7 research outputs found
Preprocessing under uncertainty
In this work we study preprocessing for tractable problems when part of the
input is unknown or uncertain. This comes up naturally if, e.g., the load of
some machines or the congestion of some roads is not known far enough in
advance, or if we have to regularly solve a problem over instances that are
largely similar, e.g., daily airport scheduling with few charter flights.
Unlike robust optimization, which also studies settings like this, our goal
lies not in computing solutions that are (approximately) good for every
instantiation. Rather, we seek to preprocess the known parts of the input, to
speed up finding an optimal solution once the missing data is known.
We present efficient algorithms that given an instance with partially
uncertain input generate an instance of size polynomial in the amount of
uncertain data that is equivalent for every instantiation of the unknown part.
Concretely, we obtain such algorithms for Minimum Spanning Tree, Minimum Weight
Matroid Basis, and Maximum Cardinality Bipartite Maxing, where respectively the
weight of edges, weight of elements, and the availability of vertices is
unknown for part of the input. Furthermore, we show that there are tractable
problems, such as Small Connected Vertex Cover, for which one cannot hope to
obtain similar results.Comment: 18 page
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
The Complexity of Finding Effectors
The NP-hard EFFECTORS problem on directed graphs is motivated by applications
in network mining, particularly concerning the analysis of probabilistic
information-propagation processes in social networks. In the corresponding
model the arcs carry probabilities and there is a probabilistic diffusion
process activating nodes by neighboring activated nodes with probabilities as
specified by the arcs. The point is to explain a given network activation state
as well as possible by using a minimum number of "effector nodes"; these are
selected before the activation process starts.
We correct, complement, and extend previous work from the data mining
community by a more thorough computational complexity analysis of EFFECTORS,
identifying both tractable and intractable cases. To this end, we also exploit
a parameterization measuring the "degree of randomness" (the number of "really"
probabilistic arcs) which might prove useful for analyzing other probabilistic
network diffusion problems as well.Comment: 28 page
Preprocessing Under Uncertainty: Matroid Intersection
We continue the study of preprocessing under uncertainty that was initiated independently by Assadi et al. (FSTTCS 2015) and Fafianie et al. (STACS 2016). Here, we are given an instance of a tractable problem with a large static/known part and a small part that is dynamic/uncertain, and ask if there is an efficient algorithm that computes an instance of size polynomial in the uncertain part of the input, from which we can extract an optimal solution to the original instance for all (usually exponentially many) instantiations of the uncertain part.
In the present work, we focus on the Matroid Intersection problem. Amongst others we present a positive preprocessing result for the important case of finding a largest common independent set in two linear matroids. Motivated by an application for intersecting two gammoids we also revisit Maximum Flow. There we tighten a lower bound of Assadi et al. and give an alternative positive result for the case of low uncertain capacity that yields a Maximum Flow instance as output rather than a matrix
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. (Proceedings of the 40th international colloquium on automata, languages and programming, ICALP 2013, part I, volume 7965 of Lecture Notes in Computer Science. Springer, Berlin, pp 196–207, 2013), 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 that 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. Our comparison of the classic dynamic programming algorithm and the improved dynamic programming algorithm that employs 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. Thus, the rank-based approach from Bodlaender et al. (2013) 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. Furthermore, we propose an alternative representation of partial solutions using weighted bit strings in order to circumvent a part of the reduction step that is computationally expensive in practice. In the experimental evaluation we find that this representation yields further significant improvements. We show that the representation can also be used for the other problems fitting in the framework of Bodlaender et al. (2013). Keywords: Experimental evaluation; Algorithmic engineering; Steiner tree; Treewidth; Dynamic programming; Exact algorithm
The Complexity of Finding Effectors
International audienceThe NP-hard Effectors problem on directed graphs is motivated by applications in network mining, particularly concerning the analysis of probabilistic information-propagation processes in social networks. In the corresponding model the arcs carry probabilities and there is a probabilistic diffusion process activating nodes by neighboring activated nodes with probabilities as specified by the arcs. The point is to explain a given network activation state as well as possible by using a minimum number of " effector nodes " ; these are selected before the activation process starts. We correct, complement, and extend previous work from the data mining community by a more thorough computational complexity analysis of Effec-tors, identifying both tractable and intractable cases. To this end, we also An extended abstract appeared in 2 Laurent Bulteau et al. exploit a parameterization measuring the " degree of randomness " (the number of 'really' probabilistic arcs) which might prove useful for analyzing other probabilistic network diffusion problems as well