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Parameterized Algorithms Randomized Techniques

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Parameterized AlgorithmsRandomized Techniques

Bart M. P. Jansen

August 18th 2014, Będlewo

- For sometasks, finding a randomizedalgorithm is mucheasierthanfinding a deterministicone
- We consideralgorithmsthat have access to a stream of uniformly random bits
- So we do notconsiderrandomlygeneratedinputs

- The actions of the algorithmdepend on the values of the random bits
- Different runs of the algorithmmaygive different outcomes, for the same input

- A Monte Carloalgorithmwithfalsenegativesandsuccessprobability is an algorithm for a decision problem that
- given a no-instance, always returns no, and
- given a yes-instance, returns yeswith probability

- Since the algorithm is always correct on no–instances, but mayfail on yes-instances, it has one-sided error
- If is a positive constant, we canrepeat the algorithm a constantnumber of times
- ensurethat the probability of failure is smaller than the probabilitythat, cosmicradiationcauses a bit to flip in memory
- (Whichwouldinvalidate even a deterministicalgorithm)

- If is not a constant, we can also boost the success probability

- Suppose we have a Monte Carlo algorithmwithone-sided error probability, which may depend on
- For example,

- If we repeat the algorithm times, the probability that all runs fail is at most
since

- Probability ≥ that the repeated algorithm is correct
- Using trials gives success probability
- For example,

- Randomlyassigncolorsto the input structure
- Ifthere is a solutionand we are luckywith the coloring, every element of the solution has received a different color
- Thenfindanalgorithmtodetectsuchcolorfulsolutions
- Solutions of elementswithpairwise different colors

- Lemma.
- Let be a set of size , and let have size
- Let be a coloring of the elements of , chosen uniformly at random
- Each element of is colored with one of colors, uniformly and independently at random
The probability that the elements of are colored with pairwise distinct colors is at least

- Each element of is colored with one of colors, uniformly and independently at random

- Proof.
- There are possible colorings
- In of them, all colors on are distinct
- We used

Input:An undirected graph and an integer Parameter:Question:Is there a simple path on vertices in ?

- A solution is a -path
- The LongestPathproblem is a restrictedversion of the problem of findingpatterns in graphs

- Color the vertices of randomly with colors
- We want todetect a colorful-pathifoneexists
- Usedynamicprogramming over subsets

- For everysubset of colorsand vertex , define
= trueiffthere is a colorfulpathwhosecolors are andthat has as anendpoint

- For everysubset of colorsand vertex , define
= trueiffthere is a colorfulpathwhosecolors are andthat has as anendpoint

- Colorful-pathif = trueforsome
,true

,false

,true

- If is a singleton set, containing some color :
= true if and only if

- If:
if

falseotherwise

- Fill the table in time

- AlgorithmLongPath(Graph, integer )
- repeat times:
- Color the vertices of uniformly at random with colors
- Fill the DP table
- if such that = true then return yes

- return no

- repeat times:
- By standard DP techniques we canconstruct the path as well
- For eachcell, store a backlink to the earliercellthatdetermineditsvalue

- Running time is is
- Bythe get-luckylemma, ifthere is a -path, itbecomescolorfulwithprobability
- If the coloringproduces a colorful-path, the DP finds it
- Bythe independent repetitionlemma, repetitions give constant success probability

- Theorem. There is a Monte Carlo algorithm for Longest Path with one-sided error that runs in time and has constant success probability

- When doingdynamicprogramming, colorcodingeffectivelyallowsustoreduce the number of statesfrom
- keeping track of allverticesvisitedby the path, , to
- keeping track of allcolorsvisitedby the path,

- The techniqueextendstofindingsize-occurrences of other “thin” patterns in graphs
- A size-patterngraph of treewidthcanbe found in time , with constant probability

Input:A host graph and pattern graph (undirected)Parameter:Question:Does have a subgraph isomorphic to ?

Does contain ?

