1,821 research outputs found
Conflict-Free Coloring of Planar Graphs
A conflict-free k-coloring of a graph assigns one of k different colors to
some of the vertices such that, for every vertex v, there is a color that is
assigned to exactly one vertex among v and v's neighbors. Such colorings have
applications in wireless networking, robotics, and geometry, and are
well-studied in graph theory. Here we study the natural problem of the
conflict-free chromatic number chi_CF(G) (the smallest k for which
conflict-free k-colorings exist). We provide results both for closed
neighborhoods N[v], for which a vertex v is a member of its neighborhood, and
for open neighborhoods N(v), for which vertex v is not a member of its
neighborhood.
For closed neighborhoods, we prove the conflict-free variant of the famous
Hadwiger Conjecture: If an arbitrary graph G does not contain K_{k+1} as a
minor, then chi_CF(G) <= k. For planar graphs, we obtain a tight worst-case
bound: three colors are sometimes necessary and always sufficient. We also give
a complete characterization of the computational complexity of conflict-free
coloring. Deciding whether chi_CF(G)<= 1 is NP-complete for planar graphs G,
but polynomial for outerplanar graphs. Furthermore, deciding whether
chi_CF(G)<= 2 is NP-complete for planar graphs G, but always true for
outerplanar graphs. For the bicriteria problem of minimizing the number of
colored vertices subject to a given bound k on the number of colors, we give a
full algorithmic characterization in terms of complexity and approximation for
outerplanar and planar graphs.
For open neighborhoods, we show that every planar bipartite graph has a
conflict-free coloring with at most four colors; on the other hand, we prove
that for k in {1,2,3}, it is NP-complete to decide whether a planar bipartite
graph has a conflict-free k-coloring. Moreover, we establish that any general}
planar graph has a conflict-free coloring with at most eight colors.Comment: 30 pages, 17 figures; full version (to appear in SIAM Journal on
Discrete Mathematics) of extended abstract that appears in Proceeedings of
the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA
2017), pp. 1951-196
A Time Hierarchy Theorem for the LOCAL Model
The celebrated Time Hierarchy Theorem for Turing machines states, informally,
that more problems can be solved given more time. The extent to which a time
hierarchy-type theorem holds in the distributed LOCAL model has been open for
many years. It is consistent with previous results that all natural problems in
the LOCAL model can be classified according to a small constant number of
complexities, such as , etc.
In this paper we establish the first time hierarchy theorem for the LOCAL
model and prove that several gaps exist in the LOCAL time hierarchy.
1. We define an infinite set of simple coloring problems called Hierarchical
-Coloring}. A correctly colored graph can be confirmed by simply
checking the neighborhood of each vertex, so this problem fits into the class
of locally checkable labeling (LCL) problems. However, the complexity of the
-level Hierarchical -Coloring problem is ,
for . The upper and lower bounds hold for both general graphs
and trees, and for both randomized and deterministic algorithms.
2. Consider any LCL problem on bounded degree trees. We prove an
automatic-speedup theorem that states that any randomized -time
algorithm solving the LCL can be transformed into a deterministic -time algorithm. Together with a previous result, this establishes that on
trees, there are no natural deterministic complexities in the ranges
--- or ---.
3. We expose a gap in the randomized time hierarchy on general graphs. Any
randomized algorithm that solves an LCL problem in sublogarithmic time can be
sped up to run in time, which is the complexity of the distributed
Lovasz local lemma problem, currently known to be and
On Derandomizing Local Distributed Algorithms
The gap between the known randomized and deterministic local distributed
algorithms underlies arguably the most fundamental and central open question in
distributed graph algorithms. In this paper, we develop a generic and clean
recipe for derandomizing LOCAL algorithms. We also exhibit how this simple
recipe leads to significant improvements on a number of problem. Two main
results are:
- An improved distributed hypergraph maximal matching algorithm, improving on
Fischer, Ghaffari, and Kuhn [FOCS'17], and giving improved algorithms for
edge-coloring, maximum matching approximation, and low out-degree edge
orientation. The first gives an improved algorithm for Open Problem 11.4 of the
book of Barenboim and Elkin, and the last gives the first positive resolution
of their Open Problem 11.10.
