1. Finding Large Set Covers Faster via the Representation Method
Jesper Nederlof
Technical University Eindhoven

2. Outline Introduction Representation Method
Problem Statements
Context
Result Statements
Representation Method
Representation Method for Set Cover
Concluding Remarks

3. Problem Statements
Set Cover (SC): Given non-empty 𝑆 1 ,…, 𝑆 𝑚 ⊆[𝑛], find smallest 𝑋⊆[𝑚] such that 𝑖∈X 𝑆 𝑖 = 𝑛
We refer to such 𝑋 as a Set Cover
Set Partition (SP): 𝑆 𝑖 ∩ 𝑆 𝑗 =∅ for all distinct 𝑖,𝑗∈𝑋
Linear Sat (LS): Given 𝑡∈ℤ, A∈ ℤ 2 𝑛×𝑚 , 𝑏∈ ℤ 2 𝑛 , 𝜔∈ ℕ 𝑚 , find 𝑥 ∈ℤ 2 𝑚 s.t. 𝐴𝑥≡𝑏 and 𝜔,𝑥 ≤𝑡
By a standard reduction (with b=1), 𝑂 ∗ (𝑐 𝑚 ) for LS implies 𝑂 ∗ ( 𝑐 𝑚 ) for SP
𝑂 ∗ () omits factors polynomial in input size

4. Set Cover versus CNF-SAT
CNF-SAT is the most studied problem in exact algo’s.
Unclear relation with Set Cover

5. Set Cover versus CNF-SAT
CNF-SAT is the most studied problem in exact algo’s.
Unclear relation with Set Cover
Set Cover with n elements, m sets
CNF-SAT with n variables, m clauses

6. Set Cover versus CNF-SAT
CNF-SAT is the most studied problem in exact algo’s.
Unclear relation with Set Cover
Set Cover with n elements, m sets
CNF-SAT with n variables, m clauses
Hypergraph problem

7. Set Cover versus CNF-SAT
CNF-SAT is the most studied problem in exact algo’s.
Unclear relation with Set Cover
Set Cover with n elements, m sets
CNF-SAT with n variables, m clauses
Hypergraph problem
Easy 𝑂 ∗ ( 2 𝑛 ) time DP algorithm
Easy 𝑂 ∗ ( 2 𝑛 ) enumeration
For 𝑖≤𝑚, 𝑋⊆[𝑛] define 𝑇 𝑖,𝑋
as the smallest number of sets
from 𝑆 1 ,…, 𝑆 𝑖 needed to cover 𝑋.

8. Set Cover versus CNF-SAT
CNF-SAT is the most studied problem in exact algo’s.
Unclear relation with Set Cover
Set Cover with n elements, m sets
CNF-SAT with n variables, m clauses
Hypergraph problem
Easy 𝑂 ∗ ( 2 𝑛 ) time DP algorithm
Easy 𝑂 ∗ ( 2 𝑛 ) enumeration

9. Set Cover versus CNF-SAT
CNF-SAT is the most studied problem in exact algo’s.
Unclear relation with Set Cover
Set Cover with n elements, m sets
CNF-SAT with n variables, m clauses
Hypergraph problem
Easy 𝑂 ∗ ( 2 𝑛 ) time DP algorithm
Easy 𝑂 ∗ ( 2 𝑛 ) enumeration
Can do in 𝑂 ∗ 𝑛 ?
Can do in 𝑂 ∗ 𝑛 ? (SETH)

10. Set Cover versus CNF-SAT
CNF-SAT is the most studied problem in exact algo’s.
Unclear relation with Set Cover
Set Cover with n elements, m sets
CNF-SAT with n variables, m clauses
Hypergraph problem
Easy 𝑂 ∗ ( 2 𝑛 ) time DP algorithm
Easy 𝑂 ∗ ( 2 𝑛 ) enumeration
Can do in 𝑂 ∗ 𝑛 ?
Can do in 𝑂 ∗ 𝑛 ? (SETH)
𝑂 ∗ ( 2 1−Ω(1/𝑑) 𝑛 ) if set sizes ≤𝑑 [K’09]
𝑂 ∗ ( 2 1−Ω(1/𝑑) 𝑛 ) if clause sizes ≤𝑑

