Parameterized Complexity

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Parameterized Complexity

• Hans Bodlaender
• IPA Basiscursus Algoritmiek

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Today

• First session
• Parameterized complexity what is it about
• FPT algorithms branching
• FPT algorithms color coding
• Hardness proofs
• Second session
• FPT algorithms Kernelisation
• FPT algorithms Iterative compression
• Third session
• Treewidth

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1

• Introduction
• Parameterized complexity
• What is it about?

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Fixed Parameter Complexity

• Many problems have a parameter
• Analysis what happens to problem when some
  parameter is small

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Motivation

• In many applications, some number can be assumed
  to be small
• Time of algorithm can be exponential in this
  small number, but should be polynomial in usual
  size of problem
• Or something is fixed

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Parameterized problem

• Given Graph G, integer k,
• Parameter k
• Question Does G have a ??? of size at least (at
  most) k?
• Examples vertex cover, independent set,
  coloring,

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Examples of parameterized problems (1)

• Graph Coloring
• Given Graph G, integer k
• Parameter k
• Question Is there a vertex coloring of G with k
  colors? (I.e., c V 1, 2, , k with for all
  v,wÎ E c(v) ¹ c(w)?)
• NP-complete, even when k3.

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Clique

• Subset W of the vertices in a graph, such that
  each pair has an edge

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Examples of parameterized problems (2)

• Clique
• Given Graph G, integer k
• Parameter k
• Question Is there a clique in G of size at least
  k?
• Solvable in O(nk) time with simple algorithm.
  Complicated algorithm gives O(n2k/3). Seems to
  require W(nf(k)) time

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Vertex cover

• Set of vertices W Í V with for all x,y Î E x Î
  W or y Î W.
• Vertex Cover problem
• Given G, find vertex cover of minimum size

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Examples of parameterized problems (3)

• Vertex cover
• Given Graph G, integer k
• Parameter k
• Question Is there a vertex cover of G of size at
  most k?
• Solvable in O(2k (nm)) time

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Three types of complexity

• When the parameter is fixed
• Still NP-complete (k-coloring, take k3)
• O(f(k) nc)
• W(nf(k))

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Fixed parameter complexity theory

• To distinguish between behavior
• O( f(k) nc)
• W( nf(k))
• Proposed by Downey and Fellows.

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Parameterized problems

• Instances of the form (x,k)
• I.e., we have a second parameter
• Decision problem (subset of 0,1 x N )

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Fixed parameter tractable problems

• FPT is the class of problems with an algorithm
  that solves instances of the form (x,k) in time
  p(x)f(k), for polynomial p and some function f.

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Hard problems

• Complexity classes
• W1 Í W2 Í Wi Í WP
• Defined in terms of Boolean circuits
• Problems hard for W1 or larger class are
  assumed not to be in FPT
• Compare with P / NP

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Examples of hard problems

• Clique and Independent Set are W1-complete
• Dominating Set is W2-complete
• Version of Satisfiability is W1-complete
• Given set of clauses, k
• Parameter k
• Question can we set (at most) k variables to
  true, and al others to false, and make all
  clauses true?

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So what is parameterized complexity about?

• Given a parameterized problem
• Establish that it is in FPT
• And then design an algorithm that is as fast as
  possible
• Or show that it is hard for W1 or higher
• Try to find a polynomial time algorithm for fixed
  parameter
• Or even show that it is NP-complete for fixed
  parameters
• Solve it with different techniques (exact or
  approximation)

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FPT techniques

• Branching algorithms
• Bounded search trees
• Greedy localization
• Color coding
• Kernelisation (local rules, global rules)
• Induction
• Iterative compression
• Extremal method
• Win/Win
• Well quasi ordering
• Structures (e.g., treewidth)

FollowingSloper/Telle taxonomy
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Disclaimer

• Here focus on techniques, not on best known
  results
• Time arguments often too crude
• Also parameterized problems are considered with
  something else as parameter
• Pair of integers, string,
• Assumed to be fixed
• Can be mapped to integer

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2

• FPT Algorithmic techniques
• Branching

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Branching

• More or less classic technique of search tree
• Counting argument helps to keep size of search
  tree small
• At each step, we recursively create a number of
  subproblems answer is yes, if and only if at
  least one subproblem has yes as answer

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Vertex cover

• First example vertex cover
• Select an edge v,w. Note that a solution
  includes v or it includes w
• If we include v, we can ignore all edges incident
  to v in subproblems

v
w
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Branching algorithm for Vertex Cover

• Recursive procedure VC(Graph G, int k)
• VC(G(V,E), k)
• If G has no edges, then return true
• If k 0, then return false
• Select an edge v,w Î E
• Compute G G V v (remove v and incident
  edges)
• Compute G G V w (remove w and incident
  edges)
• Return VC(G,k 1) or VC(G,k 1)

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Analysis of algorithm

• Correctness
• Either v or w must belong to an optimal VC
• Time analysis
• At most 2.2k recursive calls
• Each recursive call costs O(nm) time
• O(2k (nm)) time FPT

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Independent Set on planar graphs

