HaskellForMaths-0.4.0: Combinatorics, group theory, commutative algebra, non-commutative algebra

Math.Combinatorics.Matroid

Description

A module providing functions to construct and investigate (small, finite) matroids.

Synopsis

# Documentation

data TrieSet a Source

The data structure that we use to store the bases of the matroid

Constructors

 TS [(a, TrieSet a)]

Instances

 Functor TrieSet Eq a => Eq (TrieSet a) Ord a => Ord (TrieSet a) Show a => Show (TrieSet a)

data Matroid a Source

A datatype to represent a matroid. `M es bs` is the matroid whose elements are `es`, and whose bases are `bs`. The normal form is for the `es` to be in order, for each of the `bs` individually to be in order. (So the TrieSet should have the property that any path from the root to a leaf is strictly increasing).

Constructors

 M [a] (TrieSet a)

Instances

 Functor Matroid Eq a => Eq (Matroid a) Show a => Show (Matroid a)

elements :: Matroid t -> [t]Source

Return the elements over which the matroid is defined.

indeps :: Ord a => Matroid a -> [[a]]Source

Return all the independent sets of a matroid, in shortlex order.

isIndependent :: Ord a => Matroid a -> [a] -> BoolSource

isDependent :: Ord a => Matroid a -> [a] -> BoolSource

isMatroidIndeps :: Ord a => [[a]] -> BoolSource

Are these the independent sets of a matroid? (The sets must individually be ordered.)

fromIndeps :: Ord a => [a] -> [[a]] -> Matroid aSource

Construct a matroid from its elements and its independent sets.

vectorMatroid :: Fractional k => [[k]] -> Matroid IntSource

Given a matrix, represented as a list of rows, number the columns [1..], and construct the matroid whose independent sets correspond to those sets of columns which are linearly independent (or in case there are repetitions, those multisets of columns which are sets, and which are linearly independent).

vectorMatroid' :: Fractional k => [[k]] -> Matroid IntSource

Given a list of vectors (or rows of a matrix), number the vectors (rows) [1..], and construct the matroid whose independent sets correspond to those sets of vectors (rows) which are linearly independent (or in case there are repetitions, those multisets which are sets, and which are linearly independent).

cycleMatroid :: Ord a => [[a]] -> Matroid IntSource

Given the edges of an undirected graph, number the edges [1..], and construct the matroid whose independent sets correspond to those sets of edges which contain no cycle. The bases therefore correspond to maximal forests within the graph. The edge set is allowed to contain loops or parallel edges.

to1n :: Ord a => Matroid a -> Matroid IntSource

Given a matroid over an arbitrary type, relabel to obtain a matroid over the integers.

isMatroidIso :: (Ord a, Ord b) => Matroid a -> Matroid b -> BoolSource

Are the two matroids isomorphic?

matroidAuts :: Ord a => Matroid a -> [Permutation a]Source

Return the automorphisms of the matroid.

isCircuit :: Ord a => Matroid a -> [a] -> BoolSource

A circuit in a matroid is a minimal dependent set.

circuits :: Ord a => Matroid a -> [[a]]Source

Return all circuits for the given matroid, in shortlex order.

isMatroidCircuits :: Ord a => [[a]] -> BoolSource

Are the given sets the circuits of some matroid?

fromCircuits :: Ord a => [a] -> [[a]] -> Matroid aSource

Reconstruct a matroid from its elements and circuits.

isLoop :: Ord a => Matroid a -> a -> BoolSource

An element e in a matroid M is a loop if {e} is a circuit of M.

isParallel :: Ord a => Matroid a -> a -> a -> BoolSource

Elements f and g in a matroid M are parallel if {f,g} is a circuit of M.

isSimple :: Ord a => Matroid a -> BoolSource

A matroid is simple if it has no loops or parallel elements

isBase :: Ord a => Matroid a -> [a] -> BoolSource

A base or basis in a matroid is a maximal independent set.

bases :: Ord a => Matroid a -> [[a]]Source

Return all bases for the given matroid

isMatroidBases :: Ord a => [[a]] -> BoolSource

Are the given sets the bases of some matroid?

fromBases :: Ord a => [a] -> [[a]] -> Matroid aSource

Reconstruct a matroid from its elements and bases.

fundamentalCircuit :: Ord a => Matroid a -> [a] -> a -> [a]Source

Given a matroid m, a basis b, and an element e, `fundamentalCircuit m b e` returns the unique circuit contained in b union {e}, which is called the fundamental circuit of e with respect to b.

u :: Int -> Int -> Matroid IntSource

The uniform matroid U m n is the matroid whose independent sets are all subsets of [1..n] with m or fewer elements.

restriction :: Ord a => Matroid a -> [a] -> Matroid aSource

The restriction of a matroid to a subset of its elements

rankfun :: Ord a => Matroid a -> [a] -> IntSource

Given a matroid m, `rankfun m` is the rank function on subsets of its element set

rank :: Ord a => Matroid a -> IntSource

The rank of a matroid is the cardinality of a basis

fromRankfun :: Ord a => [a] -> ([a] -> Int) -> Matroid aSource

Reconstruct a matroid from its elements and rank function

closure :: Ord a => Matroid a -> [a] -> [a]Source

Given a matroid m, `closure m` is the closure operator on subsets of its element set

fromClosure :: Ord a => [a] -> ([a] -> [a]) -> Matroid aSource

Reconstruct a matroid from its elements and closure operator

isFlat :: Ord a => Matroid a -> [a] -> BoolSource

A flat in a matroid is a closed set, that is a set which is equal to its own closure

flats :: Ord a => Matroid a -> [[a]]Source

The flats of a matroid are its closed sets. They form a lattice under inclusion.

