1. From Approximative Kernelization to High Fidelity Reductions joint with Michael Fellows Ariel Kulik Frances Rosamond Technion Charles Darwin Univ. Hadas Shachnai Technion Workshop on Kernelization, Sept 2011

2. Approximative Kernelization Traditionally: used as a preprocessing tool in FPT algorithms, which does not harm the classification of the instance (as a ‘yes’ or ‘no’ w.r.t. the parameterized problem). Many FPT algorithms for NP-hard problems use kernels whose sizes are lower bounded by a function f(k) = Ω(poly(k)), where k is the parameter. Suppose that in solving an FPT problem Π, we want to obtain a kernel of smaller size (=better running time), with some compromise on its fidelity when lifting a solution for the kernelized instance back to a solution for the original instance. 2 Can we define a tradeoff between fidelity and kernel size?

3. Approximative Kernelization Let L be a parameterized problem, i.e., L consists of input pairs (x, k), where x is a problem instance, and k is the parameter. Given α ≥ 1, an α-fidelity kernelization of the problem (i) Transforms in polynomial time the input (x, k) to ‘reduced’ input (x’, k’), such that k’ ≤ k and |x’| ≤ g(k, α), and (ii) If (x, k)  L then (x’, k’)  L (iii) If (x’, k’)  L then (x, α ·k)  L 3 The special case where α = 1 is classic kernelization.

4. Approximative Kernelization Combine approximation with kernelization: While lifting up to a solution for the original problem, we may get the value k, whereas there exists a solution of value k/α. The definition refers to Minimization problems (similar for maximization problems with k/α replaced by kα). 4

5. Many 2- approximation polynomial-time algorithms Unless Unique Game Conjecture fails: No factor-(2- ε)- approximation polynomial time algorithm exists [Khot, Regev 2008]. Vertex Cover is in FPT for general graphs: can be solved in time O * (1.28 k ). 5 Application: Vertex Cover Input: An undirected graph G=(V,E), an integer k ≥ 1. Output: A subset of vertices C  V, |C| ≤ k such that each edge in E has at least one endpoint in C (if one exists).

6. Application: Vertex Cover 6 1.Reduction step: Apply standard reduction rules to obtain (G’, k’). 2. Shrinking step : Select l = ⌊ k (α-1) ⌋ independent edges, I ⊆ E. Let D ⊆ V be the subset of vertices adjacent to I. 3. Omit from G’ the subgraph induced by D; omit isolated vertices. Let G’’ be the resulting graph. 4. The kernel is G’’ with k’’=k’- l. Let G=(V,E), k ≥ 1 and α  [1,2]. GGG G’’ is α -fidelity kernel: 1)G’’ is smaller than G’, therefore the size requirement holds. 2) If (G, k)  L then (G’, k’)  L (i.e., there is a cover C, such that |C| ≤ k’).

7. Application: Vertex Cover (Cont’d) 7 1.Reduction step: Apply standard reduction rules to obtain (G’, k’). 2. Shrinking step : Select l = k (α-1) independent edges, I ⊆ E. Let D ⊆ V be the subset of vertices adjacent to I. 3. Omit from G’ the subgraph induced by D; omit isolated vertices. Let G’’ be the resulting graph. 4. The kernel is G’’ with k’’=k’- l. GGG Therefore, (G’’, k’’)  L (C\D is a cover of size no greater than k’- l ). 3) If (G’’, k’’)  L then (G, α k )  L (there is a cover C’’ of size k’’ for G’’; then, C’’ U D is a cover of size k’’+2 l = k’+ l For G’. Hence, there is a cover of size k+ l = α k for G). Kernel size is 2k(2- α ).

8. 8 Algorithm : Shrinking step v 17 v 14 v 13 v 19 v2v2 v4v4 v 12 v 18 v6v6 v3v3 v5v5 v7v7 v1v1 v8v8 v9v9 v 11 v 10 v 16 v 15 v 20 l = k( α -1) =6 D={v 1,v 2,v 4,v 6, v 7,v 9,v 12,v 14, v 15,v 16,v 18,v 19 } G’ = ({v 1,…,v 20 }, E’) k=10, α =8/5

9. 9 Algorithm : Shrinking step v 17 v 13 v3v3 v5v5 v8v8 v 11 v 10 v 20

10. 10 Algorithm : Shrinking step v5v5 v8v8 v 11 v 20 G’’= ({v 5, v 8,v 11,v 20 }, E’’)

11. 11 Algorithm : example a z y t u x c b w r v s G=(V,E), k=8

12. 12 Algorithm : example a z y t u x c b w r s v Reduction step: Omit the crown H 1 ={b,c} I 1 ={u,v,w} α =2 l = k(α -1) =8

13. 13 Algorithm : example a z y t x r s Reduction step: Omit the crown H 1 ={b,c} I 1 ={u,v,w} α =2 l = k(α-1) =8

14. 14 Algorithm : example z t s Reduction step: Omit the crown H 1 ={b,c} I 1 ={u,v,w} α =2 l = k(α -1) =8 |I| ‹ l : G’’ is a 2-fidelity kernel of size 0!