- The traditional color coding technique gives FPT algorithms for Longest Path
- Even for Subgraph Isomorphism when the pattern graph has constant treewidth

- If the pattern graph is unrestricted, we expect that no FPT algorithm exists for Subgraph Isomorphism
- It generalizes the -Clique problem
- Canonical W[1]-complete problem used to establish parameterized intractability (more later)

- If the host graph (and therefore the pattern graph ) has constant degree, there is a nice randomized FPT algorithm

- Suppose is a host graph that contains a subgraph isomorphic to a connected -vertex pattern graph
- Color the edges of uniformly independently at random with colors red () and blue ()
- If all edges of are colored red, and all other edges incident to are colored blue, it is easy to identify
- The pattern occurs as a connected component of
- Isomorphism of two -vertex graphs in time

- Let be a -vertex subgraph of graph
- A 2-coloring of isolates if the following holds:
- All edges of are red
- All other edges incident to are blue

- Observation. If the maximum degree of is , the probability that a random 2-coloring of isolates a fixed -vertex subgraph is at least
- There are at most edges incident on
- Each such edge is colored correctly with probability

- AlgorithmSubIso(Host graph, connectedpatterngraph)
- Let be the maximum degree of
- Let be the number of vertices in
- repeat times:
- Color the edges of uniformly at random with colors R, B
- foreach-vertex connected component of :
- if is isomorphicto, then return yes

- return no

- Easy toextend the algorithmtodisconnectedpatterns

- Theorem.There is a Monte Carlo algorithm for SubgraphIsomorphismwith one-sided error and constant success probability. For -vertex pattern graphs in a host graph of maximum degree , the running time is

Input:A graph and an integer Parameter:Question:Is there a set of at most adjacencies such that consists of disjoint cliques?

- Such a graph is called a -cluster graph

- -Clustering looks for a set of (non-)edges, instead of vertices
- We solve the problem on general graphs
- By randomly coloring the input, we again hope to highlight a solution with good probability, making it easier to find
- We color vertices of the graph

- A set of adjacencies is properly colored by a coloring of the vertices if:
- For all pairs , the colors of and are different

- As before, two crucial ingredients:
- What is the probability that a random coloring has the desired property?
- How to exploit that property algorithmically?

- We assign colors to the vertices and hope to obtain a property for the (non-)edges in a solution
- This allows us to save on colors

- Lemma. If the vertices of a simple graph with edges are colored independently and uniformly at random with colors, then the probability that is properly colored is at least
- Corollary. If a -Clustering instance has a solution set of adjacencies, the probability that is properly colored by a random coloring with colors is at least

For constant success probability, repetitions suffice

- Suppose properly colors a solution of
- The graph is a -cluster graph

- For ,let be the vertices colored
- As is properly colored, no (non-)edge of has both ends in
- No changes are made to by the solution
- is an induced subgraph of a -cluster graph
- For all , the graph is a -cluster graph

- consists of cliques that are not broken by the solution

- Observation. The -coloring partitions into cliques that are unbroken by the solution

- For each of the cliques into which is partitioned, guess into which of the final clusters it belongs
- For each guess, compute the cost of this solution
- Count edges between subcliques in different clusters
- Count non-edges between subcliques in the same cluster

- Total of guesses, polynomial cost computation for each
- Running time is to detect a properly colored solution, if one exists

- Using dynamic programming (exercise), this can be improved to time

1

3

3

3

2

2

- Algorithm-Cluster(graph, integer )
- Define
- repeat times:
- Color the vertices of uniformly at random with colors
- if there is a properly colored solution of size then
- return yes

- return no

- Theorem. There is a Monte Carlo algorithm for -Clustering with one-sided error and constant success probability that runs in time

- Truly random bits are very hard to come by
- Usual approach is to track radioactive decay

- Standard pseudo-random generators might work
- When spending exponential time on an answer, we do not want to get it wrong

- Luckily, we can replace most applications of randomization by deterministic constructions
- Without significant increases in the running time

- Different applications require different pseudorandom objects
- Main idea: instead of picking a random coloring , construct a family of functions
- Ensure that at least one function in has the property that we hope to achieve by the random choice

- Instead of independent repetitions of the Monte Carlo algorithm, run it once for every coloring in
- If the success probability of the random coloring is , we can often construct such a family of size

- Consider a -coloring of a universe
- A subset is split evenly by if the following holds:
- For every the sizes and differ by at most one
- All colors occur almost equally often within

- If a set of size is split evenly, then is colorful

- For , an -splitter is a family of functions from to such that:
- For every set of size , there is a function that splits evenly