fromFlats :: Ord a => [[a]] -> Matroid aSource

Reconstruct a matroid from its flats. (The flats must be given in shortlex order.)

isSpanning :: Ord a => Matroid a -> [a] -> BoolSource

A subset of the elements in a matroid is spanning if its closure is all the elements

isHyperplane :: Ord a => Matroid a -> [a] -> BoolSource

A hyperplane is a flat whose rank is one less than that of the matroid

hyperplanes :: Ord a => Matroid a -> [[a]]Source

isMatroidHyperplanes :: Ord a => [a] -> [[a]] -> BoolSource

fromHyperplanes :: Ord a => [a] -> [[a]] -> Matroid aSource

Reconstruct a matroid from its elements and hyperplanes

affineMatroid :: Fractional k => [[k]] -> Matroid IntSource

Given a list of points in k^n, number the points [1..], and construct the matroid whose independent sets correspond to those sets of points which are affinely independent.

A multiset of points in k^n is said to be affinely dependent if it contains two identical points, or three collinear points, or four coplanar points, or ... - and affinely independent otherwise.

fromGeoRep :: Ord a => [[a]] -> [[a]] -> [[a]] -> [[a]] -> Matroid aSource

fromGeoRep returns a matroid from a geometric representation consisting of dependent flats of various ranks. Given lists of dependent rank 0 flats (loops), rank 1 flats (points), rank 2 flats (lines) and rank 3 flats (planes), `fromGeoRep loops points lines planes` returns the matroid having these as dependent flats. Note that if all the elements lie in the same plane, then this should still be listed as an argument.

simpleFromGeoRep :: Ord a => [[a]] -> [[a]] -> Matroid aSource

A simple matroid has no loops or parallel elements, hence its geometric representation has no loops or dependent points. `simpleFromGeoRep lines planes` returns the simple matroid having these dependent flats.

circuitHyperplanes :: Ord a => Matroid a -> [[a]]Source

List the circuit-hyperplanes of a matroid.

relaxation :: Ord a => Matroid a -> [a] -> Matroid aSource

Given a matroid m, and a set of elements b which is both a circuit and a hyperplane in m, then `relaxation m b` is the matroid which is obtained by adding b as a new basis. This corresponds to removing b from the geometric representation of m.

transversalMatroid :: Ord a => [a] -> [[a]] -> Matroid aSource

Given a set of elements es, and a sequence as = [a1,...,am] of subsets of es, return the matroid whose independent sets are the partial transversals of the as.

dual :: Ord a => Matroid a -> Matroid aSource

The dual matroid

cocircuits :: Ord a => Matroid a -> [[a]]Source

deletion :: Ord a => Matroid a -> [a] -> Matroid aSource

contraction :: Ord a => Matroid a -> [a] -> Matroid aSource

isConnected :: Ord a => Matroid a -> BoolSource

A matroid is (2-)connected if, for every pair of distinct elements, there is a circuit containing both

dsum :: (Ord a, Ord b) => Matroid a -> Matroid b -> Matroid (Either a b)Source

The direct sum of two matroids

matroidPG :: Fractional a => Int -> [a] -> Matroid IntSource

`matroidPG n fq` returns the projective geometry PG(n,Fq), where fq is a list of the elements of Fq

matroidAG :: Fractional a => Int -> [a] -> Matroid IntSource

`matroidAG n fq` returns the affine geometry AG(n,Fq), where fq is a list of the elements of Fq

fundamentalCircuitIncidenceMatrix :: (Ord a, Num k) => Matroid a -> [a] -> [[k]]Source

Given a matroid m, the fundamental-circuit incidence matrix relative to a base b has rows indexed by the elements of b, and columns indexed by the elements not in b. The bi, ej entry is 1 if bi is in the fundamental circuit of ej relative to b, and 0 otherwise.

representations :: (Fractional fq, Ord a) => [fq] -> Matroid a -> [[[fq]]]Source

Find representations of the matroid m over fq. Specifically, this function will find one representative of each projective equivalence class of representation.

isRepresentable :: (Fractional fq, Ord a) => [fq] -> Matroid a -> BoolSource

Is the matroid representable over Fq? For example, to find out whether a matroid m is binary, evaluate `isRepresentable f2 m`.

isBinary :: Ord a => Matroid a -> BoolSource

A binary matroid is a matroid which is representable over F2

isTernary :: Ord a => Matroid a -> BoolSource

A ternary matroid is a matroid which is representable over F3

data LMR a b Source

Constructors

 L a Mid R b

Instances

 (Eq a, Eq b) => Eq (LMR a b) (Ord a, Ord b) => Ord (LMR a b) (Show a, Show b) => Show (LMR a b)

The Fano plane F7 = PG(2,F2)

F7-, the relaxation of the Fano plane by removal of a line

The Pappus configuration from projective geometry

Relaxation of the Pappus configuration by removal of a line

The Desargues configuration

The Vamos matroid V8. It is not representable over any field.

P8 is a minor-minimal matroid that is not representable over F4, F8, F16, ... . It is Fq-representable if and only if q is not a power of 2.

P8- is a relaxation of P8. It is Fq-representable if and only if q >= 4.

P8-- is a relaxation of P8-. It is a minor-minimal matroid that is not representable over F4. It is Fq-representable if and only if q >= 5.

Given a matroid m over elements es, the rank polynomial is a polynomial r(x,y), which is essentially a generating function for the subsets of es, enumerated by size and rank. It is efficiently calculated using deletion and contraction.

It has the property that r(0,0) is the number of bases in m, r(1,0) is the number of independent sets, r(0,1) is the number of spanning sets. It can also be used to derive the chromatic polynomial of a graph, the weight enumerator of a linear code, and more.