Input-Dependent and Asymptotic Approximation

Summary- Approximation algorithm for graph coloring

- Approximation algorithm for set cover

- Asymptotic approximation scheme for edge coloring

begin i:=0; U:=V;

while U ø dobegin

i:=i+1;V[i]:=ø;W:=U;H:=graph induced by W;

while W ø dobegin

v=node of minimum degree in H; insert v inV[i];delete v and its neighbors from W;U:=U-V[i]

endend

end.

Approximation of graph coloring- Polynomial-time n/logn-approximation algorithm for

MINIMUM GRAPH COLORING

ProofWe first prove that, if G is k-colorable, then the

algorithm uses at most 3|V|/logk|V| colors.

- At any iteration of the inner loop, H is k-colorable

- Hence, it contains an independent set of at least |W|/k nodes of degree at most |W|(k-1)/k

- At least |W|/k nodes will be in W at the next iteration

- At least logk|W| iterations of the inner loop

- |V[i]| is at least logk|U|

Proof (continued)

- As long as |U| |V|/logk|V|, logk|U| logk(|V|/logk|V|) (1/2) logk|V|

- When |U|<|V|/logk|V|, at most 2|V|/logk|V| colors have been used

- To color the remaining nodes|V|/logk|V| colors suffice

- That is, the algorithm uses at most 3|V|/logk|V| colors

Proof (end)

- The algorithm uses at most |V|/logm*(G)|V|, that is, at most 3nlog(m*(G))/logn colors

- The performance ratio is at most 3n/logn

MINIMUM SET COVER- INSTANCE: Collection C of subsets of a finite set S

- SOLUTION: A set cover for S, i.e., a subset C’ of C such that every element in S belongs to at least one member of C’

- MEASURE: |C’|

beginU:=S; C’:=ø;for any ci do c’i := ci ;repeat

i:=index of c’with maximum cardinality;insert ci in C’; U := U-{elements of c’i };delete all elements of ci from all c’;

until U:=øend.

Johnson’s algorithm- Polynomial-time logarithmic approximation algorithm

for MINIMUM SET COVER

Proof- We prove that the performance ratio of the algorithm

is at most ∑1 i k (1/i) where k is the maximum cardinality of the sets in C

- Let a1 ,...,a|C’| be the sequence of indices obtained by the algorithm

- Let cji be the surviving part of ci before index aj has

been chosen

- The intersection of ci and cjaj is equal to cj

i - cj+1i

- li denote largest index j such that cji is not empty

First stepFor any i, H(|ci|) 1 j C’| (|ci cj

aj |/|cj

aj|) where H(n)=

1 i n (1/i)

Proof

1 j C’| (|ci cjaj |/|cj

aj|)= 1 j C’| (|cj

i |-|cj+1i |)/|cj

aj|)

1 j li (|cji|-|cj+1

i|)/|cji|)= 1 j li

|cj+1

i|+1 k | cji | (1/|cj

i|)

1 j li 1 k | c

ji |-|c

j+1i| (1/(k+|cj+1

i|))

1 j li (H(|cj

i|)-H(|cj+1i|)= H(|c1

i|)=H(|ci|)

Second step

For any set cover C’’,

ci C’’ 1 j |C’| (|ci cjaj |/|cj

aj|) |C’|

Proof

ci C’’ 1 j |C’| (|ci cjaj |/|cj

aj|) =

1 j |C’| (1/|cjaj|) ci C’’|ci cj

aj |

1 j =|C’| (|cjaj|/|cj

aj|) = |C’|

Final stepH(|ci|) 1 j C’| (|ci cj

aj |/|cj

aj|)

and

ci C* 1 j C’| (|ci cjaj |/|cj

aj|) |C’|

imply

ci C* H(|ci|) |C’|

Since |ci | k, we have that

ci C* H(k) |C’| , that is, |C*|H(k) |C’|

Since H(k) ln k+1 ln n+1, we have that the performance ratio is logarithmic

MINIMUM EDGE COLORING- INSTANCE: Graph G=(V,E)

- SOLUTION: A coloring of E, that is, function f such that, for any pair of edges e1 and e2 that share a common endpoint, f(e1) f(e2)

- MEASURE: Number of colors, i.e., cardinality of the range of f

Vizing’s algorithm- Polynomial-time algorithm to color a graph with at

most D+1 colors, where D denotes the maximum degree of the graph

beginD:=maximum degree of G;

G’:=(V, E’:=ø); // G’ is clearly colorable with D+1 colorsrepeat

add an edge (u,v) of E to E’;extend coloring of G’ without (u,v) into coloring of G’ with at most D+1 colors; E := E-{(u,v)};

until E:=øend.

