AUXILIARY 2 Graph Coloring Hypergraphs

8/13/2019 AUXILIARY 2 Graph Coloring Hypergraphs

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FUNDAMENTAL PROBLEMS

AND

ALGORITHMS

Graph Theory and Combinational

Giovanni De Micheli

Stanford University


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Shortest/Longest path problem

Single-source shortest path problem.

Model:

Directed graph G(V, E)with Nvertices.

Weightson each edge.

A sourcevertex.

Single-source shortest path problem.

Find shortest path from the source to any vertex.

Inconsistent problem:

Negative-weighted cycles.


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Shortest path problem

Acyclic graphs: Topological sort O(N 2 ).

-

All positive weights:

Dijkstras algorithm.

Bellmans equations:

G(V, E)with Nvertices


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Dijkstras algorithm

DI JKSTRA(G(V, E, W))

{s 0=0;

for (i =1 to N)

si= w 0,i,

repeat {

select unmarked vqsuch that sqis minimal;

mark vq;

foreach (unmarked vertex v i)si= min {si, (sq+w q, i)},

}

until (all vertices are marked)

}

Apply to

Koreas map,

robot tour, etc

G(V, E)with Nvertices


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Bellman-Fords

algorithmBELLMAN_FORD(G(V, E, W))

{

s10= 0;

for (i = 1 to N)

s 1i =w 0, i;

for (j =1 to N){for (i =1 to N){

s j+1i = min {sji, (s

jk+w q, i)},

}

if (s j+1i == sji i ) return (TRUE) ;

}

return (FALSE)

}

ki


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Longestpath problem

Use shortest path algorithms:by reversing signson weights.

Modify algorithms:

by changing minwith max.

Remarks:

Dijkstras algorithm is not relevant.Inconsistent problem:

Positive-weighted cycles.


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ExampleBellman-Ford

Iteration 1:l 0=0, l 1=3, l 2= 1, l 3=.

Iteration 2:l 0=0, l 1=3, l 2=2, l 3=5.

Iteration 3:l 0=0, l 1=3, l 2=2, l 3=6.

Use shortest path algorithms:

by reversing signson

weights.

source


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Liao-Wongs

algorithm

LIAO WONG(G( V, EF, W))

{

for ( i = 1 to N)

l 1i= 0;

for ( j = 1 to |F|+ 1) {

foreach vertex vi

l j+1i= longest pathin G( V, E,W E) ;

flag = TRUE;

foreach edge ( v p, v q) F {

if ( lj+1 q < lj+ 1

p+ w p,q){

flag = FALSE;

E = E ( v0 , v q) with weight ( lj+ 1p+ w p,q)

}

}

if ( flag ) return (TRUE) ;

}

return (FALSE)

adjust


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ExampleLiao-Wong

Iteration 1:l0= 0, l 1= 3, l 2= 1, l 3= 5.

Adjust:add edge (v 0, v 1) with weight 2.

Iteration 2:l 0= 0, l 1= 3, l 2= 2, l 3= 6.

Only positive

edges from (a)

(b) adjusted by

adding longest

path from node0 to node 2

Looking for longest path

from node 0 to node 3


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Vertexcover

Given a graph G(V, E)Find a subset of the vertices

covering all the edges.

Intractable problem.

Goals:

Minimum cover.Irredundant cover:

No vertex can be removed.


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Example


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Heuristic algorithm vertex based

VERTEX_COVERV(G(V; E)){

C = ;

while (E ) do {select a vertex vV;

delete vfrom G(V, E) ;

C=C {fv} ;

}

}


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Heuristic algorithm edge based

VERTEX_COVERE(G(V, E))

{

C = ;

while (E ) do {

select an edge {u, v} E;

C=C {u, v};

delete from G(V, E) any edge incidentto either uor v;

}

}


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Graph coloring

Vertex labeling (coloring):No edge has end-point with the same label.

Intractable on general graphs.

Polynomial-time algorithms for chordal(and

interval) graphs:

Left-edge algorithm.


