1,178 research outputs found
H-free graphs, Independent Sets, and subexponential-time algorithms
It is an old open question in algorithmic graph theory to determine the complexity of the Maximum Independent Set problem on Pt-free graphs, that is, on graphs not containing any induced path on t vertices. So far, polynomial-time algorithms are known only for t ≤ 5 [Lokshtanov et al., SODA 2014, pp. 570-581, 2014]. Here we study the existence of subexponential-time algorithms for the problem: by generalizing an earlier result of Randerath and Schiermeyer for t = 5 [Discrete Appl. Math., 158 (2010), pp. 1041-1044], we show that for any t ≥ 5, there is an algorithm for Maximum Independent Set on Pt-free graphs whose running time is subexponential in the number of vertices. Scattered Set is the generalization of Maximum Independent Set where the vertices of the solution are required to be at distance at least d from each other. We give a complete characterization of those graphs H for which d-Scattered Set on H-free graphs can be solved in time subexponential in the size of the input (that is, in the number of vertices plus number of edges): If every component of H is a path, then d-Scattered Set on H-free graphs with n vertices and m edges can be solved in time 2(n+m) 1-O(1/|V (H)|), even if d is part of the input. Otherwise, assuming ETH, there is no 2o(n+m)-time algorithm for d-Scattered Set for any fixed d ≥ 3 on H-free graphs with n-vertices and m-edges. © 2016 Gábor Bacsó, Dániel Marx, and Zsolt Tuza
H-Free Graphs, Independent Sets, and Subexponential-Time Algorithms
It is an outstanding open question in algorithmic graph theory to determine the complexity of the MAXIMUM INDEPENDENT SET problem on P_t-free graphs, that is, on graphs not containing any induced path on t vertices. So far, polynomial-time algorithms are known only for t at most 5 [Lokshtanov et al., SODA 2014, 570-581, 2014]. Here we study the existence of subexponential-time algorithms for the problem: by generalizing an earlier result of Randerath and Schiermeyer for t=5 [Discrete App. Math., 158 (2010), 1041-1044], we show that for any t at least 5, there is an algorithm for MAXIMUM INDEPENDENT SET on P_t-free graphs whose running time is subexponential in the number of vertices.
SCATTERED SET is the generalization of MAXIMUM INDEPENDENT SET where the vertices of the solution are required to be at distance at least from each other. We give a complete characterization of those graphs H for which SCATTERED SET on H-free graphs can be solved in time subexponential in the size of the input (that is, in the number of vertices plus the number of edges):
* If every component of H is a path, then d-SCATTERED SET on H-free graphs with n vertices and m edges can be solved in time 2^{(n+m)^{1-O(1/|V(H)|)}}, even if d is part of the input.
* Otherwise, assuming ETH, there is no 2^{o(n+m)} time algorithm for d-SCATTERED SET for any fixed d at least 3 on H-free graphs with n vertices and m edges
Bidimensionality and Geometric Graphs
In this paper we use several of the key ideas from Bidimensionality to give a
new generic approach to design EPTASs and subexponential time parameterized
algorithms for problems on classes of graphs which are not minor closed, but
instead exhibit a geometric structure. In particular we present EPTASs and
subexponential time parameterized algorithms for Feedback Vertex Set, Vertex
Cover, Connected Vertex Cover, Diamond Hitting Set, on map graphs and unit disk
graphs, and for Cycle Packing and Minimum-Vertex Feedback Edge Set on unit disk
graphs. Our results are based on the recent decomposition theorems proved by
Fomin et al [SODA 2011], and our algorithms work directly on the input graph.
Thus it is not necessary to compute the geometric representations of the input
graph. To the best of our knowledge, these results are previously unknown, with
the exception of the EPTAS and a subexponential time parameterized algorithm on
unit disk graphs for Vertex Cover, which were obtained by Marx [ESA 2005] and
Alber and Fiala [J. Algorithms 2004], respectively.
We proceed to show that our approach can not be extended in its full
generality to more general classes of geometric graphs, such as intersection
graphs of unit balls in R^d, d >= 3. Specifically we prove that Feedback Vertex
Set on unit-ball graphs in R^3 neither admits PTASs unless P=NP, nor
subexponential time algorithms unless the Exponential Time Hypothesis fails.
Additionally, we show that the decomposition theorems which our approach is
based on fail for disk graphs and that therefore any extension of our results
to disk graphs would require new algorithmic ideas. On the other hand, we prove
that our EPTASs and subexponential time algorithms for Vertex Cover and
Connected Vertex Cover carry over both to disk graphs and to unit-ball graphs
in R^d for every fixed d
Exploring Subexponential Parameterized Complexity of Completion Problems
Let be a family of graphs. In the -Completion problem,
we are given a graph and an integer as input, and asked whether at most
edges can be added to so that the resulting graph does not contain a
graph from as an induced subgraph. It appeared recently that special
cases of -Completion, the problem of completing into a chordal graph
known as Minimum Fill-in, corresponding to the case of , and the problem of completing into a split graph,
i.e., the case of , are solvable in parameterized
subexponential time . The exploration of this
phenomenon is the main motivation for our research on -Completion.
In this paper we prove that completions into several well studied classes of
graphs without long induced cycles also admit parameterized subexponential time
algorithms by showing that:
- The problem Trivially Perfect Completion is solvable in parameterized
subexponential time , that is -Completion for , a cycle and a path on four
vertices.
