1. Appropriate algorithms for Generalized Maximum Coverage
丁文韬

2. Definition E: A set of elements B: A set of bins
Put some elements into some bins
E: A set of elements
B: A set of bins

3. Definition
L: a budget
Maximize ∑𝑃 under the condition ∑𝑊≤𝐿
P(b,e)
P/W: for each pair (b, e), there is a profit and a weight.
P(b,e)
W(b,e)
W(b,e)
W(b,e)
W’: for each bin b, there is a weight of using b.
W’(b)
W’(b)
W’(b)

4. Definition Instance: Objective: 𝐿,𝐵, 𝐸,𝑃,𝑊, 𝑊 ′
𝐴 𝑏𝑢𝑑𝑔𝑒𝑡 𝐿∈𝑁, 𝑠𝑒𝑡 𝐵, 𝑠𝑒𝑡 𝐸, 𝑓𝑢𝑛𝑡𝑖𝑜𝑛 𝑃,𝑊 𝑖𝑠 𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑜𝑛 𝐵×𝑊, ∀𝑏∈ 𝐵, 𝑒∈𝐸, 𝑃 𝑏,𝑒 >0, 𝑊 𝑏,𝑒 ≥0, 𝑓𝑢𝑛𝑡𝑖𝑜𝑛 𝑊 ′ 𝑖𝑠 𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑜𝑛 𝐵.
Objective:
𝑆= 𝛽,𝜂,𝑓 , 𝛽⊆ 𝐵, 𝜂 ⊆ 𝐸, 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑓:𝜂 →𝛽.
𝑆 𝑠𝑎𝑡𝑖𝑠𝑓𝑦 𝑊 𝑆 = 𝑏∈𝛽 𝑊‘ 𝑏 + 𝑒∈𝜂 𝑊 𝑓 𝑒 ,𝑒 ≤ 𝐿.
𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒 𝑃 𝑆 = 𝑒∈𝜂 𝑃 𝑓 𝑒 ,𝑒

5. Previous Results
Cohen R, Katzir L. The generalized maximum coverage problem[J]. Information Processing Letters, 2008, 108(1):
1− 1 𝐾 −1 1− 𝑒 − 1 α −1 −𝑎𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑖𝑜𝑛
K: maximal size of enumerated sets
α: constant for FPTAS of Knapsack
𝑒 𝑒−1 +ϵ 𝑓𝑜𝑟 𝑏𝑖𝑔 𝑒𝑛𝑜𝑢𝑔ℎ 𝐾, 𝑠𝑚𝑎𝑙𝑙 𝑒𝑛𝑜𝑢𝑔ℎ α

6. Previous Results Problem Restrictions Approx. MC* 𝑊 ′ 𝑏 =1 𝑃(𝑏,𝑒)=1
𝑊 ′ 𝑏 =1
𝑃(𝑏,𝑒)=1
e 𝑒−1
BMC*
MCKP
𝑊 ′ 𝑏 =0
FPTAS
GMC /
e 𝑒−1 +ϵ MC KP
BMC
MCKP
GMC
(*:Elements can be selected only for a subset of the bins )

7. Algorithm(1) 1 while (∃𝑒 𝑛𝑜𝑡 𝑖𝑛 𝑆,add 𝑒 to S can increase 𝑃 )
2 find 𝑏,𝐸 , 𝑊 𝑏,𝐸 +𝑊 𝑆 ≤𝐿, 𝑤𝑖𝑡ℎ max⁡( 𝑃 𝑏,𝐸 𝑊 𝑏,𝐸 ) ,
add (b, E) to S
3 while (∃ 𝑏,𝐸 , ∀𝑒 𝑊 𝑏,𝑒 ≤𝑊 𝑓 𝑒 ,𝑒 , 𝑃 𝑏,𝑒 >0)
4 find 𝑏,𝐸 𝑤𝑖𝑡ℎ max⁡( 𝑃(𝑏,𝐸)) , add (b, E) to S
5 return S
(2) can be done with a DP alg. of knapsack problem.

8. Algorithm(2) Fix a bin set B
𝑊 = 𝑊(𝑏,𝑒) , a multiple choices knapsack problem
MCKP is FPTAS

9. Algorithm Enumerate a small feasible solution 𝑆 𝐸 for Alg(1)
Enumerate a small bin set 𝐵 𝐸 for Alg(2)
Find balance between approximation rate and complexity
1− 1 𝐾 1− 𝑒 − 1 α −𝑎𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑖𝑜𝑛
K: maximal size of enumerated sets
α: constant for FPTAS of Knapsack
𝑒 𝑒−1 +ϵ 𝑓𝑜𝑟 𝑏𝑖𝑔 𝑒𝑛𝑜𝑢𝑔ℎ 𝐾, 𝑠𝑚𝑎𝑙𝑙 𝑒𝑛𝑜𝑢𝑔ℎ α

10. Experiment results 3 Implements:
Prog1 ( Alg2(MCKP) with |B|<=3, L’<=|E|^2 )
O ( B^3 * E^3 )
Prog2 ( Alg1(Greedy) with φ , L’<=|E|^3 )
O ( B^2 * E^4 )
Prog3 ( Alg1(Greedy) with |S.E|<=1, |S.B|<=1, , L’<=|E|^3 )
O ( B^3 * E^5 )

11. Experiment results Small test cases(*10):
3<=|B|<=6, 10<=|E|<=30, |E|^2 <= L <=|E|^4
β(Prog3) < β(Prog2) < β(Prog1) for all cases
Algorithm
Approx. Rate β
Prog1(KP_B3_L2)
1.05~1.2
Prog2(GR_S0_L3)
1.1~1.2
Prog3(GR_S1_L2)
1.02~1.1

12. Experiment results Larger test cases(*15):
4<=|B|<=10, 50<=|E|<=100, |E|^2 <= L <=|E|^4
Depends on |B|
B≥6 (2K) β(Prog1) > β(Prog2) for most cases
B< β(Prog1) < β(Prog2) for most cases
Algorithm
Approx. Rate β
Time
Prog1(KP_B3_L2)
1.1~1.3
Several seconds
Prog2(GR_S0_L3)
Prog3(GR_S1_L2)
Too slow on my PC

13. Experiment results Influenced by |B| and the distribution of W, P, W’
The approximation rate of Knapsack is not so important

14. Possible Improvement Heuristic method Useless states in Knapsack
Repeated enumeration

15. Q & A
Thanks for listening