128,446 research outputs found
An FPT 2-Approximation for Tree-Cut Decomposition
The tree-cut width of a graph is a graph parameter defined by Wollan [J.
Comb. Theory, Ser. B, 110:47-66, 2015] with the help of tree-cut
decompositions. In certain cases, tree-cut width appears to be more adequate
than treewidth as an invariant that, when bounded, can accelerate the
resolution of intractable problems. While designing algorithms for problems
with bounded tree-cut width, it is important to have a parametrically tractable
way to compute the exact value of this parameter or, at least, some constant
approximation of it. In this paper we give a parameterized 2-approximation
algorithm for the computation of tree-cut width; for an input -vertex graph
and an integer , our algorithm either confirms that the tree-cut width
of is more than or returns a tree-cut decomposition of certifying
that its tree-cut width is at most , in time .
Prior to this work, no constructive parameterized algorithms, even approximated
ones, existed for computing the tree-cut width of a graph. As a consequence of
the Graph Minors series by Robertson and Seymour, only the existence of a
decision algorithm was known.Comment: 17 pages, 3 figure
A theory of game trees, based on solution trees
In this paper a complete theory of game tree algorithms is presented, entirely based upon the notion of a solution tree. Two types of solution trees are distinguished: max and min solution trees respectively. We show that most game tree algorithms construct a superposition of a max and a min solution tree. Moreover, we formulate a general cut-off criterion in terms of solution trees. In the second half of this paper four well known algorithms, viz., alphabeta, SSS*, MTD and Scout are studied extensively. We show how solution trees feature in these algorithms and how the cut-off criterion is applied
Cut Tree Construction from Massive Graphs
The construction of cut trees (also known as Gomory-Hu trees) for a given
graph enables the minimum-cut size of the original graph to be obtained for any
pair of vertices. Cut trees are a powerful back-end for graph management and
mining, as they support various procedures related to the minimum cut, maximum
flow, and connectivity. However, the crucial drawback with cut trees is the
computational cost of their construction. In theory, a cut tree is built by
applying a maximum flow algorithm for times, where is the number of
vertices. Therefore, naive implementations of this approach result in cubic
time complexity, which is obviously too slow for today's large-scale graphs. To
address this issue, in the present study, we propose a new cut-tree
construction algorithm tailored to real-world networks. Using a series of
experiments, we demonstrate that the proposed algorithm is several orders of
magnitude faster than previous algorithms and it can construct cut trees for
billion-scale graphs.Comment: Short version will appear at ICDM'1
Solving Connectivity Problems Parameterized by Treedepth in Single-Exponential Time and Polynomial Space
A breakthrough result of Cygan et al. (FOCS 2011) showed that connectivity problems parameterized by treewidth can be solved much faster than the previously best known time ?^*(2^{?(twlog tw)}). Using their inspired Cut&Count technique, they obtained ?^*(?^tw) time algorithms for many such problems. Moreover, they proved these running times to be optimal assuming the Strong Exponential-Time Hypothesis. Unfortunately, like other dynamic programming algorithms on tree decompositions, these algorithms also require exponential space, and this is widely believed to be unavoidable. In contrast, for the slightly larger parameter called treedepth, there are already several examples of matching the time bounds obtained for treewidth, but using only polynomial space. Nevertheless, this has remained open for connectivity problems.
In the present work, we close this knowledge gap by applying the Cut&Count technique to graphs of small treedepth. While the general idea is unchanged, we have to design novel procedures for counting consistently cut solution candidates using only polynomial space. Concretely, we obtain time ?^*(3^d) and polynomial space for Connected Vertex Cover, Feedback Vertex Set, and Steiner Tree on graphs of treedepth d. Similarly, we obtain time ?^*(4^d) and polynomial space for Connected Dominating Set and Connected Odd Cycle Transversal
Orbitopal Fixing
The topic of this paper are integer programming models in which a subset of
0/1-variables encode a partitioning of a set of objects into disjoint subsets.
Such models can be surprisingly hard to solve by branch-and-cut algorithms if
the order of the subsets of the partition is irrelevant, since this kind of
symmetry unnecessarily blows up the search tree. We present a general tool,
called orbitopal fixing, for enhancing the capabilities of branch-and-cut
algorithms in solving such symmetric integer programming models. We devise a
linear time algorithm that, applied at each node of the search tree, removes
redundant parts of the tree produced by the above mentioned symmetry. The
method relies on certain polyhedra, called orbitopes, which have been
introduced bei Kaibel and Pfetsch (Math. Programm. A, 114 (2008), 1-36). It
does, however, not explicitly add inequalities to the model. Instead, it uses
certain fixing rules for variables. We demonstrate the computational power of
orbitopal fixing at the example of a graph partitioning problem.Comment: 22 pages, revised and extended version of a previous version that has
appeared under the same title in Proc. IPCO 200
Evolutionary trees: an integer multicommodity max-flow-min-cut theorem
In biomathematics, the extensions of a leaf-colouration of a binary tree to the whole vertex set with minimum number of colour-changing edges are extensively studied. Our paper generalizes the problem for trees; algorithms and a Menger-type theorem are presented. The LP dual of the problem is a multicommodity flow problem, for which a max-flow-min-cut theorem holds. The problem that we solve is an instance of the NP-hard multiway cut problem
Lean Tree-Cut Decompositions: Obstructions and Algorithms
The notion of tree-cut width has been introduced by Wollan in [The structure of graphs not admitting a fixed immersion, Journal of Combinatorial Theory, Series B, 110:47 - 66, 2015]. It is defined via tree-cut decompositions, which are tree-like decompositions that highlight small (edge) cuts in a graph. In that sense, tree-cut decompositions can be seen as an edge-version of tree-decompositions and have algorithmic applications on problems that remain intractable on graphs of bounded treewidth. In this paper, we prove that every graph admits an optimal tree-cut decomposition that satisfies a certain Menger-like condition similar to that of the lean tree decompositions of Thomas [A Menger-like property of tree-width: The finite case, Journal of Combinatorial Theory, Series B, 48(1):67 - 76, 1990]. This allows us to give, for every k in N, an upper-bound on the number immersion-minimal graphs of tree-cut width k. Our results imply the constructive existence of a linear FPT-algorithm for tree-cut width
Almost-Tight Distributed Minimum Cut Algorithms
We study the problem of computing the minimum cut in a weighted distributed
message-passing networks (the CONGEST model). Let be the minimum cut,
be the number of nodes in the network, and be the network diameter. Our
algorithm can compute exactly in time. To the best of our knowledge, this is the first paper that
explicitly studies computing the exact minimum cut in the distributed setting.
Previously, non-trivial sublinear time algorithms for this problem are known
only for unweighted graphs when due to Pritchard and
Thurimella's -time and -time algorithms for
computing -edge-connected and -edge-connected components.
By using the edge sampling technique of Karger's, we can convert this
algorithm into a -approximation -time algorithm for any . This improves
over the previous -approximation -time algorithm and
-approximation -time algorithm of Ghaffari and Kuhn. Due to the lower
bound of by Das Sarma et al. which holds for any
approximation algorithm, this running time is tight up to a factor.
To get the stated running time, we developed an approximation algorithm which
combines the ideas of Thorup's algorithm and Matula's contraction algorithm. It
saves an factor as compared to applying Thorup's tree
packing theorem directly. Then, we combine Kutten and Peleg's tree partitioning
algorithm and Karger's dynamic programming to achieve an efficient distributed
algorithm that finds the minimum cut when we are given a spanning tree that
crosses the minimum cut exactly once
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