IPEC Kernelization1 A tutorial on Kernelization Hans L. Bodlaender

IPEC Kernelization2 On this talk 1.Kernelization: what, why, how? 2.Connection with fixed parameter tractability: problems without kernels 3.Problems without polynomial kernels 4.Conclusions Thanks to many people for inspiration, collaboration and contribution!

IPEC Kernelization3 What to do with hard problems? Many combinatorial problems are hard, e.g., NP-complete –Arising in many contexts Approaches to deal with them: –Approximations –Special cases –Exact algorithms ILP techniques, branch and bound, sat-solvers, …

IPEC Kernelization4 Useful approach: preprocessing Before running a slow exact algorithm, preprocess / simplify the data: Transform to (hopefully smaller) equivalent instance Input Equivalent smaller input I’ Output for I’ Preprocess Solve Undo preprocessing

IPEC Kernelization5 On preprocessing Relatively fast step Attempt to obtain equivalent instance: –Answer does not change –Size may decrease Slow algorithm (like ILP-solver) uses hopefully less time on reduced instance

IPEC Kernelization6 Kernelization: central question What can we prove on the size of a reduced instance, assuming polynomial time preprocessing?

IPEC Kernelization7 What we cannot expect Proposition If P  NP, then an NP-hard problem Q has no polynomial time preprocessing algorithm A that always reduces an input to a smaller equivalent input. Proof: If A exists, then P=NP: repeat the algorithm till we have an instance of size O(1) and solve. So … instead we investigate reduced instance size as function of a parameter of the inputs.

IPEC Kernelization8 Parameterized problem Subset of  * x N: –(“real input”, “parameter”) –Decision problem –‘Part of the input is an integer, called the parameter We express quality of preprocessing as function of the parameter

IPEC Kernelization9 Classic first example: Vertex Cover Vertex cover: –Input: Graph G=(V,E), integer k –Question: Is there a vertex cover of size at most k in G, i.e., a set W  V such that for all {v,w}  E, v  W or w  W? –Parameter: k

IPEC Kernelization10 Simplification rules Input: Graph G and integer k Rule 1: remove vertex with no neighbors. Rule 2: if a vertex v has k+1 or more neighbors, then –{v must be in each vertex cover of size  k} –Remove v and all incident edges; –Set k = k – 1. Rule 3: if k < 0, then say no.

IPEC Kernelization11 Counting rule Rule 4: if earlier rules do not apply, and we have more than k 2 edges, then say no. –Each vertex has degree at most k so can cover at most k edges. So, if G has more than k 2 edges, there is no vertex cover of size at most k. Algorithm: execute rules while possible. Output of algorithm: –Sometimes no (certainly not a solution). –Sometimes a equivalent instance with at most k 2 edges (and hence O(k 2 ) vertices).

IPEC Kernelization12 “Kernel” for Vertex Cover Theorem There is a polynomial time algorithm, that given an input (G,k) of Vertex Cover, either decides on this input or gives an equivalent instance with O(k 2 ) vertices and edges. Instead of deciding, we can also transform to trivial no instance (e.g., graph with one edge and k=0). We say: –Vertex Cover has kernel with O(k2) vertices and edges.

IPEC Kernelization13 Kernel (definition) A kernelization algorithm or kernel for a parameterized problem Q is an algorithm A that maps inputs for Q to inputs for Q, such that –A uses polynomial time; –For all (x,k): (x,k)  Q if and only if A(x,k)  Q; –There is a function f such that for all (x’, k’) = A(x,k): k’  f(k); |x’|  f(k). Size and parameter of new instance bounded by function of old parameter Size and parameter of new instance bounded by function of old parameter

IPEC Kernelization14 Research questions For parameterized problems Q: –Does Q have a kernel? –If so, how small (function f) can this kernel be? Linear kernels? Polynomial kernels? Any kernels?