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Graph coloring heuristic algorithm

VERTEX_COLOR(G(V, E))

{for (i =1 to |V|) {

c =1

while (a vertex adjacent to viwith color c) do {

c = c +1;

color v iwith color c ;}

}

}


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Graph

coloring

exact

algorithm

EXACT_COLOR( G( V, E) , k)

{

repeat {

NEXT VALUE( k) ;

if ( c k== 0)

return ;

if ( k == n)

c is a proper coloring;

else

EXACT COLOR( G( V, E) , k+ 1)

}

}


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Graph

coloring

exact

algorithm

NEXT VALUE( k)

{repeat {

c k= ck+ 1;

if ( there is no adjacent vertex to v kwith the same color ck)

return ;

}until ( c k =maximum number of colors ) ;

c k= 0;

}


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Intervalgraphs

Edges represent interval intersections.

Example:

Restricted channel routing problem with no

vertical constraints.

Possible to sort the intervals by left edge.


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Example

E l


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Example


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Left-edge

algorithmLEFT EDGE( I){

Sort elements of Iin a list L with ascending order of li;

c = 0;

while (Some interval has not been colored ) do {

S =;

repeat {

s = first element in the listLwhose left edge

ls is higher than the rightmost edge in S.

S= S { s};

}until ( an element s is found );c = c + 1;

color elements of S with color c;

delete elements of Sfrom L;

}

}


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Clique partitioning and covering

A clique partition is a cover. A clique partition can be derived from a cover by

making the vertex subsets disjoint.

Intractable problem on general graphs. Heuristics:

Search for maximal cliques.

Polynomial-time algorithms for chordal graphs.

HeuristicCLIQUEPARTITION( G( V E) )


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Heuristic

algorithm

CLIQUEPARTITION( G( V, E) )

{

=;

while ( G( V, E) not empty ) do {

compute largest clique C V in G( V, E) ;

=C;

delete C from G( V, E) ;

}

}

CLI QUE( G( V, E) )

{

C = seed vertex;

repeat {

select vertex v V , vCand adjacent to all vertices of C;

if (no such vertex is found) return

C = C {v} ;

}

}


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Covering and Satisfiability

Covering problems can be cast as satisfiability.

Vertex cover.

Ex1:( x 1+ x 2) (x 2+ x 3) (x 3+ x 4) (x 4+ x 5)

Ex2:(x3+ x4) (x1+ x3) (x1+ x2) (x2+ x4) (x4+ x5)

Objective function:

Result:Ex1:x 2= 1, x 4= 1

Ex2:x 1 = 1, x 4= 1

C i bl


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Covering problem

Set covering problem:

A set S.

A collection C of subsets.

Select fewest elements of C to cover S. Intractable.

Exact method:

Branch and bound algorithm.

Heuristic methods.

E l


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Example

vertex-cover of a graph

Example


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Example

edge-cover of a hypergraph

M i i


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Matrix representation

Boolean matrix: A.

Selection Boolean vector: x. Determine x such that:

A x 1.

Select enough columns to cover all rows.

Minimize cardinality of x.

B h d b d l i h


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Branch and bound algorithm

Tree search of the solution space:

Potentially exponential search.

Use bounding function:

If the lower bound on the solution cost that can be

derived from a set of future choices exceeds the cost ofthe best solution seen so far:

Kill the search.

Good pruning may reduce run-time.

Branch andBRANCH AND BOUND{


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Branch and

bound

algorithm

_ _ {

Current best = anything;

Current cost = 1;

S= s 0;

while (S6 = ; ) do {

Select an element in s 2S;

Remove s from S ;

Make a branching decision based on s

yielding sequences f s i ; i= 1; 2; ...; mg ;

for ( i = 1 to m) {

Compute the lower bound b i of si ;

if ( b i Current cost) Kill si;else {

if ( s i is a complete solution ) {

Current best = s i,

Current cost = cost of s i,

}

elseAdd s i to set S;

}

}

}

}

Example


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Example

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AUXILIARY 2 Graph Coloring Hypergraphs