1. Exact Algorithms via Monotone Local Search
Fedor Fomin, Serge Gaspers, Daniel Lokshtanov, Saket Saurabh

2. Satisfiability U = variables F = satisfying assignments
Subset Problems
Input: Universe U (of size n), implicit family F of subsets of U. Question: Is there a set S ∈ F?
Feedback Vertex Set U = vertices of graph G F = vertex sets S of size ≤ k such that G – S is acyclic
3-Coloring U = vertices of graph G F = independent vertex sets S such that G – S is bipartite
Satisfiability U = variables F = satisfying assignments
solution

3. Algorithms for Subset Problems
Exponential Time Algorithms
O(cn) for smallest possible c < 2.
This paper
Yes – if you can remove solution vertices. ?
Parameterized Algorithms
Find solution of size at most k (if exists)
in time ckpoly(n) for smallest possible c.

4. Main Result
Theorem:1 If a subset problem can be solved in time ckpoly(n) then it can be solved in time O((2-1/c)n+o(n)).
1: terms and conditions apply

5. Corollaries

6. Algorithm Assume: ckpoly(n) time algorithm
Universe has size n, guess solution size k. Pick integer t ≤ k cleverly. Pick random set S of size t, put it in solution. Remove S from instance. Look for solution of size k - t in U – S in time ck – tnO(1)
Assume: ckpoly(n) time algorithm
Obtain: O((2-1/c)n+o(n)) time algorithm

7. 𝑘 𝑡 𝑛 𝑡 Analysis Running time: ck – tnO(1) No false positives
What is prob. of finding solution if there is one?
𝑘 𝑡 𝑛 𝑡
#ways to pick a subset of size
t from the solution (of size k)
#ways to pick a subset of size
t from the universe (of size n)

8. For constant success probability repeat 𝑛 𝑡 𝑘 𝑡 times
Analysis, part II
For constant success probability repeat 𝑛 𝑡 𝑘 𝑡 times
𝑛 𝑡 𝑘 𝑡 𝑐 𝑘−𝑡 𝑛 𝑂(1)
max 𝑘≤𝑛
min 𝑡≤𝑘
Total time:
≤𝑂( (2− 1 𝑐 ) 𝑛+𝑜(𝑛) )

9. Derandomization
A (n, k, t)-set inclusion family Q is a family of sets of size t over a universe U of size n, such that for every subset X ⊆ U of size k, there is a set S ∈ Q such that S ⊆ X.

10. Derandomization U
Sets of size k
Sets of size t Q ∅

11. Derandomization
Theorem: There exist a (n, k, t)-set inclusion families of size ≈ 𝑛 𝑡 𝑘 𝑡 . These can be constructed in output linear (ish) time.
Full de-randomization of main theorem.

12. Applications to Listing & Combinatorial Upper Bounds
«Theorem» If a family F contains at most ckpoly(n) sets of size k, then |F| ≤ O((2-1/c)n+o(n)).

13. 3-Hitting Sets can there be?
Minimal 3-Hitting Sets
Let U (|U| = n) be a universe and R = {R1… Rm} be a family of sets, |Ri| ≤ 3. A set S ⊆ U is a hitting set if Ri ∩ S ≠∅ for all Ri ∈ R A hitting set S is minimal if no proper subset of S is also a hitting set.
How many minimal
3-Hitting Sets can there be?

14. Minimal 3-Hitting Sets
Observation: There are at most 3k minimal 3-Hitting Sets of size at most k. Corollary: at most O((2-1/3)n+o(n)) = O( n) minimal 3-Hitting Sets.

15. Open Problems
How to build on top of this? Combine with measure & conquer? Can we make this work for Dominating Set?

16. Exponential Time Algorithms Parameterized Algorithms
Summary
Exponential Time Algorithms
Find solution in time O((2-1/c)n).
Parameterized Algorithms
Find solution of size at most k (if exists)
in time ckpoly(n).

17. Thank You!