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Introduction to Kernel Lower Bounds Daniel Lokshtanov

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What? Kernelization is a mathematical framework to analyze the quality of polynomial time pre- processing Until recently: Many upper bounds known. No ”non-trivial” lower bounds. This talk: Survey of recent lower bounds.

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Part I Introduction to Kernelization

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Parameterization Hard to analyze pre-processing for NP-hard problems within classical complexity. Reason: poly-time compression = poly-time solution. We consider parameterized problems. Each instance I comes with a parameter k ≤ |I| that is supposed to reflect how hard the instance is. Small k = easier instance.

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Parameterization: Example Point Line Cover IN: n points in the plane, integer k. PARAMETER: k QUESTION: Can the points be covered by k straight lines? Notice – easier to solve when k is small.

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Kernelization A f(k)-kernel for a problem P is an algorithm that: Takes as input an instance (I,k) Runs in time poly(|I|) Outputs an equivalent instance (I’,k’) with – |I’| ≤ f(k) – k’ ≤ f(k)

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Point Line Cover IN: n points in the plane, integer k. PARAMETER: k QUESTION: Can the points be covered by k straight lines?

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Point Line Cover TASK: Shoot the little devils, with only 3 shots. If some line covers 4 devils, must use it. Otherwise need 4 shots.

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k 2 - kernel for Point Line Cover R1: If some line covers more than k points delete all points on the line and decrease k by 1. R2: If no line covers at least n/k points, answer ”NO” If neither R1 nor R2 can be applied n ≤ k 2

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Edge Clique Cover IN: Graph G, integer k. PARAMETER: k QUESTION: Can the edges of G be covered by k cliques?

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4 k - Kernel for Edge Clique Cover R1: If u and v are adjacent and have same neighbours, delete v. R2: If R1 can’t be applied and n > 2 k, answer NO. If R1, R2 can’t be applied, then n < 2 k and m < 4 k.

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Recap A k 2 kernel for Point Line Cover polynomial kernel A 4 k kernel for Edge Clique Cover exponential kernel Which all parameterized problems have f(k)-kernels for some function f? Which parameterized problems have poly(k)- kernels?

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Which problems have f(k) - kernels? Theorem[Folklore]: A decidable parameterized problem P has an f(k)-kernel for some f P is fixed parameter tractable (FPT), i.e. solvable in time g(k)n O(1) for some g.

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Kernelization Complexity Q1: Does P have an f(k) kernel? P is FPT YES P is W-hard NO, unless FPT=W[1] Q2: Does P have a poly(k) kernel. poly(k) kernel YES How to say NO?

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Part II Framework for ruling out polynomial kernels

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Longest Path IN: Graph G, integer k PARAMETER: k QUESTION: Does G have a path of length k? Known: 2 k n c time algorithm [Williams 09] Does Longest Path have a polynomial kernel?

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Poly kernel for Longest Path? Suppose Longest Path has a k c kernel. Set t = kc + 1 and consider t instances with the same parameter k: (G 1,k), (G 2,k)... (G t,k) The instance (G 1 U G 2... U G t, k) is a yes instance iff some (G i, k) is. Kernelize this instance – the kernel has k c < t bits. Less than one bit per original instance, was at least one of the instances ”solved”?

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Poly Kernel for Longest Path? G 1,k G 2,k G t,k... Disjoint union G’,k’... G,k Polynomial kernel

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OR-Distillation Algorithms Detour back to classical problems. An OR-distillation algorithm for a problem L Takes as input instances I 1... I t. Runs in polynomial time Outputs an instance O of L’ such that – |O| ≤ max poly(|I i |) – O is ”yes” some I i is “yes”.

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OR-Distillation Algorithms Intuition: A distillation algorithm looks at several problem instances and pics the one ”most likely” to be a yes instance. Should not exist for NP-hard problems. Theorem [FS08]: Unless coNP ⊆ NP/poly, no NP- hard problem has an OR-distillation algorithm.

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OR-Composition algorithms: Intuition OR-Composition = ”formalization of disjoint union” OR-Composition + Kernel = OR-Distillation

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OR-Composition Algorithms Back to parameterized problems. An OR-composition algorithm for a problem P Takes as input instances I 1... I t with parameter k Runs in polynomial time Outputs an instance (O,k’) of P such that – k’ ≤ poly(k) – (O,k’) is ”yes” some (I i,k) is “yes”.