Asymptotic approximation scheme- The algorithm returns an edge-coloring with at most

D+1 colors

- The optimum is at least D

- Hence, performance ratio is at most (D+1)/m*(G) D/D+1/m*(G)=1+ 1/m*(G)

- It implies a 2-approximation

Proof- Assume G’ without (u,v) has an edge-coloring with at

most D+1 colors

- Let µ(v) denote the set of colors that are not used to color an edge incident to v

- Clearly, if the coloring uses D+1 colors, then for any v, µ(v)ø: let c(v) be one of the colors in µ(v)

Proof (continued)- Compute in polynomial-time a sequence of edges

(u,x[0]),...,(u,x[s]) such that:- x[0]=v- for any i, the color of (u,x[i]) = c(x[i-1])- there is no other edge (u,w) such that its color is equal to

c(x[s])

u

x0=v x1

x2

x3

x4

c(x0)

c(x1)

c(x2)c(x3)

First case: c(x[s]) is in µ(u)- In this case, we can simply shift the colors of the

sequence in order to obtain the new coloring of G’- That is, for any i, we color (u,x[i]) with c(x[i])

u

x0=v x1

x2

x3

x4

c(x0)

c(x1)

c(x2)c(x3)

u

x0=v x1

x2

x3

x4

c(x1)

c(x2)

c(x3)c(x4)

c(x0)

Second case: c(x[s]) is not in µ(u)- In this case, one edge (u,x[i]) must have been colored

with c(x[s]) (since the sequence is maximal)- Hence, c(x[i-1]) = c(x[s])

- We compute in polynomial time a path Pi-1 starting from x[i-1] formed by edges whose colors are, alternatively, c(u) and c(x[s])

u

x0=vx1

x2

x3

x4

c(x0)

c(x1)=c(x4)

c(x2)c(x3)

c(u) c(x4) c(u)

u

x0=vx1

x2

x3

x4

c(x0)

c(x1)=c(x4)

c(x2)c(x3)

c(u) c(x4)

c(u)

c(x4)

c(u)

u

x0=vx1

x2

x3

x4

c(x0)

c(x1)=c(x4)

c(x2)c(x3)

c(u)

c(x4) c(u)

c(x4)

c(u)

c(x4)

First subcase: Pi-1 does not end in u

- Interchange colors c(u) and c(x[s]) in the path, assign color c(u) to (u,x[i]), shift the colors of the subsequence of edges preceding (u,x[i-1])

Second subcase: Pi-1 ends in u

- Compute in polynomial time a path Ps starting from x[s] formed by edges whose colors are, alternatively, c(u) and c(x[s])- Ps does not end in u

u

x0=vx1

x2

x3

x4

c(x0)

c(x1)=c(x4)

c(x2)c(x3)

c(u) c(x4)

c(u)

c(u)c(x4)

c(u)c(x4)

c(u)c(x4)

c(u)

c(x4) u

x0=vx1

x2

x3

x4

c(x0)

c(x1)=c(x4)

c(x2)c(x3)

c(u) c(x4)

c(u)

c(u)

c(u)

c(x4)

c(u)

c(x4)

c(x4)

- Interchange colors c(u) and c(x[s]) in Ps, assign color c(u) to (u,x[s]), and shift the colors of the subsequence of edges preceding (u,x[s])

u

x0=vx1

x2

x3

x4

c(x0)

c(x1)=c(x4)

c(x2)c(x3)

c(u) c(x4)

c(u)

c(u)

c(u)c(x4)

c(u)

c(x4) u

x0=vx1

x2

x3

x4

c(x0) c(x1)

c(x2)

c(x3)

c(u) c(x4)

c(u)

c(u)

c(u)

c(x4)

c(u)

c(x4)

c(x4)

Class F-APX- Let F be a class of functions

- The class F -APX contains all NPO problems P that admit a polynomial-time algorithm A such that, for any instance x of P, R(x, A(x))) f(|x|), for a given function f F

- P is said to be f(n)-approximable

- A is said to be an f(n)-approximation algorithm

Class APTAS- The class APTAS contains all NPO problems P that

admit a polynomial-time algorithm A and a constant k such that, for any instance x of P and for any rational r, R(x, A(x,r))) r+k/m*(x)

- The time complexity of A is polynomial in |x| but not necessarily in 1/(r-1)

- A is said to be an asymptotic approximation scheme- A is clearly a (r+k)-approximation algorithm

The NPO worldNPO

APX MAXIMUM SATMINIMUM VERTEX COVERMAXIMUM CUT

PO MINIMUM PATH

O(n)-APX

O(logn)-APX

MINIMUM GRAPH COLORING

MINIMUM SET COVER

PTAS MINIMUM PARTITION

APTAS MINIMUM EDGE COLORING