43 research outputs found
Progress on the adjacent vertex distinguishing edge colouring conjecture
A proper edge colouring of a graph is adjacent vertex distinguishing if no
two adjacent vertices see the same set of colours. Using a clever application
of the Local Lemma, Hatami (2005) proved that every graph with maximum degree
and no isolated edge has an adjacent vertex distinguishing edge
colouring with colours, provided is large enough. We
show that this bound can be reduced to . This is motivated by the
conjecture of Zhang, Liu, and Wang (2002) that colours are enough
for .Comment: v2: Revised following referees' comment
Equitable orientations of sparse uniform hypergraphs
Caro, West, and Yuster studied how -uniform hypergraphs can be oriented in
such a way that (generalizations of) indegree and outdegree are as close to
each other as can be hoped. They conjectured an existence result of such
orientations for sparse hypergraphs, of which we present a proof
A Polynomial Kernel for Line Graph Deletion
The line graph of a graph is the graph whose vertex set is the
edge set of and there is an edge between if and share
an endpoint in . A graph is called line graph if it is a line graph of some
graph. We study the Line-Graph-Edge Deletion problem, which asks whether we can
delete at most edges from the input graph such that the resulting graph
is a line graph. More precisely, we give a polynomial kernel for
Line-Graph-Edge Deletion with vertices. This answers an
open question posed by Falk H\"{u}ffner at Workshop on Kernels (WorKer) in
2013.Comment: To be published in the Proceedings of the 28th Annual European
Symposium on Algorithms (ESA 2020
FPT Constant-Approximations for Capacitated Clustering to Minimize the Sum of Cluster Radii
Clustering with capacity constraints is a fundamental problem that attracted
significant attention throughout the years. In this paper, we give the first
FPT constant-factor approximation algorithm for the problem of clustering
points in a general metric into clusters to minimize the sum of cluster
radii, subject to non-uniform hard capacity constraints. In particular, we give
a -approximation algorithm that runs in time. When capacities are uniform, we obtain the following improved
approximation bounds: A (4 + )-approximation with running time
, which significantly improves over the FPT
28-approximation of Inamdar and Varadarajan [ESA 2020]; a (2 +
)-approximation with running time and a -approximation with running
time in the Euclidean space; and a (1 +
)-approximation in the Euclidean space with running time
if we are allowed to violate
the capacities by (1 + )-factor. We complement this result by showing
that there is no (1 + )-approximation algorithm running in time
, if any capacity violation is not allowed.Comment: Full version of a paper accepted to SoCG 202
A Polynomial Kernel for Paw-Free Editing
For a fixed graph , the -free-editing problem asks whether we can
modify a given graph by adding or deleting at most edges such that the
resulting graph does not contain as an induced subgraph. The problem is
known to be NP-complete for all fixed with at least vertices and it
admits a algorithm. Cai and Cai showed that the
-free-editing problem does not admit a polynomial kernel whenever or its
complement is a path or a cycle with at least edges or a -connected
graph with at least edge missing. Their results suggest that if is not
independent set or a clique, then -free-editing admits polynomial kernels
only for few small graphs , unless .
Therefore, resolving the kernelization of -free-editing for small graphs
plays a crucial role in obtaining a complete dichotomy for this problem. In
this paper, we positively answer the question of compressibility for one of the
last two unresolved graphs on vertices. Namely, we give the first
polynomial kernel for paw-free editing with vertices
Edge separators for graphs excluding a minor
We prove that every -vertex -minor-free graph of maximum degree
has a set of edges such that
every component of has at most vertices. This is best possible up
to the dependency on and extends earlier results of Diks, Djidjev, Sykora,
and Vr\v{t}o (1993) for planar graphs, and of Sykora and Vr\v{t}o (1993) for
bounded-genus graphs. Our result is a consequence of the following more general
result: The line graph of is isomorphic to a subgraph of the strong product
for some graph with treewidth at most
and
Exploiting Dense Structures in Parameterized Complexity
Over the past few decades, the study of dense structures from the perspective of approximation algorithms has become a wide area of research. However, from the viewpoint of parameterized algorithm, this area is largely unexplored. In particular, properties of random samples have been successfully deployed to design approximation schemes for a number of fundamental problems on dense structures [Arora et al. FOCS 1995, Goldreich et al. FOCS 1996, Giotis and Guruswami SODA 2006, Karpinksi and Schudy STOC 2009]. In this paper, we fill this gap, and harness the power of random samples as well as structure theory to design kernelization as well as parameterized algorithms on dense structures. In particular, we obtain linear vertex kernels for Edge-Disjoint Paths, Edge Odd Cycle Transversal, Minimum Bisection, d-Way Cut, Multiway Cut and Multicut on everywhere dense graphs. In fact, these kernels are obtained by designing a polynomial-time algorithm when the corresponding parameter is at most ?(n). Additionally, we obtain a cubic kernel for Vertex-Disjoint Paths on everywhere dense graphs. In addition to kernelization results, we obtain randomized subexponential-time parameterized algorithms for Edge Odd Cycle Transversal, Minimum Bisection, and d-Way Cut. Finally, we show how all of our results (as well as EPASes for these problems) can be de-randomized
Subidvisions de cycles orientés dans les graphes dirigés de fort nombre chromatique
An {\it oriented cycle} is an orientation of a undirected cycle.We first show that for any oriented cycle , there are digraphs containing no subdivision of (as a subdigraph) and arbitrarily large chromatic number.In contrast, we show that for any is a cycle with two blocks, every strongly connected digraph with sufficiently large chromatic number contains a subdivision of . We prove a similar result for the antidirected cycle on four vertices (in which two vertices have out-degree and two vertices have in-degree ).Un {\it cycle orienté} est l'orientation d'un cycle. Nous prouvons que pour tout cycle orienté il existe des graphes dirigés sans subdivisions de (en tant que sous graphe) et de nombre chromatique arbitrairement grand. Par ailleurs, nous prouvons que pour tout cycle a deux bloques, tout graphe dirigé fortement connexe de nombre chromatique suffisamment grand contient une subdivision de . Nous prouvons aussi un resultat semblable sur le cycle antidirigé de taille quatre (avec deux sommets de degré sortant et deux sommets de degré entrant )