- Theorem.For any one can construct an -splitter of size in time

- The special case of an -splitter is called an -perfect hash family
- Instead of trying random colorings in the Longest Path algorithm, try all colorings in a perfect hash family
- If is the vertex set of a -path, then so some function splits evenly
- Since , this causes to be colorful
- The DP then finds a colorful path

- Theorem.For any one can construct an -perfect hash family of size in time

- For an -universal set is a family of subsets of such that for any of size , all subsets of are contained in the family:
- Universal sets can be used to derandomize the random separation algorithm for Subgraph Isomorphism (exercise)

- Theorem.For any one can construct an -universal set of size in time

- For , an -coloring family is a family of functions from to with the following property:
- For every graph on the vertex set with at most edges, there is a function that properly colors

- Coloring families can be used to derandomize the chromatic coding algorithm for -Clustering
- Instead of trying random colorings, try all colorings in an -coloring family

- Theorem.For any one can construct an -coloring family of size in time

Input:A graph and an integer Parameter:Question:Is there a set of at most vertices in , such that each cycle contains a vertex of ?

(R1) Ifthere is a loop at vertex , then delete anddecreasebyone

(R2) Ifthere is anedge of multiplicitylargerthan, thenreduceitsmultiplicityto

(R3) Ifthere is a vertex of degree at most , then delete

(R4) Ifthere is a vertex of degreetwo, then delete andaddanedgebetween’s neighbors

- If (R1-R4) cannot be applied anymore, then the minimum degree is at least

- Observation. If is transformed into , then:
- fvs of size in fvs of size in
- Any feedback vertex set in is a feedback vertex set in when combined with the vertices deleted by (R1)

- We have seen a deterministicalgorithmwithruntime
- There is a simplerandomizedMonte Carlo algorithm
- In polynomial time, we canfind a size- solution withprobability at least, ifoneexists
- Repeatingthistimesgivesanalgorithmwith running time and constant successprobability

- Keyinsight is a simple procedure to select a vertex that is contained in a solution withconstant probability

- Lemma. Let be an -vertex multigraph with minimum degree at least 3. For every feedback vertex set of , more than half the edges of have at least one endpoint in .
- Proof. Consider the forest
- We prove that
- for any forest
- It suffices to prove

- Let be the edges with one end in and the other in
- Let , and be the vertices of with -degrees ,
- Every vertex of contributes to
- Every vertex of contributes to

in any forest

- Theorem. There is a randomizedpolynomial-time algorithmthat, given a Feedback Vertex Setinstance,
- either reports a failure, or
- finds a feedback vertex set in of size at most .
- If has an fvs of size , it returns a solution with probability at least

- Algorithm fvs(Graph , integer )
- Exhaustivelyapply (R1)-(R4) toobtain
- Let be the vertices with loops removed by (R1)

- if thenfailure
- if is a forest then return
- Uniformly at random, pick an edge of
- Uniformly at random, pick an endpoint of
- return fvs(

- Exhaustivelyapply (R1)-(R4) toobtain

- The algorithm outputs a feedback vertex set or failure
- Claim: If has a size-fvs, then the algorithm finds a solution with probability at least
- Proof by induction on
- Assume has a size-feedback vertex set
- By safety of (R1)-(R4), ’ has a size-’ fvs
- We have
- Since loops are in anyfvs, we have

- If, thenso’ is a forest
- Algorithmoutputswhich is a valid solution

- If, we willuse the induction hypothesis

- Case :
- Probability that random has an endpoint in is
- Probability that is
- If , then has an fvs of size
- Then, by induction, with probability recursion gives a size-(fvs of
- Sois a size-fvsof
- Byreductionrules, output is an fvs of
- Size is at most

- Probability of success is

- Theorem.There is a Monte Carlo algorithm for Feedback Vertex Setwith one-sided error and constant success probability that runs in time

- This simple, randomizedalgorithm is fasterthan the deterministicalgorithmfrom the previouslecture
- The methodgeneralizesto-minor-free deletion problems: delete vertices from the graph to ensure the resulting graph contains no member from the fixed set as a minor
- Feedback Vertex Set is -minor-free deletion

- Several variations of color coding give efficient FPT algorithms
- The general recipe is as follows:
- Randomly color the input, such that if a solution exists, one is highlighted with probability
- Show that a highlighted solution can be found in a colored instance in time

- For most problems we obtained single-exponential algorithms
- For -Clustering we obtained a subexponential algorithm

The end