2,353 research outputs found
Fast Biclustering by Dual Parameterization
We study two clustering problems, Starforest Editing, the problem of adding
and deleting edges to obtain a disjoint union of stars, and the generalization
Bicluster Editing. We show that, in addition to being NP-hard, none of the
problems can be solved in subexponential time unless the exponential time
hypothesis fails.
Misra, Panolan, and Saurabh (MFCS 2013) argue that introducing a bound on the
number of connected components in the solution should not make the problem
easier: In particular, they argue that the subexponential time algorithm for
editing to a fixed number of clusters (p-Cluster Editing) by Fomin et al. (J.
Comput. Syst. Sci., 80(7) 2014) is an exception rather than the rule. Here, p
is a secondary parameter, bounding the number of components in the solution.
However, upon bounding the number of stars or bicliques in the solution, we
obtain algorithms which run in time for p-Starforest
Editing and for p-Bicluster Editing. We
obtain a similar result for the more general case of t-Partite p-Cluster
Editing. This is subexponential in k for fixed number of clusters, since p is
then considered a constant.
Our results even out the number of multivariate subexponential time
algorithms and give reasons to believe that this area warrants further study.Comment: Accepted for presentation at IPEC 201
Half-integrality, LP-branching and FPT Algorithms
A recent trend in parameterized algorithms is the application of polytope
tools (specifically, LP-branching) to FPT algorithms (e.g., Cygan et al., 2011;
Narayanaswamy et al., 2012). However, although interesting results have been
achieved, the methods require the underlying polytope to have very restrictive
properties (half-integrality and persistence), which are known only for few
problems (essentially Vertex Cover (Nemhauser and Trotter, 1975) and Node
Multiway Cut (Garg et al., 1994)). Taking a slightly different approach, we
view half-integrality as a \emph{discrete} relaxation of a problem, e.g., a
relaxation of the search space from to such that
the new problem admits a polynomial-time exact solution. Using tools from CSP
(in particular Thapper and \v{Z}ivn\'y, 2012) to study the existence of such
relaxations, we provide a much broader class of half-integral polytopes with
the required properties, unifying and extending previously known cases.
In addition to the insight into problems with half-integral relaxations, our
results yield a range of new and improved FPT algorithms, including an
-time algorithm for node-deletion Unique Label Cover with
label set and an -time algorithm for Group Feedback Vertex
Set, including the setting where the group is only given by oracle access. All
these significantly improve on previous results. The latter result also implies
the first single-exponential time FPT algorithm for Subset Feedback Vertex Set,
answering an open question of Cygan et al. (2012).
Additionally, we propose a network flow-based approach to solve some cases of
the relaxation problem. This gives the first linear-time FPT algorithm to
edge-deletion Unique Label Cover.Comment: Added results on linear-time FPT algorithms (not present in SODA
paper
Incremental Cycle Detection, Topological Ordering, and Strong Component Maintenance
We present two on-line algorithms for maintaining a topological order of a
directed -vertex acyclic graph as arcs are added, and detecting a cycle when
one is created. Our first algorithm handles arc additions in
time. For sparse graphs (), this bound improves the best previous
bound by a logarithmic factor, and is tight to within a constant factor among
algorithms satisfying a natural {\em locality} property. Our second algorithm
handles an arbitrary sequence of arc additions in time. For
sufficiently dense graphs, this bound improves the best previous bound by a
polynomial factor. Our bound may be far from tight: we show that the algorithm
can take time by relating its performance to a
generalization of the -levels problem of combinatorial geometry. A
completely different algorithm running in time was given
recently by Bender, Fineman, and Gilbert. We extend both of our algorithms to
the maintenance of strong components, without affecting the asymptotic time
bounds.Comment: 31 page
Structural Rounding: Approximation Algorithms for Graphs Near an Algorithmically Tractable Class
We develop a framework for generalizing approximation algorithms from the structural graph algorithm literature so that they apply to graphs somewhat close to that class (a scenario we expect is common when working with real-world networks) while still guaranteeing approximation ratios. The idea is to edit a given graph via vertex- or edge-deletions to put the graph into an algorithmically tractable class, apply known approximation algorithms for that class, and then lift the solution to apply to the original graph. We give a general characterization of when an optimization problem is amenable to this approach, and show that it includes many well-studied graph problems, such as Independent Set, Vertex Cover, Feedback Vertex Set, Minimum Maximal Matching, Chromatic Number, (l-)Dominating Set, Edge (l-)Dominating Set, and Connected Dominating Set.
To enable this framework, we develop new editing algorithms that find the approximately-fewest edits required to bring a given graph into one of a few important graph classes (in some cases these are bicriteria algorithms which simultaneously approximate both the number of editing operations and the target parameter of the family). For bounded degeneracy, we obtain an O(r log{n})-approximation and a bicriteria (4,4)-approximation which also extends to a smoother bicriteria trade-off. For bounded treewidth, we obtain a bicriteria (O(log^{1.5} n), O(sqrt{log w}))-approximation, and for bounded pathwidth, we obtain a bicriteria (O(log^{1.5} n), O(sqrt{log w} * log n))-approximation. For treedepth 2 (related to bounded expansion), we obtain a 4-approximation. We also prove complementary hardness-of-approximation results assuming P != NP: in particular, these problems are all log-factor inapproximable, except the last which is not approximable below some constant factor 2 (assuming UGC)
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