11. Set Cover versus CNF-SAT
CNF-SAT is the most studied problem in exact algo’s.
Unclear relation with Set Cover
Set Cover with n elements, m sets
CNF-SAT with n variables, m clauses
Hypergraph problem
Easy 𝑂 ∗ ( 2 𝑛 ) time DP algorithm
Easy 𝑂 ∗ ( 2 𝑛 ) enumeration
Can do in 𝑂 ∗ 𝑛 ?
Can do in 𝑂 ∗ 𝑛 ? (SETH)
𝑂 ∗ ( 2 1−Ω(1/𝑑) 𝑛 ) if set sizes ≤𝑑 [K’09]
𝑂 ∗ ( 2 1−Ω(1/𝑑) 𝑛 ) if clause sizes ≤𝑑
𝑂 ∗ ( 2 1−1/ Ω(log 𝑚/𝑛 ) 𝑛 ) [cor. K’09]
𝑂 ∗ ( 2 1−1/ Ω(log 𝑚/𝑛 ) 𝑛 )

12. Set Cover versus CNF-SAT
CNF-SAT is the most studied problem in exact algo’s.
Unclear relation with Set Cover
Set Cover with n elements, m sets
CNF-SAT with n variables, m clauses
Hypergraph problem
Easy 𝑂 ∗ ( 2 𝑛 ) time DP algorithm
Easy 𝑂 ∗ ( 2 𝑛 ) enumeration
Can do in 𝑂 ∗ 𝑛 ?
Can do in 𝑂 ∗ 𝑛 ? (SETH)
𝑂 ∗ ( 2 1−Ω(1/𝑑) 𝑛 ) if set sizes ≤𝑑 [K’09]
𝑂 ∗ ( 2 1−Ω(1/𝑑) 𝑛 ) if clause sizes ≤𝑑
𝑂 ∗ ( 2 1−1/ Ω(log 𝑚/𝑛 ) 𝑛 ) [cor. K’09]
𝑂 ∗ ( 2 1−1/ Ω(log 𝑚/𝑛 ) 𝑛 )
Solving SC in 𝑂 ∗ ( 1.99 𝑚 ) (roughly) equivalent to !SETH [CDLMNOPSW’12]

13. Set Cover versus CNF-SAT
CNF-SAT is the most studied problem in exact algo’s.
Unclear relation with Set Cover
Set Cover with n elements, m sets
CNF-SAT with n variables, m clauses
Hypergraph problem
Easy 𝑂 ∗ ( 2 𝑛 ) time DP algorithm
Easy 𝑂 ∗ ( 2 𝑛 ) enumeration
Can do in 𝑂 ∗ 𝑛 ?
Can do in 𝑂 ∗ 𝑛 ? (SETH)
𝑂 ∗ ( 2 1−Ω(1/𝑑) 𝑛 ) if set sizes ≤𝑑 [K’09]
𝑂 ∗ ( 2 1−Ω(1/𝑑) 𝑛 ) if clause sizes ≤𝑑
𝑂 ∗ ( 2 1−1/ Ω(log 𝑚/𝑛 ) 𝑛 ) [cor. K’09]
𝑂 ∗ ( 2 1−1/ Ω(log 𝑚/𝑛 ) 𝑛 )
Solving SC in 𝑂 ∗ ( 1.99 𝑚 ) (roughly) equivalent to !SETH [CDLMNOPSW’12]
Counting #SC’s mod 2 in 𝑂 ∗ ( 1.99 𝑛 ) SETH hard [CDLMNOPSW’12]