Motivation for kernels Analysis of preprocessing. Kernels give new preprocessing steps. First step for FPT algorithms. IPEC Kernelization15

Compare Approximation algorithm = upper bound and lower bound heuristic + a proof of its quality. Kernel = preprocessing heuristic + a proof of its quality. IPEC Kernelization16

IPEC Kernelization17 Overview of problem behavior O(1) size kernels: problems in P. Ex: Eulerian –NP-completeness (variable parameter) Polynomial kernels Shown with algorithm. Ex.: Vertex Cover –compositionality, ppt-transformations, cross-composition Kernels, but not polynomial sized. Shown (usually) with FPT-algorithm. Ex: Long Path –W[1]-hardness XP: No kernel, polynomial if parameter is bounded. Ex.: Independent Set –NP-completeness (fixed parameter) Bad. Example: Graph Coloring is NP-complete for 3 colors

IPEC Kernelization18 How do we make kernelization algorithms General method: –Invent Safe Rules. Safe rules change an instance to an equivalent instance. –Rules should modify instances to equivalent instances that are Smaller or Give more structural insight. –Have a Counting rule or a Counting argument.

IPEC Kernelization19 Designing the algorithm Repeat until we have a (small enough) kernel: Invent safe rules. Analyse instances: if no safe rule applies, is the instance size bounded? If not, why not? Can we find a rule that avoids such situations?

IPEC Kernelization20 Example: Weighted marbles Instance: sequence of marbles and an integer k. Each marble has a positive integer cost and a color. Question: can we remove marbles of total cost at most k such that for each color, all marbles with that color are consecutive? Parameter: k Solution of cost 5=3+2

IPEC Kernelization21 Rule 1 If we have two consecutive marbles of the same color, replace it by one with the sum of the weights

IPEC Kernelization22 What we have now: Two successive marbles have a different color. But, we can have many color changes, even in a solution of cost

IPEC Kernelization23 Good colors A color is good, if there is only one marble with this color

IPEC Kernelization24 Rule 2 Suppose two successive marbles both have a good color. Give the second the color of the first times Rule 2

IPEC Kernelization25 Rule 2 Suppose two successive marbles both have a good color. Give the second the color of the first times Rule 2 Rule 2 does not make the instance smaller, but it makes it simpler: fewer colors! I.e., increases our structural insight!

IPEC Kernelization26 Algorithm While Rule 1 or Rule 2 is possible: apply the rule. Afterwards: –No 2 successive marbles of the same color. –No 2 successive marbles with a good color. The number of marbles is at most twice (+1) the number of marbles with a bad color. Can we bound the number of bad colored marbles?

IPEC Kernelization27 Rule 3: counting rule If there are at least 2k+1 bad colored marbles, say no. –Safeness: By deleting one marble, the number of bad colored marbles can decrease by at most 2 (assuming rule 1). Applying rules 1, 2, 3 while possible gives an instance with O(k) marbles. Is this a kernel for the problem? Or transform to O(1) size no-instance

IPEC Kernelization28 Rule 4 If a marble has weight > k+1, give it weight k+1. –Safeness: marble is never removed. Kernelization algorithm: –While Rules 1 – 4 are possible, apply them. Polynomial time. Gives equivalent instance with O(k log k) bits and O(k) marbles. Theorem: Weighted marbles problem has kernel of size O(k log k).

IPEC Kernelization29 Many recent results Kernelization usually algorithms of form: –Rules. Often with nontrivial correctness proofs. –Counting argument. Often nontrivial combinatorics. General techniques: meta-algorithms, crown reductions, protrusions, … Sometimes, no (small) kernel (seems to) exist: can we show this?

IPEC Kernelization30 Connection with Fixed Parameter Tractability A parameterized problem P is Fixed Parameter Tractable (  FPT) if there is an algorithm solving P that uses on inputs (x,k) in time –f(k) * |x| c –for a constant c –and some (computable) function f.

IPEC Kernelization31 Three variants of FPT Non-uniform: –For constant c: for every k, there is an algorithm that runs in O(n c ) time. Uniform: –For constant c, for a function f: there exists an algorithm that runs in f(k)n c time. Strongly uniform: –For constant c, for a computable function f: there exists an algorithm that runs in f(k)n c time.