• Given a planar graph G(V,E), integer k
• Parameter k
• Question Does G have an independent set with at
  least k vertices, i.e., a set W of size at least
  k with for all v, w Î V v,w Ï E
• NP-complete
• Easy to see that it is FPT by kernelisation
• Here O(6k n) algorithm

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The red vertices form an independent set
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Branching

• Each planar graph has a vertex of degree at most
  5
• Take vertex v of minimum degree, say with
  neighbors w1, , wr, r at most 5
• A maximum size independent set contains v or one
  of its neighbors
• Selecting a vertex is equivalent to removing it
  and its neighbors and decreasing k by one
• Create at most 6 subproblems, one for each x Î
  v, w1, , wr. In each, we set k k 1, and
  remove x and its neighbors

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v
w1
w2
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Closest string

• Given k strings s1, ,sk each of length L,
  integer d
• Parameter d
• Question is there a string s with Hamming
  distance at most d to each of s1, ,sk
• Application in molecular biology
• Here FPT algorithm
• (Gramm and Niedermeier, 2002)

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Subproblems

• Subproblems have form
• Candidate string s
• Additional integer parameter r
• We look for a solution to original problem, with
  additional condition
• Hamming distance at most r to s
• Start with s s1 and rd ( original problem)

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Branching step

• Choose an sj with Hamming distance gt d to s
• If Hamming distance of sj to s is larger than
  dr NO
• For all positions i where sj differs from s
• Solve subproblem with
• s changed at position i to value sj (i)
• r r 1
• Note we find a solution, if and only one of
  these subproblems has a solution

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Example

• Strings 01113, 02223, 01221, d3
• First position in solution will be a 0
• First subproblem (01113, 2)
• Creates three subproblems
• (02113, 1)
• (01213, 1)
• (01123, 1)

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Time analysis

• Recursion depth d
• At each level, we branch at most at d r 2d
  positions
• So, number of recursive steps at most d2d1
• Each step can be done in polynomial time O(kdL)
• Total time is O(d2d1 . kdL)
• Speed up possible by more clever branching and by
  kernelisation

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More clever branching

• Choose an sj with Hamming distance gt d to s
• If Hamming distance of si to s is larger than
  dr NO
• Choose arbitrarily d1 positions where sj differs
  from s
• Solve subproblem with
• s changed at position i to value sj (j)
• r r 1
• Note still correct, and running time can be
  made O(kL kd dd)

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Technique

• Try to find a branching rule that
• Decreases the parameter
• Splits in a bounded number of subcases
• YES, if and only if YES in at least one subcase

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3

• FPT Algorithmic techniques
• Color coding

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Color coding

• Interesting algorithmic technique to give fast
  FPT algorithms
• As example
• Long Path
• Given Graph G(V,E), integer k
• Parameter k
• Question is there a simple path in G with at
  least k vertices?

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Problem on colored graphs

• Given graph G(V,E), for each vertex v a color
  in 1,2, , k
• Question Is there a simple path in G with k
  vertices of different colors?
• Note vertices with the same colors may be
  adjacent.
• Can be solved in O(2k (nm)) time using dynamic
  programming
• Used as subroutine

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DP

• Tabulate
• (S,v) S is a set of colors, v a vertex, such
  that there is a path using vertices with colors
  in S, and ending in v
• Using Dynamic Programming, we can tabulate all
  such pairs, and thus decide if the requested path
  exists

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A randomized variant

• For each vertex v, guess a color in 1, 2, , k
• Check if there is a path of length k with only
  vertices with different colors
• Note
• If there is a path of length k, we find one with
  positive chance ( 2k /k!)
• We can do this check in O(2k nm) time
• Repeat the check many times to get good
  probability for finding the path

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Derandomization

• Instead of randomly selecting a coloring, check
  all colorings from a k-perfect family of hash
  functions
• A k-perfect family of hash functions on a set V
  is a collection H of functions V 1, ,k, such
  that for each set W Í V, Wk, there exists an h
  Î H with h(W) 1, ,k
• Algorithm
• Generate a k-perfect family of hash functions
• For each function in the set, check for the
  properly colored path of length k

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Existence and generation of k-perfect families
of hash functions

• Alon et al. (1995) collection with O(ck log n)
  functions exist and can be deterministically
  generated
• Hence O((2c)k nm log n) algorithm for Longest
  Path
• Generate all elements of k-perfect family of hash
  functions, and for each, solve the colored
  version with dynamic programming
• Refined techniques exist

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4

• Hardness proofs

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Remember Cooks theorem

• NP-completeness helps to distinguish between
  decision problems for which we have a polynomial
  time algorithm, and those for which we expect no
  such algorithm exists
• NP-hard NP-completeness reductions
• Cooks theorem first NP-complete problem used
  to prove others to be NP-complete
• Similar theory for parameterized problems by
  Downey and Fellows

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Classes

• FPT Í W1 Í W2 Í W3 Í Í Wi Í Í WP
• Theoretical reasons to believe that hierarchy is
  strict

