150 research outputs found
On Brambles, Grid-Like Minors, and Parameterized Intractability of Monadic Second-Order Logic
Brambles were introduced as the dual notion to treewidth, one of the most
central concepts of the graph minor theory of Robertson and Seymour. Recently,
Grohe and Marx showed that there are graphs G, in which every bramble of order
larger than the square root of the treewidth is of exponential size in |G|. On
the positive side, they show the existence of polynomial-sized brambles of the
order of the square root of the treewidth, up to log factors. We provide the
first polynomial time algorithm to construct a bramble in general graphs and
achieve this bound, up to log-factors. We use this algorithm to construct
grid-like minors, a replacement structure for grid-minors recently introduced
by Reed and Wood, in polynomial time. Using the grid-like minors, we introduce
the notion of a perfect bramble and an algorithm to find one in polynomial
time. Perfect brambles are brambles with a particularly simple structure and
they also provide us with a subgraph that has bounded degree and still large
treewidth; we use them to obtain a meta-theorem on deciding certain
parameterized subgraph-closed problems on general graphs in time singly
exponential in the parameter.
The second part of our work deals with providing a lower bound to Courcelle's
famous theorem, stating that every graph property that can be expressed by a
sentence in monadic second-order logic (MSO), can be decided by a linear time
algorithm on classes of graphs of bounded treewidth. Using our results from the
first part of our work we establish a strong lower bound for tractability of
MSO on classes of colored graphs
Hitting and Harvesting Pumpkins
The "c-pumpkin" is the graph with two vertices linked by c>0 parallel edges.
A c-pumpkin-model in a graph G is a pair A,B of disjoint subsets of vertices of
G, each inducing a connected subgraph of G, such that there are at least c
edges in G between A and B. We focus on covering and packing c-pumpkin-models
in a given graph: On the one hand, we provide an FPT algorithm running in time
2^O(k) n^O(1) deciding, for any fixed c>0, whether all c-pumpkin-models can be
covered by at most k vertices. This generalizes known single-exponential FPT
algorithms for Vertex Cover and Feedback Vertex Set, which correspond to the
cases c=1,2 respectively. On the other hand, we present a O(log
n)-approximation algorithm for both the problems of covering all
c-pumpkin-models with a smallest number of vertices, and packing a maximum
number of vertex-disjoint c-pumpkin-models.Comment: v2: several minor change
Hitting Forbidden Induced Subgraphs on Bounded Treewidth Graphs
For a fixed graph H, the H-IS-Deletion problem asks, given a graph G, for the minimum size of a set S ? V(G) such that G? S does not contain H as an induced subgraph. Motivated by previous work about hitting (topological) minors and subgraphs on bounded treewidth graphs, we are interested in determining, for a fixed graph H, the smallest function f_H(t) such that H-IS-Deletion can be solved in time f_H(t) ? n^{?(1)} assuming the Exponential Time Hypothesis (ETH), where t and n denote the treewidth and the number of vertices of the input graph, respectively.
We show that f_H(t) = 2^{?(t^{h-2})} for every graph H on h ? 3 vertices, and that f_H(t) = 2^{?(t)} if H is a clique or an independent set. We present a number of lower bounds by generalizing a reduction of Cygan et al. [MFCS 2014] for the subgraph version. In particular, we show that when H deviates slightly from a clique, the function f_H(t) suffers a sharp jump: if H is obtained from a clique of size h by removing one edge, then f_H(t) = 2^{?(t^{h-2})}. We also show that f_H(t) = 2^{?(t^{h})} when H = K_{h,h}, and this reduction answers an open question of Mi. Pilipczuk [MFCS 2011] about the function f_{C?}(t) for the subgraph version.
Motivated by Cygan et al. [MFCS 2014], we also consider the colorful variant of the problem, where each vertex of G is colored with some color from V(H) and we require to hit only induced copies of H with matching colors. In this case, we determine, under the ETH, the function f_H(t) for every connected graph H on h vertices: if h ? 2 the problem can be solved in polynomial time; if h ? 3, f_H(t) = 2^{?(t)} if H is a clique, and f_H(t) = 2^{?(t^{h-2})} otherwise
Hitting forbidden induced subgraphs on bounded treewidth graphs
For a fixed graph , the -IS-Deletion problem asks, given a graph ,
for the minimum size of a set such that does
not contain as an induced subgraph. Motivated by previous work about
hitting (topological) minors and subgraphs on bounded treewidth graphs, we are
interested in determining, for a fixed graph , the smallest function
such that -IS-Deletion can be solved in time assuming the Exponential Time Hypothesis (ETH), where and
denote the treewidth and the number of vertices of the input graph,
respectively.
We show that for every graph on
vertices, and that if is a clique or an independent
set. We present a number of lower bounds by generalizing a reduction of Cygan
et al. [MFCS 2014] for the subgraph version. In particular, we show that when
deviates slightly from a clique, the function suffers a sharp
jump: if is obtained from a clique of size by removing one edge, then
. We also show that
when , and this reduction answers an open question of Mi. Pilipczuk
[MFCS 2011] about the function for the subgraph version.
Motivated by Cygan et al. [MFCS 2014], we also consider the colorful variant
of the problem, where each vertex of is colored with some color from
and we require to hit only induced copies of with matching colors. In this
case, we determine, under the ETH, the function for every connected
graph on vertices: if the problem can be solved in polynomial
time; if , if is a clique, and otherwise.Comment: 24 pages, 3 figure
Close Relatives (Of Feedback Vertex Set), Revisited
At IPEC 2020, Bergougnoux, Bonnet, Brettell, and Kwon (Close Relatives of Feedback Vertex Set Without Single-Exponential Algorithms Parameterized by Treewidth, IPEC 2020, LIPIcs vol. 180, pp. 3:1-3:17) showed that a number of problems related to the classic Feedback Vertex Set (FVS) problem do not admit a 2^{o(k log k)} ? n^{?(1)}-time algorithm on graphs of treewidth at most k, assuming the Exponential Time Hypothesis. This contrasts with the 3^{k} ? k^{?(1)} ? n-time algorithm for FVS using the Cut&Count technique.
During their live talk at IPEC 2020, Bergougnoux et al. posed a number of open questions, which we answer in this work.
- Subset Even Cycle Transversal, Subset Odd Cycle Transversal, Subset Feedback Vertex Set can be solved in time 2^{?(k log k)} ? n in graphs of treewidth at most k. This matches a lower bound for Even Cycle Transversal of Bergougnoux et al. and improves the polynomial factor in some of their upper bounds.
- Subset Feedback Vertex Set and Node Multiway Cut can be solved in time 2^{?(k log k)} ? n, if the input graph is given as a cliquewidth expression of size n and width k.
- Odd Cycle Transversal can be solved in time 4^k ? k^{?(1)} ? n if the input graph is given as a cliquewidth expression of size n and width k. Furthermore, the existence of a constant ? > 0 and an algorithm performing this task in time (4-?)^k ? n^{?(1)} would contradict the Strong Exponential Time Hypothesis. A common theme of the first two algorithmic results is to represent connectivity properties of the current graph in a state of a dynamic programming algorithm as an auxiliary forest with ?(k) nodes. This results in a 2^{?(k log k)} bound on the number of states for one node of the tree decomposition or cliquewidth expression and allows to compare two states in k^{?(1)} time, resulting in linear time dependency on the size of the graph or the input cliquewidth expression
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
- …