IPEC Kernelization32 Relation between variants Non-uniform is a proper subset of uniform. –Example 1: {(x,k) | k  X} for some undecidable set of integers X is in non-uniform but not in uniform FPT. –Example 2: if w is a graph parameter that does not increase by taking minors, then Robertson-Seymour theory tells that {(G,k) | w(G)  k} is in non-uniform FPT. Uniform is proper subset of strongly uniform. –Proof by Downey and Fellows.

IPEC Kernelization33 A useful theorem with a curious proof Theorem (Folklore) A decidable parameterized problem P belongs to (uniform) FPT, if and only if it has a kernel. Proof  : If P has a kernel, then we have an FPT- algorithm: Given input (x,k), Apply kernelization and obtain (x’, k’). Now, use any algorithm to solve (x’, k’). –Answer is the same. –Running time poly(|x|) + g(f(k)). –  : …

IPEC Kernelization34 A useful theorem with a curious proof (II) Theorem (Folklore) A decidable parameterized problem P belongs to (uniform) FPT, if and only if it has a kernel. Proof continued  : If P has an algorithm A that uses f(k) n c time: Suppose we have input (x, k) with |x| = n. Run A for n c+1 steps. If A halts we have the answer (transform to O(1) size yes- or no-instance). If A does not halt, just output the original instance (x, k): we have n c+1  f(k)* n c so n  f(k).

IPEC Kernelization35 Variants Theorem (Folklore) A decidable parameterized problem P belongs to strongly uniform FPT, if and only if it has a kernel of size bounded by a computable function. Same proof. Problems in non-uniform FPT do not need to have a kernel. Practical consideration on variants: it does not matter if you use uniform or strongly uniform, as long as you don’t make mistakes…

IPEC Kernelization36 Implications of the theorem Positive: –Technique to obtain FPT-algorithms: Make small kernel. Algorithm on resulting small instance. Negative: –If we have evidence that there exists no FPT- algorithm, we also have evidence that there exists no kernel.

IPEC Kernelization37 Negative results Downey-Fellows introduce complexity classes of parameterized problems that are unlikely to have FPT algorithms, e.g. W[1]. Hardness is shown with “parameterized variant of many-one reductions”. Theorem If W[1] = FPT, then the Exponential Time Hypothesis is not valid. Corollary A parameterized problem that is W[1]- hard has no kernel, unless the ETH does not hold.

IPEC Kernelization38 Many W[1]-hard problems Many problems are W[1]-hard, e.g.: Clique, Independent Set, Dominating Set, … Canonical W[1]-complete problem: –Input: Boolean formula F in conjunctive normal form. –Question: Can we satisfy F by setting at most k variables to true? –Parameter: k. No kernels for these, unless W[1] = FPT and hence the Exponential Time Hypothesis fails.

IPEC Kernelization39 Problems with large kernels For many problems in FPT, we do not know small kernels. Consider: Long Path –Given: Graph G=(V,E), integer k. –Question: Does G have a simple path of length at least k? –Parameter: k. Is in FPT, but all known kernels have size exponential in k…

IPEC Kernelization40 Does Long Path have a kernel of polynomial size? Maybe not… Suppose we have a polynomial kernel, say with k c bits size. k k’ Size bounded by k c

IPEC Kernelization41 Long path continued Now, suppose we have a series of inputs to long path, say all with the same parameter: (G 1,k), (G 2,k), …, (G r,k). … k k k

IPEC Kernelization42 Take the disjoint union G 1  G 2  …  G r has a simple path of length k, if and only if there exists a graph G i that has a path of length k. … k k k … k k

IPEC Kernelization43 And now, apply the kernel to the union … k k k … k k k’ Size bounded by k c

IPEC Kernelization44 What happened? We have many (say r = k 2c ) instances of Long Path, and transform it to one instance of size < k c. Intuition: this cannot be possible without solving some of the instances, as we have fewer bits left than we had instances to start with… Theory (next) formalizes this idea.

IPEC Kernelization45 (Or-)Compositionality A parameterized problem Q is or-compositional, if there is an algorithm that –Receives as input a series of inputs to Q, all with the same parameter (I 1,k), …, (I r,k); –Uses polynomial time; –Outputs one input (I’,k’) to Q; –k’ bounded by polynomial in k; –(I’,k’)  Q if and only if there exists at least one j with (I j,k)  Q.

