5,370 research outputs found
Minimal surfaces from circle patterns: Geometry from combinatorics
We suggest a new definition for discrete minimal surfaces in terms of sphere
packings with orthogonally intersecting circles. These discrete minimal
surfaces can be constructed from Schramm's circle patterns. We present a
variational principle which allows us to construct discrete analogues of some
classical minimal surfaces. The data used for the construction are purely
combinatorial--the combinatorics of the curvature line pattern. A
Weierstrass-type representation and an associated family are derived. We show
the convergence to continuous minimal surfaces.Comment: 30 pages, many figures, some in reduced resolution. v2: Extended
introduction. Minor changes in presentation. v3: revision according to the
referee's suggestions, improved & expanded exposition, references added,
minor mistakes correcte
Lecture notes: Semidefinite programs and harmonic analysis
Lecture notes for the tutorial at the workshop HPOPT 2008 - 10th
International Workshop on High Performance Optimization Techniques (Algebraic
Structure in Semidefinite Programming), June 11th to 13th, 2008, Tilburg
University, The Netherlands.Comment: 31 page
Packing Chromatic Number of Distance Graphs
The packing chromatic number of a graph is the smallest
integer such that vertices of can be partitioned into disjoint classes
where vertices in have pairwise distance greater than
. We study the packing chromatic number of infinite distance graphs , i.e. graphs with the set of integers as vertex set and in which two
distinct vertices are adjacent if and only if . In
this paper we focus on distance graphs with . We improve some
results of Togni who initiated the study. It is shown that for sufficiently large odd and
for sufficiently large even . We also give a lower bound 12 for
and tighten several gaps for with small .Comment: 13 pages, 3 figure
Weighted Min-Cut: Sequential, Cut-Query and Streaming Algorithms
Consider the following 2-respecting min-cut problem. Given a weighted graph
and its spanning tree , find the minimum cut among the cuts that contain
at most two edges in . This problem is an important subroutine in Karger's
celebrated randomized near-linear-time min-cut algorithm [STOC'96]. We present
a new approach for this problem which can be easily implemented in many
settings, leading to the following randomized min-cut algorithms for weighted
graphs.
* An -time sequential algorithm:
This improves Karger's and bounds when the input graph is not extremely
sparse or dense. Improvements over Karger's bounds were previously known only
under a rather strong assumption that the input graph is simple [Henzinger et
al. SODA'17; Ghaffari et al. SODA'20]. For unweighted graphs with parallel
edges, our bound can be improved to .
* An algorithm requiring cut queries to compute the min-cut of
a weighted graph: This answers an open problem by Rubinstein et al. ITCS'18,
who obtained a similar bound for simple graphs.
* A streaming algorithm that requires space and
passes to compute the min-cut: The only previous non-trivial exact min-cut
algorithm in this setting is the 2-pass -space algorithm on simple
graphs [Rubinstein et al., ITCS'18] (observed by Assadi et al. STOC'19).
In contrast to Karger's 2-respecting min-cut algorithm which deploys
sophisticated dynamic programming techniques, our approach exploits some cute
structural properties so that it only needs to compute the values of cuts corresponding to removing pairs of tree edges, an
operation that can be done quickly in many settings.Comment: Updates on this version: (1) Minor corrections in Section 5.1, 5.2;
(2) Reference to newer results by GMW SOSA21 (arXiv:2008.02060v2), DEMN
STOC21 (arXiv:2004.09129v2) and LMN 21 (arXiv:2102.06565v1
Transversal designs and induced decompositions of graphs
We prove that for every complete multipartite graph there exist very
dense graphs on vertices, namely with as many as
edges for all , for some constant , such that can be
decomposed into edge-disjoint induced subgraphs isomorphic to~. This result
identifies and structurally explains a gap between the growth rates and
on the minimum number of non-edges in graphs admitting an
induced -decomposition
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