37,787 research outputs found
Quasiconvex Programming
We define quasiconvex programming, a form of generalized linear programming
in which one seeks the point minimizing the pointwise maximum of a collection
of quasiconvex functions. We survey algorithms for solving quasiconvex programs
either numerically or via generalizations of the dual simplex method from
linear programming, and describe varied applications of this geometric
optimization technique in meshing, scientific computation, information
visualization, automated algorithm analysis, and robust statistics.Comment: 33 pages, 14 figure
Optimality program in segment and string graphs
Planar graphs are known to allow subexponential algorithms running in time
or for most of the paradigmatic
problems, while the brute-force time is very likely to be
asymptotically best on general graphs. Intrigued by an algorithm packing curves
in by Fox and Pach [SODA'11], we investigate which
problems have subexponential algorithms on the intersection graphs of curves
(string graphs) or segments (segment intersection graphs) and which problems
have no such algorithms under the ETH (Exponential Time Hypothesis). Among our
results, we show that, quite surprisingly, 3-Coloring can also be solved in
time on string graphs while an algorithm running
in time for 4-Coloring even on axis-parallel segments (of unbounded
length) would disprove the ETH. For 4-Coloring of unit segments, we show a
weaker ETH lower bound of which exploits the celebrated
Erd\H{o}s-Szekeres theorem. The subexponential running time also carries over
to Min Feedback Vertex Set but not to Min Dominating Set and Min Independent
Dominating Set.Comment: 19 pages, 15 figure
Lombardi Drawings of Graphs
We introduce the notion of Lombardi graph drawings, named after the American
abstract artist Mark Lombardi. In these drawings, edges are represented as
circular arcs rather than as line segments or polylines, and the vertices have
perfect angular resolution: the edges are equally spaced around each vertex. We
describe algorithms for finding Lombardi drawings of regular graphs, graphs of
bounded degeneracy, and certain families of planar graphs.Comment: Expanded version of paper appearing in the 18th International
Symposium on Graph Drawing (GD 2010). 13 pages, 7 figure
Aspects of Unstructured Grids and Finite-Volume Solvers for the Euler and Navier-Stokes Equations
One of the major achievements in engineering science has been the development of computer algorithms for solving nonlinear differential equations such as the Navier-Stokes equations. In the past, limited computer resources have motivated the development of efficient numerical schemes in computational fluid dynamics (CFD) utilizing structured meshes. The use of structured meshes greatly simplifies the implementation of CFD algorithms on conventional computers. Unstructured grids on the other hand offer an alternative to modeling complex geometries. Unstructured meshes have irregular connectivity and usually contain combinations of triangles, quadrilaterals, tetrahedra, and hexahedra. The generation and use of unstructured grids poses new challenges in CFD. The purpose of this note is to present recent developments in the unstructured grid generation and flow solution technology
Counting Shortest Two Disjoint Paths in Cubic Planar Graphs with an NC Algorithm
Given an undirected graph and two disjoint vertex pairs and
, the Shortest two disjoint paths problem (S2DP) asks for the minimum
total length of two vertex disjoint paths connecting with , and
with , respectively.
We show that for cubic planar graphs there are NC algorithms, uniform
circuits of polynomial size and polylogarithmic depth, that compute the S2DP
and moreover also output the number of such minimum length path pairs.
Previously, to the best of our knowledge, no deterministic polynomial time
algorithm was known for S2DP in cubic planar graphs with arbitrary placement of
the terminals. In contrast, the randomized polynomial time algorithm by
Bj\"orklund and Husfeldt, ICALP 2014, for general graphs is much slower, is
serial in nature, and cannot count the solutions.
Our results are built on an approach by Hirai and Namba, Algorithmica 2017,
for a generalisation of S2DP, and fast algorithms for counting perfect
matchings in planar graphs
A Variant of the Maximum Weight Independent Set Problem
We study a natural extension of the Maximum Weight Independent Set Problem
(MWIS), one of the most studied optimization problems in Graph algorithms. We
are given a graph , a weight function ,
a budget function , and a positive integer .
The weight (resp. budget) of a subset of vertices is the sum of weights (resp.
budgets) of the vertices in the subset. A -budgeted independent set in
is a subset of vertices, such that no pair of vertices in that subset are
adjacent, and the budget of the subset is at most . The goal is to find a
-budgeted independent set in such that its weight is maximum among all
the -budgeted independent sets in . We refer to this problem as MWBIS.
Being a generalization of MWIS, MWBIS also has several applications in
Scheduling, Wireless networks and so on. Due to the hardness results implied
from MWIS, we study the MWBIS problem in several special classes of graphs. We
design exact algorithms for trees, forests, cycle graphs, and interval graphs.
In unweighted case we design an approximation algorithm for -claw free
graphs whose approximation ratio () is competitive with the approximation
ratio () of MWIS (unweighted). Furthermore, we extend Baker's
technique \cite{Baker83} to get a PTAS for MWBIS in planar graphs.Comment: 18 page
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