IPEC Kernelization46 Or-composition poly(t*n + k) time Q instance Q instance Q instances x1x1 k x2x2 k x.. kxtxt k n x*x* k*k* poly(k)

IPEC Kernelization47 Compositionality gives lowerbounds for kernels Theorem (B, Downey, Fellows, Hermelin + Fortnow, Santhanam, 2008) Let P be a parameterized problem that is –Or-compositional, and –“Unparameterized form” is NP-complete. Then P has no polynomial kernel unless NP  coNP/poly. Variant for and-compositionality is still open problem…

IPEC Kernelization48 Input: t instances of Longest Path. Take disjoint union, output as (G’, k). G’ has a path of length k  some G i has a path of length k. Output parameter trivially bounded in poly(k).,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k Long Path does not admit a polynomial kernel unless NP ⊆ coNP/poly Application to Long Path

Additional techniques (1) Polynomial parameter transformations (several authors): transform an argument that problem X does not have a polynomial kernel to an argument that problem Y does not have a polynomial kernel. Chen et al. (2009): no kernels of size k c n 1-  (unless NP  coNP/poly). Cross-compositions (B, Jansen, Kratsch, 2010): (composition of instances of problem X into instances of problem Y). –Composition of 2 n instances suffices. IPEC Kernelization49

Additional techniques (2) Dell and van Melkebeek (2010): extend technique to precise lower bounds, e.g.:  (k 2 ) bits for kernel for Vertex Cover (unless NP  coNP/poly). –New results by Dell and Marx, Weak composition: (Hermelin and Wu, 2011): polynomial lower bounds for several problems; super quasi polynomial lower bounds. IPEC Kernelization50

IPEC Kernelization51 Nonstandard parameters Many results on “objective parameter”, e.g. size of solution: –Minimum size of vertex cover. –Maximum length of path. –… But one can also use other parameters: –E.g., treewidth of input graph. Rich structure!

IPEC Kernelization52 Two problems Treewidth parameterized by vertex cover Input: Graph G=(V,E), integer k, vertex cover X of G. Parameter: |X|. Question: Has G treewidth at most k? Vertex cover parameterized by treewidth Input: Graph G=(V,E), integer k, tree decomposition of G Parameter: Width of tree decomposition. Question: Has G treewidth at most k?

IPEC Kernelization53 Negative result Theorem (B et al. 2010) Vertex cover parameterized by treewidth has no polynomial kernel, unless NP  coNP/poly. Refinement version is: –Or-compositional, –NP-complete, –So has no polykernel unless NP  coNP/poly. And hence problem itself has no polykernel unless Vertex cover REFINEMENT parameterized by treewidth Input: Graph G=(V,E), integer k, tree decomposition of G, vertex cover X of G of size at most k+1. Parameter: width of tree decomposition. Question: has G treewidth at most k?

IPEC Kernelization54 Positive result Theorem (B, Jansen, Kratsch, 2011) Treewidth parameterized by vertex cover has a kernel with O(|X| 3 ) vertices. Algorithm combines old preprocessing ideas! Heuristics for treewidth preprocessing: –Folklore. –B, Koster, vd Eijkhof, 2001: Preprocessing heuristics for treewidth. –Taken from (B, 1993, “Linear time algorithm”): vertices with many common neighbors.

IPEC Kernelization55 Rules for treewidth Rule 1: If k > |X|, decide yes. –G has treewidth at most (vertex cover size)+1. Rule 2: If non-adjacent v and w in X have k+2 common neighbors, add the edge {v,w}. –Taken from B, 1993 (linear time fpt algorithm for treewidth). Rule 3 (simplicial vertex rule): If the neighbors of v form a clique (v is simplicial): –If v has degree at most k, then remove v. –If v has degree more than k, then decide no. G has a clique of size k+2: treewidth(G)>k.

