Speaker: Yoni Rozenshein Instructor: Prof. Zeev Nutov

Talk outline Problem description and known approximations Interpretation in graphs (Independent Set Problem) The greedy local search method Approximation ratio Polynomial time implementation 1

Weighted k -set packing Given a collection of weighted sets of size at most k, find a maximum weight collection of disjoint sets Example with k = 3 : SetWeight { 4, 5 }16 { 2, 3, 5 }6 { 3, 4, 5 }12 { 1, 5 }5 { 4 }14 { 4, 5 } { 2, 3, 5 } { 4 } { 3, 4, 5 } { 1, 5 } { 4 } Set packing 2 Weight

Hardness of the problem For k = 2 we get the Maximum (Weighted) Matching Problem, which admits a polynomial time algorithm [Edmonds 1965] The problem is NP-hard for k ≥ 3 [Karp 1972] Reducible to k -SAT k = 3 : 3 -Dimensional Matching 3

Approximation algorithms OPT(I) – weight of the best solution (on instance I) ALG(I) – weight of a given algorithm’s solution Algorithm’s approximation ratio on instance I: Algorithm’s approximation ratio: We seek an algorithm that minimizes the ratio 4

The greedy algorithm Repeatedly choose maximum-weight set S and delete from the family all sets that intersect S Very fast; performance ratio k : SetWeight { 1, 2, 3, …, k }1 + ε { 1 }1 { 2 }1 …… { k }1 Greedy choice Intersects... AA, A 1, A 2, …, A n BB, B 1, B 2, …, B m …… ZZ, Z 1, Z 2, …, Z l 5

Local search heuristic Replace a subset of the solution with a better subset Repeat until “locally optimal” How are the improvements chosen? Polynomial running time? Performance ratio k – 1 + ε [Arkin and Hassin 1997] Can we improve on these ratios? 6

Interpretation in graphs The sets’ intersection graph: Nodes correspond to sets Edges correspond to sets sharing elements Set packing is a particular case of Maximum Weight Independent Set in intersection graphs What characterizes intersection graphs arising from k -set packing instances? SetWeight A = { 4, 5 }16 B = { 2, 3, 5 }6 C = { 3, 4, 5 }12 D = { 1, 5 }5 E = { 4 }14 A B E D C 7

The characterization The graph is k + 1 -claw free From now on, we consider the “Independent Set Problem in k + 1 -claw free graphs” 1 2 k + 1 Example with k = 3 8 k + 1-claw Proof: a.At most k elements in parent node b.At least one in each child node c.Pigeon-hole principle Proof: a.At most k elements in parent node b.At least one in each child node c.Pigeon-hole principle

Approach: Greedy local search Previous algorithms combined How are improvements chosen? Polynomial running time? G REEDY -L OCAL -S EARCH (G) I ← G REEDY (G) while I is not locally optimal do I’ ← local improvement of I I ← I’ end while output I G REEDY -L OCAL -S EARCH (G) I ← G REEDY (G) while I is not locally optimal do I’ ← local improvement of I I ← I’ end while output I 9

Local improvement scheme Improvement: Pick some v I, add some of v’s neighbors, and delete any interfering nodes The payoff factor is 10 A B E D C Local Improvement Pick B Add D Delete A, B A B E D C Pick B Add D Delete A, B Pick B Add D Delete A, B A B E D C

First algorithm: A NY I MP Polynomial running time? For now, we will analyze the approximation ratio only A NY I MP (G, α) I ← G REEDY (G) while I is not locally optimal do I’ ← any improvement of I with payoff factor ≥ α I ← I’ end while output I A NY I MP (G, α) I ← G REEDY (G) while I is not locally optimal do I’ ← any improvement of I with payoff factor ≥ α I ← I’ end while output I 11

Analysis of the approximation ratio Projection: “Distance” between I and OPT proj(I) and w(I) maintain equilibrium 12 proj(I)w(I)w(I)Φ(I) ≈ x ⋅ proj(I) + y ⋅ w(I) min Φ(I)

Projection A B E D C A B E D C 13

Projection properties Equilibrium property: Local optimum property: Result: (for any intermediate I) (for the final I) 14

Second algorithm: B EST I MP B EST I MP (G) I ← G REEDY (G) while I is not locally optimal do I’ ← an improvement of I with the highest payoff factor I ← I’ end while output I B EST I MP (G) I ← G REEDY (G) while I is not locally optimal do I’ ← an improvement of I with the highest payoff factor I ← I’ end while output I 15

Potential function In A NY I MP ’s potential*, d was constant (I i, d i ): Sub-sequence of local improvements Largest such that d i (payoff factor) is descending: 2, 5, 3, 2, 4, 1.5, 3, 2, 1.2, 4, 3, * d 1 = 2d 2 = 1.5d 3 = 1.2d 4 = 1.1 (Greedy) I 0 I1I1 I2I2 I3I3 I 4 (Final)

Potential properties Equilibrium property: Weight evaluation property: Result: (for i = 1, 2, …) (for the final I ) 17

Running time analysis Reminder: Individual steps run in polynomial time Number of improvements? G REEDY -L OCAL -S EARCH (G) I ← G REEDY (G) while I is not locally optimal do I’ ← local improvement of I I ← I’ end while output I G REEDY -L OCAL -S EARCH (G) I ← G REEDY (G) while I is not locally optimal do I’ ← local improvement of I I ← I’ end while output I 18

Polynomial-time approximation scheme Given an algorithm with approximation ratio ρ, produce a polynomial time algorithm with approximation ratio ρ + ε Well-known example: Knapsack The running time may depend strongly on ε (For example: Polynomial in 1/ ε ) Our greedy local search algorithm already runs in pseudo-polynomial time 19

PTAS: Weight scaling Each improvement increases weight by 1 Number of improvements at most: 20

Weight scaling – cont’d Special case: If we get within r of OPT'', we get within r(1 – ε) of OPT 21

22 Questions?