Logarithmic time NC algorithms for comparability graphs and circle graphs

A maximum linear matroid parity set is called a basic matroid parity set, if its size is the rank of the matroid. We show that determining the existence of a common base (basic matroid parity set) for linear matroid intersection (linear matroid parity) is in NC2, provided that there are polynomial number of common bases (basic matroid parity sets). For graphic matroids, we show that finding a common base for matroid intersection is in NC2, if the number of common bases is polynomial bounded. To our knowledge, these algorithms are the first deterministic NC algorithms for matroid intersection and matroid parity. We also give a new RNC2 algorithm that finds a common base for graphic matroid intersection. We prove that if there is a black-box NC algorithm for Polynomial Identity Testing (PIT), then there is an NC algorithm to determine the existence of a common base (basic matroid parity set) for linear matroid intersection (linear matroid parity).

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NC-ALGORITHMS FOR OPERATIONS ON BINARY DECISION DIAGRAMS

Parallel Processing Letters ◽

10.1142/s0129626493000022 ◽

1993 ◽

Vol 03 (01) ◽

pp. 3-12 ◽

Cited By ~ 33

Author(s):

DETLEF SIELING ◽

INGO WEGENER

Keyword(s):

Binary Decision Diagrams ◽

Pattern Generation ◽

Decision Diagrams ◽

Equivalence Test ◽

Ordered Binary Decision Diagrams ◽

Binary Decision ◽

Verification Test ◽

Logical Synthesis ◽

Redundancy Test ◽

Nc Algorithms

(Ordered) binary decision diagrams are a powerful representation for Boolean functions and are widely used in logical synthesis, verification, test pattern generation or as part of CAD tools. NC-algorithms are presented for the most important operations on this representation, e.g. evaluation for a given input, minimization, satisfiability, redundancy test, replacement of variables by constants or functions, equivalence test and synthesis. The algorithms have logarithmic run time on CRCW COMMON PRAMs with a polynomial number of processors.

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New clique and independent set algorithms for circle graphs

Discrete Applied Mathematics ◽

10.1016/0166-218x(92)90200-t ◽

1992 ◽

Vol 36 (1) ◽

pp. 1-24 ◽

Cited By ~ 21

Author(s):

Alberto Apostolico ◽

Mikhail J. Atallah ◽

Susanne E. Hambrusch

Keyword(s):

Independent Set ◽

Circle Graphs

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MINIMUM WEIGHT FEEDBACK VERTEX SETS IN CIRCLE n-GON GRAPHS AND CIRCLE TRAPEZOID GRAPHS

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830911001243 ◽

2011 ◽

Vol 03 (03) ◽

pp. 323-336 ◽

Cited By ~ 2

Author(s):

FANICA GAVRIL

Keyword(s):

Minimum Weight ◽

Polynomial Time Algorithm ◽

Time Algorithm ◽

Circle Graphs ◽

Intersection Model ◽

The Family ◽

Positive Weights ◽

Vertex Set ◽

Vertex Sets ◽

Trapezoid Graphs

A circle n-gon is the region between n or fewer non-crossing chords of a circle, no chord connecting the arcs between two other chords; the sides of a circle n-gon are either chords or arcs of the circle. A circle n-gon graph is the intersection graph of a family of circle n-gons in a circle. The family of circle trapezoid graphs is exactly the family of circle 2-gon graphs and the family of circle graphs is exactly the family of circle 1-gon graphs. The family of circle n-gon graphs contains the polygon-circle graphs which have an intersection representation by circle polygons, each polygon with at most n chords. We describe a polynomial time algorithm to find a minimum weight feedback vertex set, or equivalently, a maximum weight induced forest, in a circle n-gon graph with positive weights, when its intersection model by n-gon-interval-filaments is given.

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