348 research outputs found
Parameterized lower bound and NP-completeness of some -free Edge Deletion problems
For a graph , the -free Edge Deletion problem asks whether there exist
at most edges whose deletion from the input graph results in a graph
without any induced copy of . We prove that -free Edge Deletion is
NP-complete if is a graph with at least two edges and has a component
with maximum number of vertices which is a tree or a regular graph.
Furthermore, we obtain that these NP-complete problems cannot be solved in
parameterized subexponential time, i.e., in time ,
unless Exponential Time Hypothesis fails.Comment: 15 pages, COCOA 15 accepted pape
Hierarchies of Inefficient Kernelizability
The framework of Bodlaender et al. (ICALP 2008) and Fortnow and Santhanam
(STOC 2008) allows us to exclude the existence of polynomial kernels for a
range of problems under reasonable complexity-theoretical assumptions. However,
there are also some issues that are not addressed by this framework, including
the existence of Turing kernels such as the "kernelization" of Leaf Out
Branching(k) into a disjunction over n instances of size poly(k). Observing
that Turing kernels are preserved by polynomial parametric transformations, we
define a kernelization hardness hierarchy, akin to the M- and W-hierarchy of
ordinary parameterized complexity, by the PPT-closure of problems that seem
likely to be fundamentally hard for efficient Turing kernelization. We find
that several previously considered problems are complete for our fundamental
hardness class, including Min Ones d-SAT(k), Binary NDTM Halting(k), Connected
Vertex Cover(k), and Clique(k log n), the clique problem parameterized by k log
n
Parameterized Complexity Dichotomy for Steiner Multicut
The Steiner Multicut problem asks, given an undirected graph G, terminals
sets T1,...,Tt V(G) of size at most p, and an integer k, whether
there is a set S of at most k edges or nodes s.t. of each set Ti at least one
pair of terminals is in different connected components of G \ S. This problem
generalizes several graph cut problems, in particular the Multicut problem (the
case p = 2), which is fixed-parameter tractable for the parameter k [Marx and
Razgon, Bousquet et al., STOC 2011].
We provide a dichotomy of the parameterized complexity of Steiner Multicut.
That is, for any combination of k, t, p, and the treewidth tw(G) as constant,
parameter, or unbounded, and for all versions of the problem (edge deletion and
node deletion with and without deletable terminals), we prove either that the
problem is fixed-parameter tractable or that the problem is hard (W[1]-hard or
even (para-)NP-complete). We highlight that:
- The edge deletion version of Steiner Multicut is fixed-parameter tractable
for the parameter k+t on general graphs (but has no polynomial kernel, even on
trees). We present two proofs: one using the randomized contractions technique
of Chitnis et al, and one relying on new structural lemmas that decompose the
Steiner cut into important separators and minimal s-t cuts.
- In contrast, both node deletion versions of Steiner Multicut are W[1]-hard
for the parameter k+t on general graphs.
- All versions of Steiner Multicut are W[1]-hard for the parameter k, even
when p=3 and the graph is a tree plus one node. Hence, the results of Marx and
Razgon, and Bousquet et al. do not generalize to Steiner Multicut.
Since we allow k, t, p, and tw(G) to be any constants, our characterization
includes a dichotomy for Steiner Multicut on trees (for tw(G) = 1), and a
polynomial time versus NP-hardness dichotomy (by restricting k,t,p,tw(G) to
constant or unbounded).Comment: As submitted to journal. This version also adds a proof of
fixed-parameter tractability for parameter k+t using the technique of
randomized contraction
On the Threshold of Intractability
We study the computational complexity of the graph modification problems
Threshold Editing and Chain Editing, adding and deleting as few edges as
possible to transform the input into a threshold (or chain) graph. In this
article, we show that both problems are NP-complete, resolving a conjecture by
Natanzon, Shamir, and Sharan (Discrete Applied Mathematics, 113(1):109--128,
2001). On the positive side, we show the problem admits a quadratic vertex
kernel. Furthermore, we give a subexponential time parameterized algorithm
solving Threshold Editing in time,
making it one of relatively few natural problems in this complexity class on
general graphs. These results are of broader interest to the field of social
network analysis, where recent work of Brandes (ISAAC, 2014) posits that the
minimum edit distance to a threshold graph gives a good measure of consistency
for node centralities. Finally, we show that all our positive results extend to
the related problem of Chain Editing, as well as the completion and deletion
variants of both problems
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