- The problems known in the literature as Pseudosplit Completion, the case
where , and Threshold Completion, where , are also solvable in time .
We complement our algorithms for -Completion with the following
lower bounds:
- For , , , and
, -Completion cannot be solved in time
unless the Exponential Time Hypothesis (ETH) fails.
Our upper and lower bounds provide a complete picture of the subexponential
parameterized complexity of -Completion problems for .Comment: 32 pages, 16 figures, A preliminary version of this paper appeared in
the proceedings of STACS'1
Independent Set, Induced Matching, and Pricing: Connections and Tight (Subexponential Time) Approximation Hardnesses
We present a series of almost settled inapproximability results for three
fundamental problems. The first in our series is the subexponential-time
inapproximability of the maximum independent set problem, a question studied in
the area of parameterized complexity. The second is the hardness of
approximating the maximum induced matching problem on bounded-degree bipartite
graphs. The last in our series is the tight hardness of approximating the
k-hypergraph pricing problem, a fundamental problem arising from the area of
algorithmic game theory. In particular, assuming the Exponential Time
Hypothesis, our two main results are:
- For any r larger than some constant, any r-approximation algorithm for the
maximum independent set problem must run in at least
2^{n^{1-\epsilon}/r^{1+\epsilon}} time. This nearly matches the upper bound of
2^{n/r} (Cygan et al., 2008). It also improves some hardness results in the
domain of parameterized complexity (e.g., Escoffier et al., 2012 and Chitnis et
al., 2013)
- For any k larger than some constant, there is no polynomial time min
(k^{1-\epsilon}, n^{1/2-\epsilon})-approximation algorithm for the k-hypergraph
pricing problem, where n is the number of vertices in an input graph. This
almost matches the upper bound of min (O(k), \tilde O(\sqrt{n})) (by Balcan and
Blum, 2007 and an algorithm in this paper).
We note an interesting fact that, in contrast to n^{1/2-\epsilon} hardness
for polynomial-time algorithms, the k-hypergraph pricing problem admits
n^{\delta} approximation for any \delta >0 in quasi-polynomial time. This puts
this problem in a rare approximability class in which approximability
thresholds can be improved significantly by allowing algorithms to run in
quasi-polynomial time.Comment: The full version of FOCS 201
Beyond Bidimensionality: Parameterized Subexponential Algorithms on Directed Graphs
We develop two different methods to achieve subexponential time parameterized
algorithms for problems on sparse directed graphs. We exemplify our approaches
with two well studied problems.
For the first problem, {\sc -Leaf Out-Branching}, which is to find an
oriented spanning tree with at least leaves, we obtain an algorithm solving
the problem in time on directed graphs
whose underlying undirected graph excludes some fixed graph as a minor. For
the special case when the input directed graph is planar, the running time can
be improved to . The second example is a
generalization of the {\sc Directed Hamiltonian Path} problem, namely {\sc
-Internal Out-Branching}, which is to find an oriented spanning tree with at
least internal vertices. We obtain an algorithm solving the problem in time
on directed graphs whose underlying
undirected graph excludes some fixed apex graph as a minor. Finally, we
observe that for any , the {\sc -Directed Path} problem is
solvable in time , where is some
function of \ve.
Our methods are based on non-trivial combinations of obstruction theorems for
undirected graphs, kernelization, problem specific combinatorial structures and
a layering technique similar to the one employed by Baker to obtain PTAS for
planar graphs
(Total) Vector Domination for Graphs with Bounded Branchwidth
Given a graph of order and an -dimensional non-negative
vector , called demand vector, the vector domination
(resp., total vector domination) is the problem of finding a minimum
such that every vertex in (resp., in ) has
at least neighbors in . The (total) vector domination is a
generalization of many dominating set type problems, e.g., the dominating set
problem, the -tuple dominating set problem (this is different from the
solution size), and so on, and its approximability and inapproximability have
been studied under this general framework. In this paper, we show that a
(total) vector domination of graphs with bounded branchwidth can be solved in
polynomial time. This implies that the problem is polynomially solvable also
for graphs with bounded treewidth. Consequently, the (total) vector domination
problem for a planar graph is subexponential fixed-parameter tractable with
respectto , where is the size of solution.Comment: 16 page
Optimality program in segment and string graphs
Planar graphs are known to allow subexponential algorithms running in time
or for most of the paradigmatic
problems, while the brute-force time is very likely to be
asymptotically best on general graphs. Intrigued by an algorithm packing curves
in by Fox and Pach [SODA'11], we investigate which
problems have subexponential algorithms on the intersection graphs of curves
(string graphs) or segments (segment intersection graphs) and which problems
have no such algorithms under the ETH (Exponential Time Hypothesis). Among our
results, we show that, quite surprisingly, 3-Coloring can also be solved in
time on string graphs while an algorithm running
in time for 4-Coloring even on axis-parallel segments (of unbounded
length) would disprove the ETH. For 4-Coloring of unit segments, we show a
weaker ETH lower bound of which exploits the celebrated
Erd\H{o}s-Szekeres theorem. The subexponential running time also carries over
to Min Feedback Vertex Set but not to Min Dominating Set and Min Independent
Dominating Set.Comment: 19 pages, 15 figure
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