35 research outputs found
Complexity Framework for Forbidden Subgraphs III: When Problems Are Tractable on Subcubic Graphs
For any finite set H = {H1,. .. , Hp} of graphs, a graph is H-subgraph-free if it does not contain any of H1,. .. , Hp as a subgraph. In recent work, meta-classifications have been studied: these show that if graph problems satisfy certain prescribed conditions, their complexity can be classified on classes of H-subgraph-free graphs. We continue this work and focus on problems that have polynomial-time solutions on classes that have bounded treewidth or maximum degree at most 3 and examine their complexity on H-subgraph-free graph classes where H is a connected graph. With this approach, we obtain comprehensive classifications for (Independent) Feedback Vertex Set, Connected Vertex Cover, Colouring and Matching Cut. This resolves a number of open problems. We highlight that, to establish that Independent Feedback Vertex Set belongs to this collection of problems, we first show that it can be solved in polynomial time on graphs of maximum degree 3. We demonstrate that, with the exception of the complete graph on four vertices, each graph in this class has a minimum size feedback vertex set that is also an independent set
Edge Multiway Cut and Node Multiway Cut are NP-complete on subcubic graphs
We show that Edge Multiway Cut (also called Multiterminal Cut) and Node
Multiway Cut are NP-complete on graphs of maximum degree (also known as
subcubic graphs). This improves on a previous degree bound of . Our
NP-completeness result holds even for subcubic graphs that are planar
The Parameterised Complexity of Integer Multicommodity Flow
The Integer Multicommodity Flow problem has been studied extensively in the
literature. However, from a parameterised perspective, mostly special cases,
such as the Disjoint Paths problem, have been considered. Therefore, we
investigate the parameterised complexity of the general Integer Multicommodity
Flow problem. We show that the decision version of this problem on directed
graphs for a constant number of commodities, when the capacities are given in
unary, is XNLP-complete with pathwidth as parameter and XALP-complete with
treewidth as parameter. When the capacities are given in binary, the problem is
NP-complete even for graphs of pathwidth at most 13. We give related results
for undirected graphs. These results imply that the problem is unlikely to be
fixed-parameter tractable by these parameters.
In contrast, we show that the problem does become fixed-parameter tractable
when weighted tree partition width (a variant of tree partition width for edge
weighted graphs) is used as parameter
Complexity Framework for Forbidden Subgraphs II: When Hardness Is Not Preserved under Edge Subdivision
For a fixed set of graphs, a graph is -subgraph-free
if does not contain any as a (not necessarily induced)
subgraph. A recently proposed framework gives a complete classification on
-subgraph-free graphs (for finite sets ) for problems that
are solvable in polynomial time on graph classes of bounded treewidth,
NP-complete on subcubic graphs, and whose NP-hardness is preserved under edge
subdivision. While a lot of problems satisfy these conditions, there are also
many problems that do not satisfy all three conditions and for which the
complexity -subgraph-free graphs is unknown.
In this paper, we study problems for which only the first two conditions of
the framework hold (they are solvable in polynomial time on classes of bounded
treewidth and NP-complete on subcubic graphs, but NP-hardness is not preserved
under edge subdivision). In particular, we make inroads into the classification
of the complexity of four such problems: -Induced Disjoint Paths,
-Colouring, Hamilton Cycle and Star -Colouring. Although we do not
complete the classifications, we show that the boundary between polynomial time
and NP-complete differs among our problems and differs from problems that do
satisfy all three conditions of the framework. Hence, we exhibit a rich
complexity landscape among problems for -subgraph-free graph classes
Prevalence of Hepatitis B and Hepatitis C Virus Infection in Patients with Advanced Renal Failure: A Tertiary Care Centre Study from North Indian Population
Abstract: Viral hepatitis (Hepatitis B Virus (HBV) & Hepatitis C Virus (HCV)) related liver disease is a leading cause of morbidity and mortality especially in the patients with advanced renal failure who are treated with dialysis, and this is due to high number of blood transfusion sessions and/or cross contamination from the dialysis circuits. Aims & Objectives: This study aimed to determine the prevalence of HBV and HCV infections in patients with advanced renal failure (ARF). Materials & Methods: A cross-sectional study was done in joint collaboration of Department of Nephrology and Department of Gastroenterology, KGMU, Lucknow, from June 2018 to June 2020 among, CRF patients. Clinical data such as age, gender, duration of dialysis; number of transfusions, Serum sample was collected from each patient. Serological markers for HBV and HCV were determined with ELISA by using commercial diagnostic kits. HCV-RNA and HBV-DNA were determined quantitatively by polymerase chain reaction (PCR) assay. Results: A total 934 patients with advanced renal failure attended the nephrology OPD. Out of 934 patients, 65 (6.96%) patients screened positive for HBV/HCV infection. The results of this study also showed that the prevalence of viral hepatitis infection in the haemodialysis (HD) and without HD patients is 8.25% and 6.3% respectively. Conclusion: It has been found that viral infections, particularly HBV and HCV infections are common in advanced renal failure patients who are on HD
