Approximative Kernelization : On the Trade-off between Fidelity and Kernel Size

1. ApproximativeKernelization: On the Trade-off between Fidelity and Kernel Size Hadas ShachnaiTechnion Workshop on Kernelization, Nov 2010 joint with Michael Fellows and Frances RosamondCharles Darwin University

2. Kernelization – Fidelity vs. Kernel Size 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. Can we define a tradeoff between fidelity and kernel size?

3. ApproximativeKernelization 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 kernelizationof 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/α)  L then (x’, k’)  L (iii) If (x’, k’)  L then (x, k)  L The special case where α = 1 is classic kernelization.

4. ApproximativeKernelization 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/αreplaced by kα).

5. Many 2- approximation polynomial-time algorithms A 3/2- approximation known for maximum degree four [Hochbaum 1983]. 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*(1.28k). Application: Vertex Cover Input: An undirected graph G=(V,E), an integer k ≥ 1. Output: A subset of vertices C  V, |C| ≤ k suchthat each edge in E has at least one endpoint in C (if one exists).

6. Application: Vertex Cover Let G=(V,E), k ≥ 1 and α  [1,2]. Initially C=Ф Reduction step: Apply reduction rules to (G, k/α). The resulting instance is (Ĝ, ), where =k/α –h, and h=|C|. If ‹ 0 return failure, else (a) Let l= 2(1 – 1/ α)k. Find a maximum matching M in Ĝ. (b) Partition the edges in M to m ≥ 1sets, each (except maybe the last) contains lvertices. Denote the vertex sets by {S1,…, Sm}. (c) Shrinking step: C= C U S1. Omit from Ĝ the vertices in S1 and all neighboring edges. 4. Omit from the resulting graph, G’, isolated vertices. Return G’ with parameter k’= - | S1| /2.

7. k=10, α=10/9 v1 v12 v2 Ĝ = ({v1,…,v20},Ế) =k/α –3=6 v13 v3 Algorithm : Shrinking step v14 v4 l =2(1 – 1/ α)k =4 v15 v5 v16 v6 v17 v7 S1={v1,v2,v12,v14} v18 v8 v9 v19 v10 v20 v11

8. G’ = (V’, E’) V’= {v3,v4,…,v20} k’= -2=4 v13 v3 Algorithm : Shrinking step v4 v15 v5 v16 v6 v17 v7 v18 v8 v9 v19 v10 v20 v11

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

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

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

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

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

14. Reduction step: Omit the crown H={b,c} I={u,v,w} r y s Algorithm : example z t a u α =1 b l =2(1 – 1/α)k =0 v c w x

15. Analysis: α-fidelity We show that the algorithm satisfies the properties of α-fidelity kernelization. The transformation from G to G’ is polynomial. If (G, k/α)  L then (G’, k’)  L We notethat if there is a vertex cover of size k/α for G, there is a cover of size = k/α -h for Ĝ, and there is a cover for G’ of size k’= - | S1| /2. 3. If (G’, k’)  L then (G, k)  L Assume that there is a vertex cover C(G’) of size k’ for G’. Consider the cover C*= C(G’) U S1 U C, where C is the cover found in the Reduction step.

16. Analysis : α-fidelity (Cont’d) Then, |C*| =|C(G’) U S1 U C | = k’ + | S1| + |C| = k/α – h - |S1| /2 + | S1| + h = k/α + |S1| /2 ≤ k/α + (1 – 1/α)k = k Last inequality follows from the definition of l.

17. Suppose there is a cover of size k/α for G, then the number of vertices in Ĝ is at most 2k/α (using, e.g., crown rules). Distinguish between two cases: If |M| ≥l /2 = (1-1/α)k then the number of vertices in G’ is at most 2k/ α - l = 2k/ α – 2k(1 – 1/ α)= 2k(2- α)/ α. If |M| ‹l /2, then S1 contains all the matched vertices in M, therefore G’ is empty. It follows that the kernel size is at most 2k(2- α)/ α. Analysis: Kernel Size

18. 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) • Links between approximation and kernelization • Exploit polynomial time approximation results in kernelization (Bevern, Moser and Niedermeier, 2010) 18 18

19. Future work • Explore further approximative kernelization: • Better tradeoff for vertex cover? (Current algorithm • does not optimize on kernel size.) • Define tradeoffs for other FPT problems • A general framework for combining exact reduction rules with approximation algorithms to guarantee α-fidelity, for any α ≥ 1. 19 19