19,825 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
On the (non-)existence of polynomial kernels for Pl-free edge modification problems
Given a graph G = (V,E) and an integer k, an edge modification problem for a
graph property P consists in deciding whether there exists a set of edges F of
size at most k such that the graph H = (V,E \vartriangle F) satisfies the
property P. In the P edge-completion problem, the set F of edges is constrained
to be disjoint from E; in the P edge-deletion problem, F is a subset of E; no
constraint is imposed on F in the P edge-edition problem. A number of
optimization problems can be expressed in terms of graph modification problems
which have been extensively studied in the context of parameterized complexity.
When parameterized by the size k of the edge set F, it has been proved that if
P is an hereditary property characterized by a finite set of forbidden induced
subgraphs, then the three P edge-modification problems are FPT. It was then
natural to ask whether these problems also admit a polynomial size kernel.
Using recent lower bound techniques, Kratsch and Wahlstrom answered this
question negatively. However, the problem remains open on many natural graph
classes characterized by forbidden induced subgraphs. Kratsch and Wahlstrom
asked whether the result holds when the forbidden subgraphs are paths or cycles
and pointed out that the problem is already open in the case of P4-free graphs
(i.e. cographs). This paper provides positive and negative results in that line
of research. We prove that parameterized cograph edge modification problems
have cubic vertex kernels whereas polynomial kernels are unlikely to exist for
the Pl-free and Cl-free edge-deletion problems for large enough l
The Complexity of Rationalizing Network Formation
We study the complexity of rationalizing network formation. In this problem we fix an underlying model describing how selfish parties (the vertices) produce a graph by making individual decisions to form or not form incident edges. The model is equipped with a notion of stability (or equilibrium), and we observe a set of "snapshots" of graphs that are assumed to be stable. From this we would like to infer some unobserved data about the system: edge prices, or how much each vertex values short paths to each other vertex. We study two rationalization problems arising from the network formation model of Jackson and Wolinsky [14]. When the goal is to infer edge prices, we observe that the rationalization problem is easy. The problem remains easy even when rationalizing prices do not exist and we instead wish to find prices that maximize the stability of the system. In contrast, when the edge prices are given and the goal is instead to infer valuations of each vertex by each other vertex, we prove that the rationalization problem becomes NP-hard. Our proof exposes a close connection between rationalization problems and the Inequality-SAT (I-SAT) problem. Finally and most significantly, we prove that an approximation version of this NP-complete rationalization problem is NP-hard to approximate to within better than a 1/2 ratio. This shows that the trivial algorithm of setting everyone's valuations to infinity (which rationalizes all the edges present in the input graphs) or to zero (which rationalizes all the non-edges present in the input graphs) is the best possible assuming P ≠ NP To do this we prove a tight (1/2 + δ) -approximation hardness for a variant of I-SAT in which all coefficients are non-negative. This in turn follows from a tight hardness result for MAX-LlN_(R_+) (linear equations over the reals, with non-negative coefficients), which we prove by a (non-trivial) modification of the recent result of Guruswami and Raghavendra [10] which achieved tight hardness for this problem without the non-negativity constraint. Our technical contributions regarding the hardness of I-SAT and MAX-LIN_(R_+) may be of independent interest, given the generality of these problem
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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