- An improved distributed algorithm for the Lov\'{a}sz Local Lemma, which
gets closer to a conjecture of Chang and Pettie [FOCS'17], and moreover leads
to improved distributed algorithms for problems such as defective coloring and
-SAT.Comment: 37 page
An Efficient Reduction of a Gammoid to a Partition Matroid
Our main contribution is a polynomial-time algorithm to reduce a k-colorable gammoid to a (2k-2)-colorable partition matroid. It is known that there are gammoids that can not be reduced to any (2k-3)-colorable partition matroid, so this result is tight. We then discuss how such a reduction can be used to obtain polynomial-time algorithms with better approximation ratios for various natural problems related to coloring and list coloring the intersection of matroids
Packing Hamilton Cycles Online
It is known that w.h.p. the hitting time for the random
graph process to have minimum degree coincides with the hitting time
for edge disjoint Hamilton cycles. In this paper we prove an online
version of this property. We show that, for a fixed integer , if
random edges of are presented one by one then w.h.p. it is possible to
color the edges online with colors so that at time ,
each color class is Hamiltonian.Comment: Minor change
Combinatorial Properties of Triangle-Free Rectangle Arrangements and the Squarability Problem
We consider arrangements of axis-aligned rectangles in the plane. A geometric
arrangement specifies the coordinates of all rectangles, while a combinatorial
arrangement specifies only the respective intersection type in which each pair
of rectangles intersects. First, we investigate combinatorial contact
arrangements, i.e., arrangements of interior-disjoint rectangles, with a
triangle-free intersection graph. We show that such rectangle arrangements are
in bijection with the 4-orientations of an underlying planar multigraph and
prove that there is a corresponding geometric rectangle contact arrangement.
Moreover, we prove that every triangle-free planar graph is the contact graph
of such an arrangement. Secondly, we introduce the question whether a given
rectangle arrangement has a combinatorially equivalent square arrangement. In
addition to some necessary conditions and counterexamples, we show that
rectangle arrangements pierced by a horizontal line are squarable under certain
sufficient conditions.Comment: 15 pages, 13 figures, extended version of a paper to appear at the
International Symposium on Graph Drawing and Network Visualization (GD) 201
Rainbow Turan Methods for Trees
The rainbow Turan number, a natural extension of the well-studied traditionalTuran number, was introduced in 2007 by Keevash, Mubayi, Sudakov and Verstraete. The rainbow Tur ́an number of a graph F , ex*(n, F ), is the largest number of edges for an n vertex graph G that can be properly edge colored with no rainbow F subgraph. Chapter 1 of this dissertation gives relevant definitions and a brief history of extremal graph theory. Chapter 2 defines k-unique colorings and the related k-unique Turan number and provides preliminary results on this new variant. In Chapter 3, we explore the reduction method for finding upper bounds on rainbow Turan numbers and use this to inform results for the rainbow Turan numbers of specific families of trees. These results are used in Chapter 4 to prove that the rainbow Turan numbers of all trees are linear in n, which correlates to a well-known property of the traditional Turan numbers of trees. We discuss improvements to the constant term in Chapters 4 and 5, and conclude with a discussion on avenues for future work
Book of Abstracts of the Sixth SIAM Workshop on Combinatorial Scientific Computing
Book of Abstracts of CSC14 edited by Bora UçarInternational audienceThe Sixth SIAM Workshop on Combinatorial Scientific Computing, CSC14, was organized at the Ecole Normale Supérieure de Lyon, France on 21st to 23rd July, 2014. This two and a half day event marked the sixth in a series that started ten years ago in San Francisco, USA. The CSC14 Workshop's focus was on combinatorial mathematics and algorithms in high performance computing, broadly interpreted. The workshop featured three invited talks, 27 contributed talks and eight poster presentations. All three invited talks were focused on two interesting fields of research specifically: randomized algorithms for numerical linear algebra and network analysis. The contributed talks and the posters targeted modeling, analysis, bisection, clustering, and partitioning of graphs, applied in the context of networks, sparse matrix factorizations, iterative solvers, fast multi-pole methods, automatic differentiation, high-performance computing, and linear programming. The workshop was held at the premises of the LIP laboratory of ENS Lyon and was generously supported by the LABEX MILYON (ANR-10-LABX-0070, Université de Lyon, within the program ''Investissements d'Avenir'' ANR-11-IDEX-0007 operated by the French National Research Agency), and by SIAM
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