14. Set Cover versus CNF-SAT
CNF-SAT is the most studied problem in exact algo’s.
Unclear relation with Set Cover
Set Cover with n elements, m sets
CNF-SAT with n variables, m clauses
Hypergraph problem
Easy 𝑂 ∗ ( 2 𝑛 ) time DP algorithm
Easy 𝑂 ∗ ( 2 𝑛 ) enumeration
Can do in 𝑂 ∗ 𝑛 ?
Can do in 𝑂 ∗ 𝑛 ? (SETH)
𝑂 ∗ ( 2 1−Ω(1/𝑑) 𝑛 ) if set sizes ≤𝑑 [K’09]
𝑂 ∗ ( 2 1−Ω(1/𝑑) 𝑛 ) if clause sizes ≤𝑑
𝑂 ∗ ( 2 1−1/ Ω(log 𝑚/𝑛 ) 𝑛 ) [cor. K’09]
𝑂 ∗ ( 2 1−1/ Ω(log 𝑚/𝑛 ) 𝑛 )
Solving SC in 𝑂 ∗ ( 1.99 𝑚 ) (roughly) equivalent to !SETH [CDLMNOPSW’12]
Counting #SC’s mod 2 in 𝑂 ∗ ( 1.99 𝑛 ) SETH hard [CDLMNOPSW’12]
Only handful of papers
Very well studied
Dream: Perhaps clever DP techniques can break SETH!?

15. Result Statements Main Theorem
There is a Monte Carlo algorithm that determines if there is a Set Cover of size at most 𝜎𝑛 in 𝑂 ∗ −Ω( 𝜎 4 ) 𝑛 time.
Main Theorem
Generalizes set of instances solvable in 𝑂 ∗ ( 2−𝜖 𝑛 ) time
If set sizes ≤𝑑, we may assume 𝜎≥1/𝑑
Our techniques are different than from [K’09], but give a worse dependence in this special case
The star in the 𝑂 ∗ () can be omitted under some conditions
Gives a speedup even if 𝑚 is exponential in 𝑛

16. Result Statements Main Theorem
There is a Monte Carlo algorithm that determines if there is a Set Cover of size at most 𝜎𝑛 in 𝑂 ∗ −Ω( 𝜎 4 ) 𝑛 time.
Obtained by applying the representation method in a novel way
Additionally, we show rep. method also useful for SP if 𝑚≤𝑛:
There is an 𝑂 ∗ ( 𝑚 ) time Monte Carlo algorithm for Linear Sat.
Theorem
Previous fastest algorithm runs in 𝑂 ∗ ( 2 𝑚/2 ) time
For SP, running time comes close to previous branching algo’s:
DP’02 (Drori & Peleg): 𝑂 ∗ ( 𝑚 ) time

17. Representation Method [HJ’10]
Say we seek for a set in 𝒮⊆ [𝑚] 𝑠 as follows:
List [𝑚] 𝑠/2 , check if 𝐴∪B∈𝒮 for all pairs 𝐴,𝐵 listed
𝑆∈𝒮 gives rise to [𝑠] 𝑠/2 ≈ 2 𝑠 pairs (representatives)
Thus we can restrict our search in various ways
Only consider
needle

18. Representation Method [HJ’10]
Say we seek for a set in 𝒮⊆ [𝑚] 𝑠 as follows:
List [𝑚] 𝑠/2 , check if 𝐴∪B∈𝒮 for all pairs 𝐴,𝐵 listed
𝑆∈𝒮 gives rise to [𝑠] 𝑠/2 ≈ 2 𝑠 pairs (representatives)
Thus we can restrict our search in various ways
Seems bad idea: 𝑚 𝑠/2 2 / 2 𝑠 ≥ 𝑚 𝑠
Suppose 𝐴 determines 𝐵: about 𝑚 𝑠/2 / 2 𝑠 time!
Indeed this has been useful particularly for the `Meet in the Middle’ attack (where indeed 𝐴 determines 𝐵)
We show the method is also useful elsewhere

