Super-polynomial time approximability of inapproximable problems - PowerPoint PPT Presentation

Introduction Min Independent Dominating Set Max Induced Path/Forest/Tree Max Minimal Vertex Cover Super-polynomial time approximability of inapproximable problems ´ Edouard Bonnet, Michael Lampis, Vangelis Paschos SZTAKI, Hungarian Academy of Sciences LAMSADE UniversitÃľ Paris Dauphine STACS, Feb 18, 2016

Introduction Min Independent Dominating Set Max Induced Path/Forest/Tree Max Minimal Vertex Cover time exponent n Optimal under ETH? n / ρ − 1 ( r ) approximation ratio r ρ ( n ) Consider Time-Approximation Trade-offs for Clique.

Introduction Min Independent Dominating Set Max Induced Path/Forest/Tree Max Minimal Vertex Cover time exponent n Optimal under ETH? n / ρ − 1 ( r ) approximation ratio ρ ( n ) r Clique is ˜ Θ( n ) -approximable in P and optimally solvable in λ n .

Introduction Min Independent Dominating Set Max Induced Path/Forest/Tree Max Minimal Vertex Cover time exponent n Optimal under ETH? n / ρ − 1 ( r ) approximation ratio r ρ ( n ) n / r . Clique is r -approximable in time 2

Introduction Min Independent Dominating Set Max Induced Path/Forest/Tree Max Minimal Vertex Cover time exponent n Optimal under ETH? n / ρ − 1 ( r ) approximation ratio r ρ ( n ) Is this the correct algorithm? For every r ?

Introduction Min Independent Dominating Set Max Induced Path/Forest/Tree Max Minimal Vertex Cover Minimization subset problems I , n

Introduction Min Independent Dominating Set Max Induced Path/Forest/Tree Max Minimal Vertex Cover Minimization subset problems I , n � n / r

Introduction Min Independent Dominating Set Max Induced Path/Forest/Tree Max Minimal Vertex Cover Minimization subset problems I , n � n / r

Introduction Min Independent Dominating Set Max Induced Path/Forest/Tree Max Minimal Vertex Cover Minimization subset problems I , n � n / r

Introduction Min Independent Dominating Set Max Induced Path/Forest/Tree Max Minimal Vertex Cover Minimization subset problems I , n � n / r ◮ If a solution is found, it is an optimal solution.

Introduction Min Independent Dominating Set Max Induced Path/Forest/Tree Max Minimal Vertex Cover Minimization subset problems I , n � n / r ◮ If a solution is found, it is an optimal solution. ◮ If not, any feasible solution is an r -approximation.

Introduction Min Independent Dominating Set Max Induced Path/Forest/Tree Max Minimal Vertex Cover Weakly monotone maximization subset problems I , n

Introduction Min Independent Dominating Set Max Induced Path/Forest/Tree Max Minimal Vertex Cover Weakly monotone maximization subset problems I , n � n / r

Introduction Min Independent Dominating Set Max Induced Path/Forest/Tree Max Minimal Vertex Cover Weakly monotone maximization subset problems I , n � n / r

Introduction Min Independent Dominating Set Max Induced Path/Forest/Tree Max Minimal Vertex Cover Weakly monotone maximization subset problems I , n � n / r

Introduction Min Independent Dominating Set Max Induced Path/Forest/Tree Max Minimal Vertex Cover Weakly monotone maximization subset problems I , n � n / r ◮ If a solution is found, it is an r -approximation.

Introduction Min Independent Dominating Set Max Induced Path/Forest/Tree Max Minimal Vertex Cover Weakly monotone maximization subset problems I , n � n / r ◮ If a solution is found, it is an r -approximation. ◮ If not, there is no feasible solution.

Introduction Min Independent Dominating Set Max Induced Path/Forest/Tree Max Minimal Vertex Cover The r -approximation takes time � n n log ( er ) / r ) . O ∗ ( � ) = O ∗ (( en n / r ) = O ∗ (( er ) n / r ) = O ∗ ( 2 n / r ) n / r

Introduction Min Independent Dominating Set Max Induced Path/Forest/Tree Max Minimal Vertex Cover The r -approximation takes time � n n log ( er ) / r ) . O ∗ ( � ) = O ∗ (( en n / r ) = O ∗ (( er ) n / r ) = O ∗ ( 2 n / r ) n / r Can we improve this time to O ∗ ( 2 n / r ) ?

Introduction Min Independent Dominating Set Max Induced Path/Forest/Tree Max Minimal Vertex Cover The r -approximation takes time � n n log ( er ) / r ) . O ∗ ( � ) = O ∗ (( en n / r ) = O ∗ (( er ) n / r ) = O ∗ ( 2 n / r ) n / r Can we improve this time to O ∗ ( 2 n / r ) ? ◮ In this talk we don’t care! (?? sort of) ◮ Bottom line: r n / r is a Base-line Trade-off . ◮ When can we do better? ◮ When is it optimal?

Introduction Min Independent Dominating Set Max Induced Path/Forest/Tree Max Minimal Vertex Cover Min Asymmetric Traveling Salesman Problem

Introduction Min Independent Dominating Set Max Induced Path/Forest/Tree Max Minimal Vertex Cover Min ATSP in polytime ◮ O ( log n ) -approximation [FGM ’82]. log n ◮ O ( log log n ) -approximation [AGMOS ’10]. Our goal: Theorem ∀ r � n, Min ATSP is log r-approximable in time O ∗ ( 2 n / r ) .

