123 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
Finding Hamiltonian Cycle in Graphs of Bounded Treewidth: Experimental Evaluation
The notion of treewidth, introduced by Robertson and Seymour in their seminal Graph Minors series, turned out to have tremendous impact on graph algorithmics. Many hard computational problems on graphs turn out to be efficiently solvable in graphs of bounded treewidth: graphs that can be sweeped with separators of bounded size. These efficient algorithms usually follow the dynamic programming paradigm.
In the recent years, we have seen a rapid and quite unexpected development of involved techniques for solving various computational problems in graphs of bounded treewidth. One of the most surprising directions is the development of algorithms for connectivity problems that have only single-exponential dependency (i.e., 2^{{O}(t)}) on the treewidth in the running time bound, as opposed to slightly superexponential (i.e., 2^{{O}(t log t)}) stemming from more naive approaches. In this work, we perform a thorough experimental evaluation of these approaches in the context of one of the most classic connectivity problem, namely Hamiltonian Cycle
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
Determinantal Sieving
We introduce determinantal sieving, a new, remarkably powerful tool in the
toolbox of algebraic FPT algorithms. Given a polynomial on a set of
variables and a linear matroid of
rank , both over a field of characteristic 2, in
evaluations we can sieve for those terms in the monomial expansion of which
are multilinear and whose support is a basis for . Alternatively, using
evaluations of we can sieve for those monomials whose odd support
spans . Applying this framework, we improve on a range of algebraic FPT
algorithms, such as:
1. Solving -Matroid Intersection in time and -Matroid
Parity in time , improving on (Brand and Pratt,
ICALP 2021)
2. -Cycle, Colourful -Path, Colourful -Linkage in undirected
graphs, and the more general Rank -Linkage problem, all in
time, improving on respectively (Fomin et al., SODA 2023)
3. Many instances of the Diverse X paradigm, finding a collection of
solutions to a problem with a minimum mutual distance of in time
, improving solutions for -Distinct Branchings from time
to (Bang-Jensen et al., ESA 2021), and for Diverse
Perfect Matchings from to (Fomin et al.,
STACS 2021)
All matroids are assumed to be represented over a field of characteristic 2.
Over general fields, we achieve similar results at the cost of using
exponential space by working over the exterior algebra. For a class of
arithmetic circuits we call strongly monotone, this is even achieved without
any loss of running time. However, the odd support sieving result appears to be
specific to working over characteristic 2
LIPIcs, Volume 244, ESA 2022, Complete Volume
LIPIcs, Volume 244, ESA 2022, Complete Volum
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