15. 15 Algorithm : example a z y t u x c b w r v s G=(V,E), k=8

16. 16 Algorithm : example a z y t u x c b w r s v Reduction step: Omit the crown H 1 ={b,c} I 1 ={u,v,w} α =1 l = k( α-1 ) =0

17. What Happens If We Switch the Order? 17 1. Reduction step: Apply standard reduction rules to obtain (G’, k’). 2. Shrinking step : Select l = k (α-1) independent edges, I ⊆ E. Let D ⊆ V be the subset of vertices adjacent to I. 3. Omit from G’ the subgraph induced by D; omit isolated vertices. Let G’’ be the resulting graph. 4. The kernel is G’’ with k’’=k’- l. GGG

18. What Happens If We Switch the Order? 18 1. Shrinking step : Select l = k (α-1) independent edges, I ⊆ E. Let D ⊆ V be the subset of vertices adjacent to I. 2. Omit from G the subgraph induced by D; omit isolated vertices. Let G’ be the resulting graph with k’=k- l. 3. Reduction step: Apply standard reduction rules to obtain from G’ a kernel (G’’, k’’). GGG G’’ is α -fidelity kernel. Eliminates the assumption on linearity of kernelization (we previously assumed that a cover of size k’+ l for G’ implies a cover of size k+ l for G).

19. 19 1. Shrinking step : Select l = k (α-1) independent edges, I ⊆ E. Let D ⊆ V be the subset of vertices adjacent to I. 2. Omit from G the subgraph induced by D; omit isolated vertices. Let G’ be the resulting graph with k’=k- l. 3. Reduction step: Apply standard reduction rules to obtain from G’ a kernel (G’’, k’’). A parameterized Approximation Algorithm Replace kernelization by any FPT algorithm.

20. Main contribution of Steps 1. and 2. is in decreasing the value of k (tradeoff with fidelity). 20 1. Shrinking step : Select l = k (α-1) independent edges, I ⊆ E. Let D ⊆ V be the subset of vertices adjacent to I. 2. Omit from G the subgraph induced by D; omit isolated vertices. Let G’ be the resulting graph with k’=k- l. 3. Solution step: Run FPT algorithm on G’. A parameterized Approximation Algorithm

21. Let L be a parameterized problem, i.e., L consists of input pairs (x, k), where x is a problem instance, and k is the parameter. Given α ≥ 1, an α -fidelity shrinking of order h (i) Transforms in polynomial time the input (x, k) to ‘reduced’ input (x’, k’), such that k’ ≤ h(k), (ii) If (x, k)  L then (x’, k’)  L (iii) If (x’, k’)  L then (x, α ·k)  L 21 A Generic Approach: α-fidelity Shrinking Define simple reduction steps as a key building block to obtain α -fidelity shrinking for various problems

22. Given a parameterized problem L, a transformation r: U ─ > U is (a,b)-reduction step if, for any (x, k)  L, and (x’, k’) = r(x, k): (i) k’ = k - a, (ii) If (x, k)  L then (x’, k’)  L (iii) For any integer n ≥ 0, if (x’, k’+n)  L then (x, k+b+n)  L 22 Obtaining α-fidelity Shrinking Given a parameterized problem L, an (a,b)-reduction step r, and α ≤ 1+b/a, such that r can be evaluated in polynomial time, there is α -fidelity shrinking of order k(b+ a - α a)/b for L.

23. 23 Approximations via α -fidelity Shrinking: Some Examples Problem Kernel sizeRunning timeBest FPT algorithm Vertex cover 2(2- α)k 1.273 (2- α)k 1.273 k [CKX’06] Connected vertex cover No poly- kernel 2 k(2- α) 2 k [CN+ 11] 3-Hitting set 2.076 k [W’07] (All problems parameterized by solution size, k)

24. 24 How Powerful Is The Approach?

25. 25 Related Work FPT approximation Obtain a solution of value g(k) for a problem parameterized by k (e.g., Downey, F, McCartin and R, 2008; Many more..) Parameterized approximations for NP-hard problems by moderately exponential time algorithms Improve best known approximation ratios for subgraph maximization, minimum covering (Bourgeois, Escoffier and Paschos, 2009) β-approximation algorithms for vertex cover, β  (1,2), through accelerated branching (Fernau, Brankovic and Cakic, 2009) β-approximation algorithms for Hitting sets ( Fernau, 2011)

26. 26 Related Work (Cont’d) Links between approximation and kernelization: Exploit polynomial time approximation results in kernelization (Bevern, Moser and Niedermeier, 2010)

27. 27 What’s next?  Explore further the Generic approach for approximations based on α -fidelity shrinking : efficient application for other problems (e.g. Feedback Vertex Set, Edge Dominating Set..) ?  Can non-linear reduction (kernelization) rules be used to obtain better order (i.e., decreased running time)?  Extend the approach to problems with no FPT algorithm.