63 research outputs found
The Effect of Planarization on Width
We study the effects of planarization (the construction of a planar diagram
from a non-planar graph by replacing each crossing by a new vertex) on
graph width parameters. We show that for treewidth, pathwidth, branchwidth,
clique-width, and tree-depth there exists a family of -vertex graphs with
bounded parameter value, all of whose planarizations have parameter value
. However, for bandwidth, cutwidth, and carving width, every graph
with bounded parameter value has a planarization of linear size whose parameter
value remains bounded. The same is true for the treewidth, pathwidth, and
branchwidth of graphs of bounded degree.Comment: 15 pages, 6 figures. To appear at the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
Planar Induced Subgraphs of Sparse Graphs
We show that every graph has an induced pseudoforest of at least
vertices, an induced partial 2-tree of at least vertices, and an
induced planar subgraph of at least vertices. These results are
constructive, implying linear-time algorithms to find the respective induced
subgraphs. We also show that the size of the largest -minor-free graph in
a given graph can sometimes be at most .Comment: Accepted by Graph Drawing 2014. To appear in Journal of Graph
Algorithms and Application
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)
Optimal Algorithms for Hitting (Topological) Minors on Graphs of Bounded Treewidth
For a fixed collection of graphs F, the F-M-DELETION problem consists in, given a graph G and an integer k, decide whether there exists a subset S of V(G) of size at most k such that G-S does not contain any of the graphs in F as a minor. We are interested in the parameterized complexity of F-M-DELETION when the parameter is the treewidth of G, denoted by tw. Our objective is to determine, for a fixed F}, the smallest function f_F such that F-M-DELETION can be solved in time f_F(tw)n^{O(1)} on n-vertex graphs. Using and enhancing the machinery of boundaried graphs and small sets of representatives introduced by Bodlaender et al. [J ACM, 2016], we prove that when all the graphs in F are connected and at least one of them is planar, then f_F(w) = 2^{O(wlog w)}. When F is a singleton containing a clique, a cycle, or a path on i vertices, we prove the following asymptotically tight bounds:
- f_{K_4}(w) = 2^{Theta(wlog w)}.
- f_{C_i}(w) = 2^{Theta(w)} for every i4.
- f_{P_i}(w) = 2^{Theta(w)} for every i5.
The lower bounds hold unless the Exponential Time Hypothesis fails, and the superexponential ones are inspired by a reduction of Marcin Pilipczuk [Discrete Appl Math, 2016]. The single-exponential algorithms use, in particular, the rank-based approach introduced by Bodlaender et al. [Inform Comput, 2015]. We also consider the version of the problem where the graphs in F are forbidden as topological minors, and prove essentially the same set of results holds
5-Approximation for ?-Treewidth Essentially as Fast as ?-Deletion Parameterized by Solution Size
The notion of ?-treewidth, where ? is a hereditary graph class, was recently introduced as a generalization of the treewidth of an undirected graph. Roughly speaking, a graph of ?-treewidth at most k can be decomposed into (arbitrarily large) ?-subgraphs which interact only through vertex sets of size ?(k) which can be organized in a tree-like fashion. ?-treewidth can be used as a hybrid parameterization to develop fixed-parameter tractable algorithms for ?-deletion problems, which ask to find a minimum vertex set whose removal from a given graph G turns it into a member of ?. The bottleneck in the current parameterized algorithms lies in the computation of suitable tree ?-decompositions.
We present FPT-approximation algorithms to compute tree ?-decompositions for hereditary and union-closed graph classes ?. Given a graph of ?-treewidth k, we can compute a 5-approximate tree ?-decomposition in time f(?(k)) ? n^?(1) whenever ?-deletion parameterized by solution size can be solved in time f(k) ? n^?(1) for some function f(k) ? 2^k. The current-best algorithms either achieve an approximation factor of k^?(1) or construct optimal decompositions while suffering from non-uniformity with unknown parameter dependence. Using these decompositions, we obtain algorithms solving Odd Cycle Transversal in time 2^?(k) ? n^?(1) parameterized by bipartite-treewidth and Vertex Planarization in time 2^?(k log k) ? n^?(1) parameterized by planar-treewidth, showing that these can be as fast as the solution-size parameterizations and giving the first ETH-tight algorithms for parameterizations by hybrid width measures
5-Approximation for -Treewidth Essentially as Fast as -Deletion Parameterized by Solution Size
The notion of -treewidth, where is a hereditary
graph class, was recently introduced as a generalization of the treewidth of an
undirected graph. Roughly speaking, a graph of -treewidth at most
can be decomposed into (arbitrarily large) -subgraphs which
interact only through vertex sets of size which can be organized in a
tree-like fashion. -treewidth can be used as a hybrid
parameterization to develop fixed-parameter tractable algorithms for
-deletion problems, which ask to find a minimum vertex set whose
removal from a given graph turns it into a member of . The
bottleneck in the current parameterized algorithms lies in the computation of
suitable tree -decompositions.
We present FPT approximation algorithms to compute tree
-decompositions for hereditary and union-closed graph classes
. Given a graph of -treewidth , we can compute a
5-approximate tree -decomposition in time
whenever -deletion parameterized by solution size can be solved in
time for some function . The current-best
algorithms either achieve an approximation factor of or construct
optimal decompositions while suffering from non-uniformity with unknown
parameter dependence. Using these decompositions, we obtain algorithms solving
Odd Cycle Transversal in time parameterized by
-treewidth and Vertex Planarization in time parameterized by -treewidth, showing that
these can be as fast as the solution-size parameterizations and giving the
first ETH-tight algorithms for parameterizations by hybrid width measures.Comment: Conference version to appear at the European Symposium on Algorithms
(ESA 2023
Hitting forbidden subgraphs in graphs of bounded treewidth
We study the complexity of a generic hitting problem H-Subgraph Hitting,
where given a fixed pattern graph and an input graph , the task is to
find a set of minimum size that hits all subgraphs of
isomorphic to . In the colorful variant of the problem, each vertex of
is precolored with some color from and we require to hit only
-subgraphs with matching colors. Standard techniques shows that for every
fixed , the problem is fixed-parameter tractable parameterized by the
treewidth of ; however, it is not clear how exactly the running time should
depend on treewidth. For the colorful variant, we demonstrate matching upper
and lower bounds showing that the dependence of the running time on treewidth
of is tightly governed by , the maximum size of a minimal vertex
separator in . That is, we show for every fixed that, on a graph of
treewidth , the colorful problem can be solved in time
, but cannot be solved in time
, assuming the Exponential Time
Hypothesis (ETH). Furthermore, we give some preliminary results showing that,
in the absence of colors, the parameterized complexity landscape of H-Subgraph
Hitting is much richer.Comment: A full version of a paper presented at MFCS 201
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