Introduction to Kernel Lower Bounds

Daniel Lokshtanov

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.

Part I

Introduction to Kernelization

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.

Parameterization: Example

Point Line Cover

IN: n points in the plane, integer k.PARAMETER: kQUESTION: Can the points be covered by k

straight lines?

Notice – easier to solve when k is small.

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)

Point Line Cover

IN: n points in the plane, integer k.PARAMETER: kQUESTION: Can the points be covered by k

straight lines?

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.

k2 - 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 ≤ k2

Edge Clique Cover

IN: Graph G, integer k.PARAMETER: kQUESTION: Can the edges of G be covered by k

cliques?

4k - 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 > 2k, answer NO.

If R1, R2 can’t be applied, then n < 2k and m < 4k.

Recap

A k2 kernel for Point Line Cover polynomial kernel

A 4k 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?

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)nO(1) for some g.

Kernelization Complexity

Q1: Does P have an f(k) kernel?P is FPT YESP is W-hard NO, unless FPT=W[1]

Q2: Does P have a poly(k) kernel.poly(k) kernel YESHow to say NO?

Part II

Framework for ruling out polynomial kernels

Longest Path

IN: Graph G, integer kPARAMETER: kQUESTION: Does G have a path of length k?

Known: 2knc time algorithm [Williams 09]Does Longest Path have a polynomial kernel?

Poly kernel for Longest Path?

Suppose Longest Path has a kc kernel.

Set t = kc + 1 and consider t instances with the same parameter k: (G1,k), (G2,k) ... (Gt,k)

The instance (G1 U G2 ... U Gt, k) is a yes instance iff some (Gi, k) is.

Kernelize this instance – the kernel has kc < t bits. Less than one bit per original instance, was at least one of the instances ”solved”?

Poly Kernel for Longest Path?

G1,k G2,k Gt,k

...Disjoint union

G’,k’

... G,k

Polynomial kernel

OR-Distillation Algorithms

Detour back to classical problems.

An OR-distillation algorithm for a problem L• Takes as input instances I1... It.• Runs in polynomial time• Outputs an instance O of L’ such that– |O| ≤ max poly(|Ii|)

– O is ”yes” some Ii is “yes”.

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.

OR-Composition algorithms: Intuition

OR-Composition = ”formalization of disjoint union”

OR-Composition + Kernel = OR-Distillation

OR-Composition Algorithms

Back to parameterized problems.

An OR-composition algorithm for a problem P• Takes as input instances I1 ... It with parameter k • Runs in polynomial time• Outputs an instance (O,k’) of P such that– k’ ≤ poly(k)– (O,k’) is ”yes” some (Ii,k) is “yes”.

OR-Composition for Longest Path

G1,k G2,k Gt,k

...Disjoint union

... G,k

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

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⊆ .

I1 I2 I3 Itt instances of size n...

I1,1 I2,1 I3,2 It,n...

Parameterization

Group by parameter

OR-Composition

O1,k1 O2,k2 On,kn...n instances instead of t. ki ≤ poly(n)

O1,k1 O3,k2 On,kn...n instances instead of t. ki ≤ poly(n)

Kernelization

O’1,k’1 O’2,k’2 O’n,k’n...n instances of sizepoly(n) each.

Forget parameter

O’1 O’2 O’n...n instances of sizepoly(n) each. This is one instance to OR(L)of size poly(n)

Recap II

NP-hard + OR-composition = no poly kernel.

Longest Path has no polynomial kernelLongest Cycle has no polynomial kernel...

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.

AND-Compositions

• Some interesting problems have AND-compositions;– treewidth – pathwidth – ...width – vertex ranking

• Under ”AND-Distillation Conjecture” they have no polynomial kernel.

Open Problem

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

Part III

Kernel lower bounds for more problems

Next

Polynomial Parameter Transformations: Reductions to show kernel lower bouds

”Non-trivial” OR-Composition algorithms

k-k-Paths

IN: Graph G, integer kPARAMETER: kQUESTION: Does G contain k vertex-disjoint k-

paths?

Disjoint union doesnt work as OR-composition. Other way to show no poly kernel?

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.

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.

Proof of Theorem [BTY09]:

I,k I’,k’

O’,k’O*,k*

PPT

Kernel

P-timereduction

A

A

B

B

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-completenessgives reductionback.

Non-trivial Compositions?

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

Bounded Universe Set Cover

IN: Set family F={S1...Sm} over a universe U of size k, integer t

PARAMETER: kQUESTION: 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⊆ .

Steiner Tree

IN: Graph G=(V,E), subset S V⊆ of size k, integer tPARAMETER: tQUESTION: Is there a subtree T on ≤ t vertices of G,

containing S?

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

Connected Vertex Cover

IN: Graph G=(V,E) integer k.PARAMETER: kQUESTION: 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.

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

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.

Colored Bounded Universe Set Cover

IN: t set families F1={A1...Aa}, F2={B1...Bb}, Ft={X1...Xc} over a universe U of size k, integer t

PARAMETER: kQUESTION: Is there a family F’ = {Ai,Bj, ... Xl} of

size t containing one set of each color, such that F’ covers U?

Composition, recap

An OR-composition algorithm for a problem P• Takes as input instances I1 ... It with parameter k • Runs in polynomial time• Outputs an instance (O,k’) of P such that– k’ ≤ poly(k)– (O,k’) is ”yes” some (Ii,k) is “yes”.

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(2k|F|).

Trick: If t ≥ 2k then t2k|F| is polynomial, so wlog t < 2k.

Composition for CBUSC

Plan: Glue the instances together on the universe.

BA B C CA

Universe

Sets

BA B C CA

Universe

Sets

BA B CA

Universe

Sets

C

BA B CA C

Composition for CBUSC

GOOD: If one input is YES YESBAD: Can have NO + NO YES

Need to make sure: A solution picks sets from the same instance.

ID’s and boxes

ID’s: Every instance gets a unique identification number from 0 to 2k-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.

Boxes and ID’s

A box is a gadget containing k elements.

RED-BLUE box

BLUE-RED box

101100 101100101000

The red-blue and blue-red boxes together make sure that the blue and red solution vertices come from the same instance

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(k3), still poly(k).

Theorem [DLS09]: Colored Bounded Universe Set Cover has no poly(k) kernel unless coNP ⊆NP/poly.

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

BA B C CA

Universe

Sets

Universe

Sets

More Universe

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⊆ .

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

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

THANK YOU!