2,745 research outputs found
Cut and Count and Representative Sets on Branch Decompositions
Recently, new techniques have been introduced to speed up dynamic programming algorithms on tree decompositions for connectivity problems: the \u27Cut and Count\u27 method and a method called the rank-based approach, based on representative sets and Gaussian elimination. These methods respectively give randomised and deterministic algorithms that are single exponential in the treewidth, and polynomial, respectively linear in the number of vertices. In this paper, we adapt these methods to branch decompositions yielding algorithms, both randomised and deterministic, that are in many cases faster than when tree decompositions would be used.
In particular, we obtain the currently fastest randomised algorithms for several problems on planar graphs. When the involved weights are O(n^{O(1)}), we obtain faster randomised algorithms on planar graphs for Steiner Tree, Connected Dominating Set, Feedback Vertex Set and TSP, and a faster deterministic algorithm for TSP. When considering planar graphs with arbitrary real weights, we obtain faster deterministic algorithms for all four mentioned problems
Solving weighted and counting variants of connectivity problems parameterized by treewidth deterministically in single exponential time
It is well known that many local graph problems, like Vertex Cover and
Dominating Set, can be solved in 2^{O(tw)}|V|^{O(1)} time for graphs G=(V,E)
with a given tree decomposition of width tw. However, for nonlocal problems,
like the fundamental class of connectivity problems, for a long time we did not
know how to do this faster than tw^{O(tw)}|V|^{O(1)}. Recently, Cygan et al.
(FOCS 2011) presented Monte Carlo algorithms for a wide range of connectivity
problems running in time $c^{tw}|V|^{O(1)} for a small constant c, e.g., for
Hamiltonian Cycle and Steiner tree. Naturally, this raises the question whether
randomization is necessary to achieve this runtime; furthermore, it is
desirable to also solve counting and weighted versions (the latter without
incurring a pseudo-polynomial cost in terms of the weights).
We present two new approaches rooted in linear algebra, based on matrix rank
and determinants, which provide deterministic c^{tw}|V|^{O(1)} time algorithms,
also for weighted and counting versions. For example, in this time we can solve
the traveling salesman problem or count the number of Hamiltonian cycles. The
rank-based ideas provide a rather general approach for speeding up even
straightforward dynamic programming formulations by identifying "small" sets of
representative partial solutions; we focus on the case of expressing
connectivity via sets of partitions, but the essential ideas should have
further applications. The determinant-based approach uses the matrix tree
theorem for deriving closed formulas for counting versions of connectivity
problems; we show how to evaluate those formulas via dynamic programming.Comment: 36 page
Truth Table Invariant Cylindrical Algebraic Decomposition by Regular Chains
A new algorithm to compute cylindrical algebraic decompositions (CADs) is
presented, building on two recent advances. Firstly, the output is truth table
invariant (a TTICAD) meaning given formulae have constant truth value on each
cell of the decomposition. Secondly, the computation uses regular chains theory
to first build a cylindrical decomposition of complex space (CCD) incrementally
by polynomial. Significant modification of the regular chains technology was
used to achieve the more sophisticated invariance criteria. Experimental
results on an implementation in the RegularChains Library for Maple verify that
combining these advances gives an algorithm superior to its individual
components and competitive with the state of the art
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
On Directed Feedback Vertex Set parameterized by treewidth
We study the Directed Feedback Vertex Set problem parameterized by the
treewidth of the input graph. We prove that unless the Exponential Time
Hypothesis fails, the problem cannot be solved in time on general directed graphs, where is the treewidth of
the underlying undirected graph. This is matched by a dynamic programming
algorithm with running time .
On the other hand, we show that if the input digraph is planar, then the
running time can be improved to .Comment: 20
Permutation combinatorics of worldsheet moduli space
52 pages, 21 figures52 pages, 21 figures; minor corrections, "On the" dropped from title, matches published version52 pages, 21 figures; minor corrections, "On the" dropped from title, matches published versio
Parameterized Complexity of Secluded Connectivity Problems
The Secluded Path problem models a situation where a sensitive information
has to be transmitted between a pair of nodes along a path in a network. The
measure of the quality of a selected path is its exposure, which is the total
weight of vertices in its closed neighborhood. In order to minimize the risk of
intercepting the information, we are interested in selecting a secluded path,
i.e. a path with a small exposure. Similarly, the Secluded Steiner Tree problem
is to find a tree in a graph connecting a given set of terminals such that the
exposure of the tree is minimized. The problems were introduced by Chechik et
al. in [ESA 2013]. Among other results, Chechik et al. have shown that Secluded
Path is fixed-parameter tractable (FPT) on unweighted graphs being
parameterized by the maximum vertex degree of the graph and that Secluded
Steiner Tree is FPT parameterized by the treewidth of the graph. In this work,
we obtain the following results about parameterized complexity of secluded
connectivity problems.
We give FPT-algorithms deciding if a graph G with a given cost function
contains a secluded path and a secluded Steiner tree of exposure at most k with
the cost at most C.
We initiate the study of "above guarantee" parameterizations for secluded
problems, where the lower bound is given by the size of a Steiner tree.
We investigate Secluded Steiner Tree from kernelization perspective and
provide several lower and upper bounds when parameters are the treewidth, the
size of a vertex cover, maximum vertex degree and the solution size. Finally,
we refine the algorithmic result of Chechik et al. by improving the exponential
dependence from the treewidth of the input graph.Comment: Minor corrections are don
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