Approximate Knapsack

1. Approximate Knapsack Get as much value as you can into the knapsack

2. Approximate Knapsack • Ingredients: • Instances: The volume V of the knapsack. The volume and price of n objects<<v1,p1>,<v2,p2>,… ,<vn,pn>>. • Solutions: A set of objects that fit in the knapsack. • i.e. iS vi V • Cost of Solution: The total value of objects in set. • i.e. iS pi • Goal: Get as much value as you can into the knapsack.

3. Approximate Knapsack • Dynamic Programming Running time = ( V × n ) = ( 2#bits in V× n ) • Poly time if size of knapsack is small • Exponential time if size is an arbitrary integer.

4. Approximate Knapsack • Dynamic Programming Running time = ( V × n ) = ( 2#bits in V× n ) • NP-Complete • Approximate Algorithm • In poly-time (n3/), solution can be found • that is perfect in iS vi V • (1+)as good as optimal wrt iS pi • Eg,  = .001, Time is 1000n3

5. No Approximate Knapsack • Subinstance: V’[0..V], i[0..n], knapsack(V’,i) = maximize iS pi • subject to S  {1..i} and iS vi V • Recurrence Relation • knapsack(V’,i) = max( knapsack(V’,i-1), knapsack(V’-vi,i-1)+pi ) Yes

6. same + pi same No Approximate Knapsack  V’[0..V], i[0..n], knapsack(V’,i) 1 0 2 V’-vi V’ V OptSol price 0 1 2 i-1 Yes i Take best of best. Our price? n

7. Approximate Knapsack  V’[0..V], i[0..n], knapsack(V’,i) 1 0 2 V’ V OptSol price 0 1 2 i-1 i Time = O(nV) n

8. Approximate Knapsack • Ingredients: (strange version) • Instances: The price P wanted from the knapsack. The volume and price of n objects<<v1,p1>,<v2,p2>,… ,<vn,pn>>. • Solutions: A set of objects with total value P. • i.e. iS pi≥ P • Cost of Solution: The total volume of objects in set. • i.e. iS vi • Goal: Minimize the volume needed to obtain this value P. into the knapsack.

9. No Approximate Knapsack • Subinstance: P’[0..P], i[0..n], knapsack’(P’,i) = minimize iS vi • subject to S  {1..i} and iS pi≥ P • Recurrence Relation • knapsack’(P’,i) = min( knapsack’(P’,i-1), knapsack’(P’-pi,i-1)+vi ) Yes

10. same + pi same No Approximate Knapsack  V’[0..V], i[0..n], knapsack’(P’,i) 1 0 2 P’-pi P’ P OptSol volume 0 1 2 i-1 Yes i Take best of best. Our volume? n

11. Approximate Knapsack Original problem knapsack(V,n) 1 0 2 P’ P OptSol volume 0 1 2 i Find largest pricenot using more than V volume Time = O(nP) n P = i pi

12. Approximate Knapsack • Dynamic Programming Running time = ( V × n ) = ( 2#bits in V× n ) • Poly time if size of knapsack is small • Exponential time if size is an arbitrary integer. • Strange Dynamic Programming Running time = ( P × n ) = ( 2#bits in P× n ) • Poly time if prices are small • Exponential time if prices are arbitrary integers.

13. Approximate Knapsack • Approximation Algorithm: • Given V, <<v1,p1>,<v2,p2>,… ,<vn,pn>>, &  • Let k be some # of bits chosen later • Let p’i = pi with low k bits zeroed • eg pi = 101101101011p’i = 1011011000002 k=4 • (Lost precision in prices, but that is ok) • Solve knapsack using the strange algorithm.

14. Approximate Knapsack Original problem knapsack(V,n) 1 0 2 P’ P OptSol volume 0 1 2 i Find largest pricenot using more than V volume n P = i pi

15. Approximate Knapsack Original problem knapsack(V,n) 1 0 2 P’ P OptSol volume 0 1 2 i Time Only every 2k th column matters = O(n P/2k) n

16. Approximate Knapsack pi • Let Palg = • be the price returned by our alg. iSalg pi • Let Popt = • be the price returned by our alg. iSopt • Approximation Algorithm: • Let Salg be the set of items selected by our alg. • Let Sopt be the set of items in the optimal sol. • Need Palg ≥ Popt (1-)

17. Approximate Knapsack Palg = pi iSalg ≥ p’i ≥ p’i ≥ (pi -2k) iSalg iSopt iSopt because rounded down because Salg is an optimal solutionfor the <vi,p’i> problem because rounded by at most 2k. = Popt – n 2k

18. Approximate Knapsack n2k = 1 - Popt Popt Popt Popt 2k = n = O( n2 P ) Popt n = O( n3) Time  Palg (need) ≥ Popt– n2k = 1 -  Time = O(n P/2k) P = i pi Popt ≥ P maxi pi ≥ done