Complexity Framework for Forbidden Subgraphs IV: The Steiner Forest Problem
We study Steiner Forest on -subgraph-free graphs, that is, graphs that do
not contain some fixed graph as a (not necessarily induced) subgraph. We
are motivated by a recent framework that completely characterizes the
complexity of many problems on -subgraph-free graphs. However, in contrast
to e.g. the related Steiner Tree problem, Steiner Forest falls outside this
framework. Hence, the complexity of Steiner Forest on -subgraph-free graphs
remained tantalizingly open. In this paper, we make significant progress
towards determining the complexity of Steiner Forest on -subgraph-free
graphs. Our main results are four novel polynomial-time algorithms for
different excluded graphs that are central to further understand its
complexity. Along the way, we study the complexity of Steiner Forest for graphs
with a small -deletion set, that is, a small set of vertices such that
each component of has size at most . Using this parameter, we give two
noteworthy algorithms that we later employ as subroutines. First, we prove
Steiner Forest is FPT parameterized by when (i.e. the vertex cover
number). Second, we prove Steiner Forest is polynomial-time solvable for graphs
with a 2-deletion set of size at most 2. The latter result is tight, as the
problem is NP-complete for graphs with a 3-deletion set of size 2
Computing Subset Vertex Covers in H-Free Graphs
We consider a natural generalization of Vertex Cover: the Subset Vertex Cover problem, which is to decide for a graph G=(V,E), a subset T⊆V and integer k, if V has a subset S of size at most k, such that S contains at least one end-vertex of every edge incident to a vertex of T. A graph is H-free if it does not contain H as an induced subgraph. We solve two open problems from the literature by proving that Subset Vertex Cover is NP-complete on subcubic (claw,diamond)-free planar graphs and on 2-unipolar graphs, a subclass of 2P3-free weakly chordal graphs. Our results show for the first time that Subset Vertex Cover is computationally harder than Vertex Cover (under P ≠NP). We also prove new polynomial time results. We first give a dichotomy on graphs where G[T] is H-free. Namely, we show that Subset Vertex Cover is polynomial-time solvable on graphs G, for which G[T] is H-free, if H=sP1+tP2 and NP-complete otherwise. Moreover, we prove that Subset Vertex Cover is polynomial-time solvable for (sP1+P2+P3)-free graphs and bounded mim-width graphs. By combining our new results with known results we obtain a partial complexity classification for Subset Vertex Cover on H-free graphs
Complexity Framework for Forbidden Subgraphs: When Hardness Is Not Preserved under Edge Subdivision
For a fixed set H of graphs, a graph G is H-subgraph-free if G does not contain any H∈H as a (not necessarily induced) subgraph. A recently proposed framework gives a complete classification on H-subgraph-free graphs (for finite sets H) for problems that are solvable in polynomial time on graph classes of bounded treewidth, NP-complete on subcubic graphs, and whose NP-hardness is preserved under edge subdivision. While a lot of problems satisfy these conditions, there are also many problems that do not satisfy all three conditions and for which the complexity H-subgraph-free graphs is unknown. In this paper, we study problems for which only the first two conditions of the framework hold (they are solvable in polynomial time on classes of bounded treewidth and NP-complete on subcubic graphs, but NP-hardness is not preserved under edge subdivision). In particular, we make inroads into the classification of the complexity of four such problems: k-Induced Disjoint Paths, C5-Colouring, Hamilton Cycle and Star 3-Colouring. Although we do not complete the classifications, we show that the boundary between polynomial time and NP-complete differs among our problems and differs from problems that do satisfy all three conditions of the framework. Hence, we exhibit a rich complexity landscape among problems for H-subgraph-free graph classes
Complexity Framework for Forbidden Subgraphs {III:}: When Problems Are Tractable on Subcubic Graphs
For any finite set H = {H1, . . ., Hp} of graphs, a graph is H-subgraph-free if it does not contain any of H1, . . ., Hp as a subgraph. In recent work, meta-classifications have been studied: these show that if graph problems satisfy certain prescribed conditions, their complexity can be classified on classes of H-subgraph-free graphs. We continue this work and focus on problems that have polynomial-time solutions on classes that have bounded treewidth or maximum degree at most 3 and examine their complexity on H-subgraph-free graph classes where H is a connected graph. With this approach, we obtain comprehensive classifications for (Independent) Feedback Vertex Set, Connected Vertex Cover, Colouring and Matching Cut. This resolves a number of open problems. We highlight that, to establish that Independent Feedback Vertex Set belongs to this collection of problems, we first show that it can be solved in polynomial time on graphs of maximum degree 3. We demonstrate that, with the exception of the complete graph on four vertices, each graph in this class has a minimum size feedback vertex set that is also an independent set