, using a mathematical approach. Also, customer’s value corresponding to the benefits is considered. Finally, the Steiner tree methodology is adapted to a multi-objective model optimized by an ant colony optimization method. The approach is applicable for all digital products, such as mobile, tablet, laptop, etc. An example is worked out to illustrate the proposed approach
Lower Bounds for QBFs of Bounded Treewidth
The problem of deciding the validity (QSAT) of quantified Boolean formulas
(QBF) is a vivid research area in both theory and practice. In the field of
parameterized algorithmics, the well-studied graph measure treewidth turned out
to be a successful parameter. A well-known result by Chen in parameterized
complexity is that QSAT when parameterized by the treewidth of the primal graph
of the input formula together with the quantifier depth of the formula is
fixed-parameter tractable. More precisely, the runtime of such an algorithm is
polynomial in the formula size and exponential in the treewidth, where the
exponential function in the treewidth is a tower, whose height is the
quantifier depth. A natural question is whether one can significantly improve
these results and decrease the tower while assuming the Exponential Time
Hypothesis (ETH). In the last years, there has been a growing interest in the
quest of establishing lower bounds under ETH, showing mostly problem-specific
lower bounds up to the third level of the polynomial hierarchy. Still, an
important question is to settle this as general as possible and to cover the
whole polynomial hierarchy. In this work, we show lower bounds based on the ETH
for arbitrary QBFs parameterized by treewidth (and quantifier depth). More
formally, we establish lower bounds for QSAT and treewidth, namely, that under
ETH there cannot be an algorithm that solves QSAT of quantifier depth i in
runtime significantly better than i-fold exponential in the treewidth and
polynomial in the input size. In doing so, we provide a versatile reduction
technique to compress treewidth that encodes the essence of dynamic programming
on arbitrary tree decompositions. Further, we describe a general methodology
for a more fine-grained analysis of problems parameterized by treewidth that
are at higher levels of the polynomial hierarchy
Solving two-stage stochastic network design problems to optimality
The Steiner tree problem (STP) is a central and well-studied graph-theoretical combinatorial optimization problem which plays an important role in various applications. It can be stated as follows: Given a weighted graph and a set of terminal vertices, find a subset of edges which connects the terminals at minimum cost. However, in real-world applications the input data might not be given with certainty or it might depend on future decisions. For the STP, for example, edge costs representing the costs of establishing links may be subject to inflations and price deviations. In this thesis we tackle data uncertainty by using the concept of stochastic programming and we study the two-stage stochastic version of the Steiner tree problem (SSTP). Thereby, a set of scenarios defines the possible outcomes of a random variable; each scenario is given by its realization probability and defines a set of terminals and edge costs. A feasible solution consists of a subset of edges in the first stage and edge subsets for all scenarios (second stage) such that each terminal set is connected. The objective is to find a solution that minimizes the expected cost. We consider two approaches for solving the SSTP to optimality: combinatorial algorithms, in particular fixed-parameter tractable (FPT) algorithms, and methods from mathematical programming.
Regarding the combinatorial algorithms we develop a linear-time algorithm for trees, an FPT algorithm parameterized by the number of terminals, and we consider treewidth-bounded graphs where we give the first FPT algorithm parameterized by the combination of treewidth and number of scenarios.
The second approach is based on deriving strong integer programming (IP) formulations for the SSTP. By using orientation properties we introduce new semi-directed cut- and flow-based IP formulations which are shown to be stronger than the undirected models from the literature. To solve these models to optimality we use a decomposition-based two-stage branch&cut algorithm, which is improved by a fast and efficient method for strengthening the optimality cuts. Moreover, we develop new and stronger integer optimality cuts. The computational performance is evaluated in a comprehensive computational study, which shows the superiority of the new formulations, the benefit of the decomposition, and the advantage of using the strengthened optimality cuts.
The Steiner forest problem (SFP) is a related problem where sets of terminals need to be connected. On the one hand, the SFP is a generalization of the STP and on the other hand, we show that the SFP is a special case of the SSTP. Therefore, our results are transferable to the SFP and we present the first FPT algorithm for treewidth-bounded graphs and we model new and stronger (semi-)directed cut- and flow-based IP formulations for the SFP.
In the second part of this thesis we consider the two-stage stochastic survivable network design problem, an extension of the SSTP where pairs of vertices may demand a higher connectivity. Similarly to the first part we introduce new and stronger semi-directed cut-based models, apply the same decomposition along with the cut strengthening technique, and argue the validity of the newly introduced integer optimality cuts. A computational study shows the benefit, robustness, and good performance of the decomposition and the cut strengthening method