IPEC Kernelization56 Counting If no rule applies: –Associate each vertex z in V-X to a non-edge {v,w} with v, w neighbors of z. We can associate at most k+1 vertices to a pair in X. So, |V-X| = O(k |X| 2 ) = O(|X| 3 ). QED

IPEC Kernelization57 Remark We saw a cubic vertex kernel for Treewidth parameterized by vertex cover. Polynomial kernel for parameterization of treewidth by feedback vertex set. –More and more complicated rules. –Some rules generalize Almost simplicial vertex rule from B, Koster, vd Eijkhof Interaction between preprocessing heuristics and kernelization.

IPEC Kernelization58 Disjoint cycles –Given: Graph G=(V,E), integer k. –Question: Does G contain k vertex disjoint cycles? –Parameter: k. NP-complete, FPT, but does it has a polynomial kernel?? Resembles Feedback Vertex Set, but behaves differently! –Feedback vertex set Given: Graph G, integer k. Question: Is there a set of k vertices W such that G-W has no cycle? Parameter: k. –FVS has O(k 2 ) kernel (Thomassé)

IPEC Kernelization59 PPT-transformation A polynomial-parameter-time transformation (ppt- transformation) P to Q is an algorithm –which takes an instance (x,k) of P as input, –uses time polynomial in |x| + k, –outputs an instance (x’, k’) of Q with (x,k) ∈ P  (x’, k’) ∈ Q, k’ is polynomial in k. Theorem: If P has a ppt-transformation to Q, Q is NP- complete, P is in NP, and P has no polynomial kernel, then Q has no polynomial kernel.

IPEC Kernelization60 Proof Theorem: If P has a ppt-transformation to Q, Q is NP-complete, P is in NP, and P has no polynomial kernel, then Q has no polynomial kernel. Proof Suppose Q has a polynomial kernel. Build a polynomial kernel for P as follows: –Take input (x,k) for P. –Transform (x,k) to input (y,l) for Q with ppt-transformation. –Use kernel on (y,l): gives equivalent (y’,l’) for Q with polynomial size bound on |y|. –NP-completeness gives transformation from Q to P: apply it to (y’,l’) gives equivalent (x’,k’) with |x’| polynomially bounded in |y’|+l’, which is polynomially bounded in (x,k).

IPEC Kernelization61 Intermediate problem: Disjoint Factors Disjoint Factors –Given: Integer k, string s on alphabet {1, 2, …, k}. –Question: Can we find disjoint substrings s 1, s 2, …, s k in s such that s i starts and ends with i? –Parameter: k Disjoint Factors is NP-complete. Solvable with Dynamic Programming in 2 k |s| time. Next: compositionality

IPEC Kernelization62 Disjoint Factors is compositional: proof by example Number of instances r can be bounded by 2 k otherwise we can solve them all in polynomial time. Take log r new characters, and build new string, like (example for r=4): –b a s 1 a s 2 a b a s 3 a s 4 a b –New characters “eat” all but one instance, in which we must then find the other factors: b a s 1 a s 2 a b a s 3 a s 3 a b Corollary: Disjoint Factors has no polynomial kernel unless NP  coNP/poly.

IPEC Kernelization63 PPT-transformation from Disjoint Factors to Disjoint Cycles Disjoint Cycles does not admit a polynomial kernel unless NP ⊆ coNP/poly

IPEC Kernelization64 Overview of problem behavior O(1) size kernels: problems in P. Ex: Eulerian Graph –NP-completeness (variable parameter) Polynomial kernels Shown with algorithm. Ex.: Vertex Cover –compositionality, ppt-transformations, cross-composition Kernels, but not polynomial sized. Shown (usually) with FPT-algorithm. Ex: Long Path –W[1]-hardness XP: No kernel, polynomial if parameter is bounded. Ex.: Independent Set –NP-completeness (fixed parameter) Bad. Example: Graph Coloring is NP-complete for 3 colors

IPEC Kernelization65 Conclusions Kernelization: –Allows to give preprocessing algorithms with a provable guarantee on the size of resulting instances. –Generates new rules for preprocessing. –Reveals new complexity structure in combinatorial problems. Lively research area: –Many kernelization algorithms, including meta-results and kernel races. –New lower bound techniques. –Many nice open problems …