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OR-Composition for Longest Path G 1,k G 2,k G t,k... Disjoint union... G,k

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Ruling Out Polynomial Kernels Theorem [BDFH08]: If a parameterization P of an NP-hard * problem L has a composition algorithm, then P has no polynomial kernel unless coNP ⊆ NP/poly. Corollary [BDFH08]: Longest Path has no polynomial kernel unless coNP ⊆ NP/poly. * Originally proved only for NP-complete. New statement/proof by Holger Dell

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Proof of [BDFH08]-Theorem Given OR-Composition + Kernel for P we give an OR-distillation for L into OR(L). By [FS08] this implies that coNP ⊆ NP/poly.

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I1I1 I2I2 I3I3 ItIt t instances of size n... I 1,1 I 2,1 I 3,2 I t,n... Parameterization Group by parameter OR-Composition O 1,k 1 O 2,k 2 O n,k n... n instances instead of t. k i ≤ poly(n)

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O 1,k 1 O 3,k 2 O n,k n... n instances instead of t. k i ≤ poly(n) Kernelization O’ 1,k’ 1 O’ 2,k’ 2 O’ n,k’ n... n instances of size poly(n) each. Forget parameter O’ 1 O’ 2 O’ n... n instances of size poly(n) each. This is one instance to OR(L) of size poly(n)

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Recap II NP-hard + OR-composition = no poly kernel. Longest Path has no polynomial kernel Longest Cycle has no polynomial kernel...

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AND-Distillations / Compositions We can define AND-Distillation / Composition similarly to the OR case AND-Composition + Kernel = AND-Distillation Conjecture [BDFH08]: No NP-hard problem has an AND-Distillation.

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AND-Compositions Some interesting problems have AND- compositions; – treewidth – pathwidth –...width – vertex ranking Under ”AND-Distillation Conjecture” they have no polynomial kernel.

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Open Problem Relate the ”AND-Distillation” conjecture to a reasonable assumption in classical / parameterized complexity

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Part III Kernel lower bounds for more problems

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Next Polynomial Parameter Transformations: Reductions to show kernel lower bouds ”Non-trivial” OR-Composition algorithms

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k-k-Paths IN: Graph G, integer k PARAMETER: k QUESTION: Does G contain k vertex-disjoint k- paths? Disjoint union doesnt work as OR-composition. Other way to show no poly kernel?

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Polynomial Parameter Transformations A Polynomial Parameter Transformation (PPT) from A to B is an algorithm that: Takes as input an instance (I,k) of A Runs in polynomial time Outputs an instance (O,k’) of B such that – k’ ≤ poly(k) – (O,k’) is ”yes” for B (I,k) is “yes” for A.

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Reduction between problems Theorem [BTY09]: If there is a PPT from A to B, and a P-time reduction from B to A* then: B has a poly(k) kernel A has a poly(k) kernel *If B is NP and A is NP-hard, a trivial p-time reduction exists.

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Proof of Theorem [BTY09]: I,k I’,k’ O’,k’ O*,k* PPT Kernel P-time reduction A A B B

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Back to k-k-Paths Theorem [L09]: to k-k-Paths have no polynomial kernel unless coNP ⊆ NP/poly G,k k-Path G,k k-k-Paths k-1 paths of length k NP-completeness gives reduction back.

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Non-trivial Compositions? Next, excluding polynomial kernels for: – Bounded Universe Set Cover – Connected Vertex Cover (2-approximable!) – Steiner Tree

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Bounded Universe Set Cover IN: Set family F={S 1...S m } over a universe U of size k, integer t PARAMETER: k QUESTION: Is there a subfamily F’ ⊆ F of size ≤ t such that F’ covers U? Theorem [DLS09]: Bounded Universe Set Cover has no poly(k) kernel unless coNP ⊆ NP/poly.

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Steiner Tree IN: Graph G=(V,E), subset S ⊆ V of size k, integer t PARAMETER: t QUESTION: Is there a subtree T on ≤ t vertices of G, containing S?

