4312 ITSD ITSD Advanced Programming II
Approximation Algorithms
Motivation Motivation
Many important problems are NPcomplete We cant abandon them But we can certainly try different approaches
If n is small, an exponential solution is OK Special cases might be identified Solve for a near-optimal solution
A Well-known Approximation Well-known
n 1 1 e = = lim n x = 0 x! x = 0 x!
111 1 1 e= + + + + 1 2 6 24 120
Another Approximation Another
x 2 k +1 k sin x = lim (1) n (2k + 1)! k =0
n
x3 x5 x7 sin x = x + + 3! 5! 7!
Two Classes of Problems Two
Return a minimum
Traveling salesman (minimize hamiltonian cycle)
Return a maximum
Subset-sum (maximize up to limit)
Getting Close Getting
Lets allow a approximation ratio, (n) C is cost using our approx. algorithm C* is optimal cost
C C * max , ( n) C* C
0 < C C* (for maximizing problem) 0 < C* C (for minimizing problem)
Traveling Salesman Traveling
Given
G(V, E) Cost c(u,v) 1, (u,v) E Cost c( A) = c(u , v)
( u ,v ) A
Where A E
Find a hamiltonian cycle (tour) that minimizes c(A) on G
TSP Approximation (1977) TSP
Given G(V,E)
Pick root r V Generate MST(G,r) L = pre-order traverse MST(G,r) Return hamiltonian cycle generated by visiting vertices in the order L
This is a polynomial time 2-approximation algorithm
(n)=2 i.e. its max runtime is 2C*
TSP Approx Visualized TSP
a d e b c h f g c h b f a d e g
abchdefg
a
d e
a
d e
b c h
f
g
b
f
g
c(A)=19.074
c h
C*=14.715
Subset-Sum Subset-Sum
Defined by (S, t) A set of positive integers, S={x1, x2, , xn} A positive integer, t Is there a subset of S that adds up exactly to t? Many practical applications
Truck packing without exceeding weight limit Waiter holding meals
NP-complete Solution Start NP-complete
with empty list L For each x in S
L += (L + x)
Ex S = { 1, 4, 5 } P1 = { 0, 1 } P2 = { 0, 1, 4, 5 } P3 = { 0, 1, 4, 5, 6, 9, 10 }
Subset-sum Approximation Subset-sum
Same as before Except, trim list after each iteration
Select trimming factor: 0 < d < 1 Remove y and keep z such that
y z y 1+ d
Remove elements larger than t
Return largest in list
Subset-sum Approx Ex Subset-sum
d = 0.1 L = { 10, 11, 12, 15, 20, 21, 22, 23, 24, 29 } L = { 10, 12, 15, 20, 23, 29 } - or L = { 11, 15, 22, 24, 29 }
Trimming Example
d = 0.1 L = { 10, 11, 12, 15, 20, 21, 22, 23, 24, 29 } L = { 10, 12, 15, 20, 23, 29 } - or L = { 11, 15, 22, 24, 29 }
Subset-sum Approx Ex Subset-sum
S = { 104, 102, 201, 101 }, t = 308, = 0.40 d = /(2n) = 0.40/8 = 0.05 P1 = { 0, 104 } P2 = { 0, 102, 104, 206 } P3 = { 0, 102, 201, 206, 303, 407 } P4 = { 0, 101, 102, 201, 302, 303, 404 } Optimal: 101 + 102 + 104 = 307
Backtracking Backtracking
N-queens problem = correctly place N queens onto an NxN board such that no queen attacks any other queen. Q Q Q ...