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

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

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.

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

Corollaries

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

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)

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) )

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.

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

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.

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)).

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?

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(1.66667n) minimal 3-Hitting Sets.

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

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).

Thank You!