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Steiner Tree Theorem [DLS09]: Steiner Tree has no poly(k) kernel unless coNP ⊆ NP/poly. Proof: PPT from Bounded Universe Set Cover Universe Terminals Sets Non-Terminals

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Connected Vertex Cover IN: Graph G=(V,E) integer k. PARAMETER: k QUESTION: Is there a set S of at most k vertices such that G[S] is connected and every edge if G has at least one endpoint in S.

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Connected Vertex Cover Theorem [DLS09]: Connected Vertex Cover has no poly(k) kernel unless coNP ⊆ NP/poly. Proof: PPT from Steiner Tree Terminals Non-Terminals

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Bounded Universe Set Cover Theorem [DLS09]: Bounded Universe Set Cover has no poly(k) kernel unless coNP ⊆ NP/poly. Proof plan: – Composition for ”Colored Bounded Universe Set Cover” – PPT from Colored Bounded Universe Set Cover to Bounded Universe Set Cover.

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Colored Bounded Universe Set Cover IN: t set families F 1 ={A 1...A a }, F 2 ={B 1...B b }, F t ={X 1...X c } over a universe U of size k, integer t PARAMETER: k QUESTION: Is there a family F’ = {A i,B j,... X l } of size t containing one set of each color, such that F’ covers U?

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Composition, recap An OR-composition algorithm for a problem P Takes as input instances I 1... I t with parameter k Runs in polynomial time Outputs an instance (O,k’) of P such that – k’ ≤ poly(k) – (O,k’) is ”yes” some (I i,k) is “yes”.

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Composition for CBUSC Task: Given t instances of CBUSC all of size ≤ n and parameter k, output in polynomial time one ”equivalent” CBUSC instance. Theorem [FKW04]: CBUSC instances with |U|=k can be solved in time O(2 k |F|). Trick: If t ≥ 2 k then t2 k |F| is polynomial, so wlog t < 2 k.

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Composition for CBUSC Plan: Glue the instances together on the universe. BABCCA Universe Sets BABCCA Universe Sets B A BC A Universe Sets C B A BC A C

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Composition for CBUSC GOOD: If one input is YES YES BAD: Can have NO + NO YES Need to make sure: A solution picks sets from the same instance.

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ID’s and boxes ID’s: Every instance gets a unique identification number from 0 to 2 k -1, written in binary (k bits!) Identification Check: Will check that for every pair of colors, the two solution verties of these colors come from the same instance = have the same ID.

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Boxes and ID’s A box is a gadget containing k elements. RED-BLUE box BLUE-RED box 101100 101000 The red-blue and blue-red boxes together make sure that the blue and red solution vertices come from the same instance

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Composition for CBUSC Modified plan: Glue the instances together on the universe. Add two boxes for every pair of colors. Universe size increases to O(k 3 ), still poly(k). Theorem [DLS09]: Colored Bounded Universe Set Cover has no poly(k) kernel unless coNP ⊆ NP/poly.

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No kernel for Bounded Universe Set Cover Theorem [DLS09]: Bounded Universe Set Cover has no poly(k) kernel unless coNP ⊆ NP/poly. PPT from CBUSC to BUSC BABCCA Universe Sets Universe Sets More Universe

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Epilogue Compositions and Polynomial Parameter Transformations are tools to show kernel lower bounds. Longest Path and Connected Vertex Cover are FPT but have no polynomial kernel unless coNP ⊆ NP/poly.

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List of FPT problems with no poly(k) kernels unless coNP ⊆ NP/poly. [HN06+FS08] k-Variable CNF-SAT [BDFH08] Longest Path, Longest Cycle [BTY09] Vertex Disjoint Paths, Cycles [DLS09] Bounded Universe Hitting Set, Bounded Universe Set Cover, Connected Vertex Cover, Steiner Tree, Capacitated Vertex Cover [KW09] Windmill-free Edge-Deletion [KW09’] Cases of MinOnesSat [JLS??] Dogson Score

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List of FPT problems with no poly(k) kernels unless AND-Distillation fails. [BDFH08] Treewidth, Pathwidth, Cutwidth, your-favourite width, and all sorts of stuff parameterized by them. [Z09] Vertex Ranking

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