19. Representation Method for Set Cover

20. Representation Method for Set Cover
Partition
Representation Method for Set Cover
Let 𝒮⊆ [𝑚] 𝑠 be the set of all set partitions (𝑠=𝜎𝑛)
Assume 𝑆 𝑖 ≤𝜖𝑛 for a small 𝜖>0 (depending on 𝜎)
Suppose 𝑋∈𝒮, partition 𝑋 1 ∪ 𝑋 2 =𝑋
Let 𝑁 𝑋 𝑗 = 𝑖∈ 𝑋 𝑗 𝑆 𝑖 , then 𝑁 𝑋 1 ,𝑁( 𝑋 2 ) partitions 𝑛
Claim: Ω(2 𝜎𝑛 ) subsets 𝑌⊆[𝑛] s.t. 𝑌 =(0.5± 𝑂 𝜖 )𝑛, and 𝑌=𝑁 𝑋 1 = 𝑛 ∖𝑁( 𝑋 2 ), for some partition 𝑋 1 ,𝑋 2 of 𝑋.
There are 2 𝜎𝑛 pairs 𝑋 1 , 𝑋 2
𝑁( 𝑋 1 ),𝑁( 𝑋 2 ) determine 𝑋 1 , 𝑋 2 since 𝑁 𝑁( 𝑋 𝑖 ) =𝑋 𝑖
𝑌 concentrated around n/2 by Hoeffding and 𝑆 𝑖 ≤𝜖𝑛
Pf.
Assumes 𝑋 𝑖 has no empty sets
We refer to a Y as in the claim as a witness halve
Sets 𝑆 1 ,…, 𝑆 𝑚 X
𝑿 𝟏
𝑿 𝟐
𝑵( 𝑿 𝟏 )
𝑵( 𝑿 𝟐 )
Elements [𝑛] 𝒀=

21. Representation Method for Set Cover
Partition
Representation Method for Set Cover
So if 𝑋 exists, for some l= 0.5± 𝑂 𝜖 𝑛, there exist Ω 2 𝜎𝑛 /𝑛 witness halves of size l
We can guess l and sample 𝒲⊆ 𝑛 𝑙 , by including every 𝑙-sized set with probability 2 −𝜎𝑛
For some 𝑙, 𝒲 contains a witness halve with good probability
Given 𝒲⊆ 𝑛 𝑙 , we need to determine whether there exist 𝑌∈𝒲 and 𝑋 1 , 𝑋 2 ∈ [𝑚] 𝑠/2 s.t. 𝑁 𝑋 1 =𝑌,𝑁 𝑋 2 = 𝑛 ∖𝑌
Denote ↓𝒲≔ {𝑃:∃𝑄∈𝒲∧𝑃⊆𝑄}
↑𝒲≔ {𝑃:∃𝑄∈𝒲∧𝑃⊇𝑄}
This can be done with standard DP in time O ∗ ( ↑𝒲 + ↓𝒲 )

22. Representation Method for Set Cover
Partition
Representation Method for Set Cover
Choosing 𝜖≈ 𝜎 4 , ↑𝒲 + ↓𝒲 is 𝑂 ∗ −Ω( 𝜎 4 ) 𝑛
We want to stay close to the middle layer!
The claimed running time follows
The analysis might be improvable to have better dependence on 𝜎, but for significant improvements additional algorithmic ideas seem needed
The algorithm works for Set Cover using a standard reduction

23. Concluding Remarks + open questions
𝑂 ∗ −Ω( 𝜎 4 ) 𝑛 time for set covers of size 𝜎𝑛
Additive 1-apx for chromatic nr. in same time
Set Cover with set sizes adding up to 1+𝜖 𝑛 faster?
Witness halves might be unbalanced
Subset Sum with target 𝑡 in time 𝑂 ∗ ( 𝑡 1−𝜖 )?
Lower bound assuming SETH?
Perhaps in a special case?
More applications of representation method?
Exact complexity of Set Cover still open
Can reduce general case to special case solved in this work?
Thanks for listening!