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Approximative Kernelization: On the Trade-off between Fidelity and Kernel Size joint with Michael Fellows and Frances Rosamond Charles Darwin University Hadas Shachnai Technion Workshop on Kernelization, Nov 2010

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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. 2 Can we define a tradeoff between fidelity and kernel size?

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

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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

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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 * (1.28 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).

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Application: Vertex Cover 6 1.Initially C=Ф 2.Reduction step: Apply reduction rules to (G, k/α). The resulting instance is (Ĝ, ), where = k/α –h, and h=|C|. 3.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 ≥ 1 sets, each (except maybe the last) contains l vertices. Denote the vertex sets by {S 1,…, S m }. (c) Shrinking step: C= C U S 1. Omit from Ĝ the vertices in S 1 and all neighboring edges. 4. Omit from the resulting graph, G’, isolated vertices. Return G’ with parameter k’= - | S 1 | /2. Let G=(V,E), k ≥ 1 and α [1,2].

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7 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 = 2(1 – 1/ α)k =4 S 1 ={v 1,v 2,v 12,v 14 } Ĝ = ( {v 1,…,v 20 }, Ế ) = k/α –3=6 k=10, α =10/9

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8 Algorithm : Shrinking step v 17 v 13 v 19 v4v4 v 18 v6v6 v3v3 v5v5 v7v7 v8v8 v9v9 v 11 v 10 v 16 v 15 v 20 G’ = ( V’, E’) V’= {v 3,v 4,…,v 20 } k’= - 2=4

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

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

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

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

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

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

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Analysis: α-fidelity We show that the algorithm satisfies the properties of α- fidelity kernelization. 1.The transformation from G to G’ is polynomial. 2.If (G, k/α) L then (G’, k’) L We note that 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’= - | S 1 | /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 S 1 U C, where C is the cover found in the Reduction step. 15

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

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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: (i) 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. (ii) If |M| ‹ l / 2, then S 1 contains all the matched vertices in M, therefore G’ is empty. It follows that the kernel size is at most 2k(2- α)/ α. 17 Analysis: Kernel Size

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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)

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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.

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