A Fixed-Parameter Algorithm for the Minimum Weight Triangulation Problem Based on Small Graph Separators

Improved Fixed-Parameter Algorithm for the Minimum Weight 3-SAT Problem

WALCOM: Algorithms and Computation - Lecture Notes in Computer Science ◽

10.1007/978-3-642-36065-7_25 ◽

2013 ◽

pp. 265-273 ◽

Cited By ~ 2

Author(s):

Venkatesh Raman ◽

Bal Sri Shankar

Keyword(s):

Minimum Weight ◽

Sat Problem ◽

Fixed Parameter ◽

Fixed Parameter Algorithm

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FIXED PARAMETER ALGORITHMS FOR THE MINIMUM WEIGHT TRIANGULATION PROBLEM

International Journal of Computational Geometry & Applications ◽

10.1142/s0218195908002581 ◽

2008 ◽

Vol 18 (03) ◽

pp. 185-220 ◽

Cited By ~ 3

Author(s):

MAGDALENE GRANTSON BORGELT ◽

CHRISTIAN BORGELT ◽

CHRISTOS LEVCOPOULOS

Keyword(s):

Dynamic Programming ◽

Time Complexity ◽

Minimum Weight ◽

Simple Polygon ◽

Divide And Conquer ◽

The Core ◽

Fixed Parameter ◽

Minimum Weight Triangulation ◽

Divide And Conquer Scheme

We discuss and compare four fixed parameter algorithms for finding the minimum weight triangulation of a simple polygon with (n – k) vertices on the perimeter and k vertices in the interior (hole vertices), that is, for a total of n vertices. All four algorithms rely on the same abstract divide-and-conquer scheme, which is made efficient by a variant of dynamic programming. They are essentially based on two simple observations about triangulations, which give rise to triangle splits and paths splits. While each of the first two algorithms uses only one of these split types, the last two algorithms combine them in order to achieve certain improvements and thus to reduce the time complexity. By discussing this sequence of four algorithms we try to bring out the core ideas as clearly as possible and thus strive to achieve a deeper understanding as well as a simpler specification of these approaches. In addition, we implemented all four algorithms in Java and report results of experiments we carried out with this implementation.

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A better subgraph of the minimum weight triangulation

Lecture Notes in Computer Science - Computing and Combinatorics ◽

10.1007/bfb0030865 ◽

1995 ◽

pp. 452-455 ◽

Cited By ~ 1

Author(s):

Bo-Ting Yang

Keyword(s):

Minimum Weight ◽

Minimum Weight Triangulation

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Multiplicative Parameterization Above a Guarantee

ACM Transactions on Computation Theory ◽

10.1145/3460956 ◽

2021 ◽

Vol 13 (3) ◽

pp. 1-16

Author(s):

Fedor V. Fomin ◽

Petr A. Golovach ◽

Daniel Lokshtanov ◽

Fahad Panolan ◽

Saket Saurabh ◽

...

Keyword(s):

Minimum Size ◽

Input Graph ◽

Fixed Parameter Tractable ◽

Long Cycle ◽

Fixed Parameter ◽

Parameterized Problem ◽

Fixed Constant ◽

Fixed Parameter Algorithm ◽

Np Hardness ◽

Additive Form

Parameterization above a guarantee is a successful paradigm in Parameterized Complexity. To the best of our knowledge, all fixed-parameter tractable problems in this paradigm share an additive form defined as follows. Given an instance ( I,k ) of some (parameterized) problem π with a guarantee g(I) , decide whether I admits a solution of size at least (or at most) k + g(I) . Here, g(I) is usually a lower bound on the minimum size of a solution. Since its introduction in 1999 for M AX SAT and M AX C UT (with g(I) being half the number of clauses and half the number of edges, respectively, in the input), analysis of parameterization above a guarantee has become a very active and fruitful topic of research. We highlight a multiplicative form of parameterization above (or, rather, times) a guarantee: Given an instance ( I,k ) of some (parameterized) problem π with a guarantee g(I) , decide whether I admits a solution of size at least (or at most) k · g(I) . In particular, we study the Long Cycle problem with a multiplicative parameterization above the girth g(I) of the input graph, which is the most natural guarantee for this problem, and provide a fixed-parameter algorithm. Apart from being of independent interest, this exemplifies how parameterization above a multiplicative guarantee can arise naturally. We also show that, for any fixed constant ε > 0, multiplicative parameterization above g(I) 1+ε of Long Cycle yields para-NP-hardness, thus our parameterization is tight in this sense. We complement our main result with the design (or refutation of the existence) of fixed-parameter algorithms as well as kernelization algorithms for additional problems parameterized multiplicatively above girth.

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