Introduction Min Independent Dominating Set Max Induced Path/Forest/Tree Max Minimal Vertex Cover A circuit cover of minimum length can be found in polytime.

Introduction Min Independent Dominating Set Max Induced Path/Forest/Tree Max Minimal Vertex Cover Pick any vertex in each cycle and recurse.

Introduction Min Independent Dominating Set Max Induced Path/Forest/Tree Max Minimal Vertex Cover This can only decrease the total length (triangle inequality).

Introduction Min Independent Dominating Set Max Induced Path/Forest/Tree Max Minimal Vertex Cover n / r . ratio = recursion depth: log n for polytime; log r for time 2

Introduction Min Independent Dominating Set Max Induced Path/Forest/Tree Max Minimal Vertex Cover Is this optimal? NO! Is this close to optimal? No idea!

Introduction Min Independent Dominating Set Max Induced Path/Forest/Tree Max Minimal Vertex Cover Inapproximability in super-polynomial time (Randomized) Exponential Time Hypothesis: There is no (randomized) 2 o ( n ) -time algorithm solving 3-SAT. Theorem (CLN ’13) 1 / 2 − ε , Under randomized ETH, ∀ ε > 0 , for all sufficiently big r < n n 1 − ε / r 1 + ε . Max Independent Set is not r-approximable in time 2

Introduction Min Independent Dominating Set Max Induced Path/Forest/Tree Max Minimal Vertex Cover Inapproximability in super-polynomial time (Randomized) Exponential Time Hypothesis: There is no (randomized) 2 o ( n ) -time algorithm solving 3-SAT. Theorem (CLN ’13) 1 / 2 − ε , Under randomized ETH, ∀ ε > 0 , for all sufficiently big r < n n 1 − ε / r 1 + ε . Max Independent Set is not r-approximable in time 2 SAT formula φ with N variables � graph G with r 1 + ε N 1 + ε vertices ◮ φ satisfiable ⇒ α ( G ) ≈ rN 1 + ε . ◮ φ unsatisfiable ⇒ α ( G ) ≈ r ε N 1 + ε .

Introduction Min Independent Dominating Set Max Induced Path/Forest/Tree Max Minimal Vertex Cover Inapproximability in super-polynomial time Goal: Assuming ETH, Π is not r -approximable in time 2 o ( n / f ( r ) )

Introduction Min Independent Dominating Set Max Induced Path/Forest/Tree Max Minimal Vertex Cover Inapproximability in super-polynomial time Goal: Assuming ETH, Π is not r -approximable in time 2 o ( n / f ( r ) ) SAT formula φ with N variables � I instance of Π s.t. ◮ |I| ≈ f ( r ) N ◮ φ satisfiable ⇒ val (Π) ≈ a ◮ φ unsatisfiable ⇒ val (Π) ≈ ra

Introduction Min Independent Dominating Set Max Induced Path/Forest/Tree Max Minimal Vertex Cover Min Independent Dominating Set

Introduction Min Independent Dominating Set Max Induced Path/Forest/Tree Max Minimal Vertex Cover Inapproximability in polytime [I ’91, H ’93] C 1 C 2 C 3 C 4 C 5 ¬ x 1 ¬ x 2 ¬ x 3 ¬ x 4 x 1 x 2 x 3 x 4 Satifiable CNF formula with N variables and CN clauses

Introduction Min Independent Dominating Set Max Induced Path/Forest/Tree Max Minimal Vertex Cover Inapproximability in polytime [I ’91, H ’93] C 1 C 2 C 3 C 4 C 5 ¬ x 1 ¬ x 2 ¬ x 3 ¬ x 4 x 1 x 2 x 3 x 4 MIDS of size N

Introduction Min Independent Dominating Set Max Induced Path/Forest/Tree Max Minimal Vertex Cover Inapproximability in polytime [I ’91, H ’93] C 1 C 2 C 3 C 4 C 5 ¬ x 1 ¬ x 2 ¬ x 3 ¬ x 4 x 1 x 2 x 3 x 4 Unsatifiable CNF formula with N variables and CN clauses

Introduction Min Independent Dominating Set Max Induced Path/Forest/Tree Max Minimal Vertex Cover Inapproximability in polytime [I ’91, H ’93] C 1 C 2 C 3 C 4 C 5 x 1 ¬ x 1 x 2 ¬ x 2 x 3 ¬ x 3 x 4 ¬ x 4 MIDS of size greater than rN

Introduction Min Independent Dominating Set Max Induced Path/Forest/Tree Max Minimal Vertex Cover Inapproximability in polytime [I ’91, H ’93] C 1 C 2 C 3 C 4 C 5 x 1 ¬ x 1 x 2 ¬ x 2 x 3 ¬ x 3 x 4 ¬ x 4 9998 Set r = N 9998 ≈ n 10000 � n 0 . 999 As n = 2 N + CrN 2 ≈ N 1000

Introduction Min Independent Dominating Set Max Induced Path/Forest/Tree Max Minimal Vertex Cover (In)approximability in subexponential time Our goal: Theorem Under ETH, ∀ ε > 0 , ∀ r � n, n 1 − ε / r 1 + ε ) . MIDS is not r-approximable in time O ∗ ( 2 n log ( er ) / r ) . almost matching the r -approximation in time O ∗ ( 2