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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

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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?

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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) 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.

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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 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.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’).

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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 ).

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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

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

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

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

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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

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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

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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!

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

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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

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What Happens If We Switch the Order? 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

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What Happens If We Switch the Order? 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).

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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.

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Main contribution of Steps 1. and 2. is in decreasing the value of k (tradeoff with fidelity) 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

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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

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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.

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

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24 How Powerful Is The Approach?

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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)

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

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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.

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