EMI 2 – Matching problemEMI 2 – Matching problem

Javier Belenguer FaguásCatalin Costin Stanciu

Maria Cabezuelo SepúlvedaJaime Barrachina Verdia

The problem:

Let G = {{a,b,c,d,e},{e,f,g,h,i,j}} be complete

(this means that for every x from X and y from Y there exists a (x,y) that pertains to E)

H is a weighted graph whose weights are given by:

f g h i j

a 3 5 5 4 1 5

b 2 2 0 2 2 2

c 2 4 4 1 0 4

d 0 1 1 0 0 1

e 1 2 1 3 3 3

0 0 0 0 0

The graph:

Step 1 → u = {a}; V(T) = {a}; E(T) = 0; V(T,E) = 0;Step 2 → Ad(a) = {g,h};

g → M-insaturated → go to Step 3Step 3 → (a – g); V(T) = E(T) = 0; go to Step 1;

Step 1 → u = {b}; V(T) = {b}; E(T) = 0; V(T,E) = 0;Step 2 → Ad(b) = {f,g,i,j};

f → M-insaturated → go to Step 3Step 3 → (b-f); V(T) = E(T) = 0; go to Step 1;

Step 1 → u = {c}; V(T) = {c}; E(T) = 0; V(T,E) = 0;Step 2 → Ad(c) = {g,h};

g → M-saturated ^ g pertains to V(T)→ go to Step 4;Step 4 → (g,a) edge; V(T)={a,c,g}; E(T)={cg,ga};

V(T,E)={g}; Go to Step 2;h → M-insaturated → Step 3 → (h,c) → go to Step 1

Step 1 → u = {d}; V(T) = {d}; E(T) = {}; V(T,E) = {d};Step 2 → Ad(d) = {g,h};

g → M-saturated ^g does not pertain to V(T); → go to 4Step 4 → (g,a) edge → V(T) = {d,g,a}; E(T) = {dg,ga};

V(T,E) = {d,a};h → M-saturated ^h does not pertain to V(T): → go to 4Step 4 → (h,c) edge → V(T) = {a,c,d,g,h};

V(T,E) = {d,a,c}; E(T) = {dh,hc,dg,ga}; go to Step 2

We choose a; Ad(a) = {g,h};g → M-saturated ^h does not pertain to V(T)

→ go to Step 5;Step 5 → No perfect matching.

Now we recalculate the labels:

f g h i j L1 L2

a 3 5 5 4 1 5(e) 4

b 2 2 0 2 2 2 2

c 2 4 4 1 0 4(e) 3

d 0 1 1 0 0 1(e) 0

e 1 2 1 3 3 3 3

L1 0 0(o) 0(o) 0 0

L2 0 1 1 0 0

alpha = min(L(x) + L(y) – p(x,y)) = 1 Odd → L(v) + alphaEven → L(v) - alpha

Step 1 → u = {e}; V(T) = {d,g,h,a,c,e}; V(T,E) = {d,c,a,e};Step 2 → Ad(e) = {i,j};

j → M-insaturated → go to Step 3Step 3 → (e-j) →there's an M-augmenting path → Go to 1

Step 1 → u = {d}; V(T) = {d}; V(T,E) = {};Step 2 → Ad(d) = {g,h};

g → M-saturated ^ doesn't belong to V(T) → go to Step 4Step 4 → (g,a) pertains to M → V(T)={g,a,d}

V(T,E)={a,d} ; E(T)={dg,ga}; return to step 2h→ M-saturated ^ doesn't belong to V(T) → go to Step 4Step 4 → (h,c) pertains to M → V(T)={g,a,d,c,h}

V(T,E)={a,d,c} ; E(T)={dg,ga,dh,hc}; return to step 2

Step 2 →v= a; Ad(a) = {g,h,i}; i → M-insaturated → go to Step 3Step 3 →There's a path d-g-a-i ^ (ga) belongs to MDelete the edge (ga) from M ; Add (ai) to M;Delete V(T) , E(T), V(T,E); go to Step 1;

Step 1 → u=d; V(T)={d}; V(T,E)={d}; E(T)=0;Step 2 → v=d; Ad(d)={g,h};

g → M-insaturated → go to Step 3;Step 3 → (dg) is an M-augmenting path;Add (dg) to M; Delete V(T), V(T,E), E(T);Go to step 1;

Step 2 →v= a; Ad(a) = {g,h,i}; i → M-insaturated → go to Step 3Step 3 →There's a path d-g-a-i ^ (ga) belongs to MDelete the edge (ga) from M ; Add (ai) to M;Delete V(T) , E(T), V(T,E); go to Step 1;

Step 1 → All x are saturated so M is a perfect matching