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Parameterized m-reduction

• Let L, L be parameterized problems.
• A standard parameterized m-reduction transforms
  an input (I,k) of L to an input (f(I,k), g(k)) of
  L
• (I,k) Î L if and only if (f(I,k), g(k)) Î L
• f uses time p(I) h(k) for a polynomial p, and
  some function h
• Note time may be exponential or worse in k
• Note the parameter only depends on parameter,
  not on rest of the input

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A Complete Problem

• Classes W1, are defined in terms of circuits
  (definition skipped here)
• Short Turing Machine Acceptance
• Given A non-deterministic Turing machine M,
  input x, integer k
• Parameter k
• Question Does M accept x in a computation with
  at most k steps?
• Short Turing Machine Acceptance is W1-complete
  (compare Cook)
• Note easily solvable in O(nkc) time

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More complete problems for W1

• Weighted q-CNF Satisfiability
• Given Boolean formula in CNF, such that each
  clause has at most q literals, integer k
• Parameter k
• Question Can we satisfy the formula by making at
  most k literals true?
• For each fixed q gt 1, Weighted q-CNF
  Satisfiability is complete for W1.

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Hard problems

• Independent Set, Clique W1-complete
• Dominating Set W2-complete
• Longest Common Subsequence III W1-complete
  (complex reduction to Clique)
• Given set of k strings S1, , Sk, integer m
• Parameter k, m
• Question is there a string S of length m that is
  a subsequence of each string Si, 1 i k?

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Example of reduction

• Precedence constrained K-processor scheduling
• Instance set of tasks T, each taking 1 unit of
  time, partial order lt on tasks, deadline D,
  number of processors K
• Parameter K
• Question can we carry out the tasks on K
  processors, such that
• If task1 lt task2, then task1 is carried out
  before task2
• At most one task per time step per processor
• All tasks finished at most at time D

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Transform from Dominating Set

• Let G(V,E), k be instance of DS
• Write n V, c n2, D knc 2n.
• Take the following tasks and precedences
• Floor D tasks in series

1
2
3




D-2
D-1
D
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Floor gadgets

• For all j of the form j n-1 ac bn (0 a lt
  kn, 1 b n), take a task that must happen on
  time j (parallel to the jth floor vertex)

1
2

j-1
j
j1


D-1
D
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Selector paths

• We take k paths of length D-n1
• Each models a vertex from the dominating set
• To some vertices on the path, we also take
  parallel vertices
• If vi, vj Ï E, and i ¹ j, then place a vertex
  parallel to the n-1acin-jth vertex for all a, 0
  a lt kn

1








D-n-1

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Lemma and Theorem

• Lemma We can schedule this set of tasks with
  deadline D and 2k processors, if and only if G
  has a dominating set of size at most k
• Theorem Precedence constrained k-processor
  scheduling is W2-hard
• Note size of instance must depend in polynomial
  way on size of G (and hence on k lt V)
• It is allowed to use transformations where new
  parameter is exponential in old parameter

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Conclusions

• Fixed parameter proofs method of showing that a
  problem probably has no FPT-algorithms
• Often complicated proof ?
• But not always ?

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5

• FPT Algorithmic techniques
• Kernelisation

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Kernelisation

• Preprocessing rules reduce starting instance to
  one of size f(k)
• Should work in polynomial time
• Then use any algorithm to solve problem on kernel
• Time will be p(n) g(f(k))

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Simple exampleIndependent Set on planar graphs

• A planar graph has an independent set of at least
  n/4 vertices
• 4-color the graph, and take the set of vertices
  with the most frequent color
• Kernelisation algorithm
• If n ³ 4k, then return YES
• Else return G

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Vertex cover observations that helps for
kernelisation

• If v has degree at least k1, then v belongs to
  each vertex cover in G of size at most k.
• If v is not in the vertex cover, then all its
  neighbors are in the vertex cover.
• If all vertices have degree at most k, then a
  vertex cover has at least m/k vertices.
• (mE). Any vertex covers at most k edges.

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Kernelisation for Vertex Cover

• H G ( S Æ )
• While there is a vertex v in H of degree at least
  k1 do
• Remove v and its incident edges from H
• k k 1 ( S S v )
• If k lt 0 then return false
• If H has at least k21 edges, then return false
• Solve vertex cover on (H,k) with some algorithm

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Time

• Kernelisation step can be done in O(nm) time
• After kernelisation, we must solve the problem on
  a graph with at most k2 edges, e.g., with
  branching this gives
• O( n m 2k k2) time
• O( kn 2k k2) time can be obtained by noting
  that there is no solution when m gt kn.

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Maximum Satisfiability

• Given Boolean formula in conjunctive normal
  form integer k
• Parameter k
• Question Is there a truth assignment that
  satisfies at least k clauses?
• Denote number of clauses C

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Reducing the number of clauses

• If C ³ 2k, then answer is YES
• Look at arbitrary truth assignment, and the
  mirror truth assignment where we flip each value
• Each clause is satisfied in one of these two
  assignment
• So, one assignment satisfies at least half of all
  clauses

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Bounding number of long clauses

• Long clause has at least k literals
• Short clause has at most k-1 literals
• Let L be number of long clauses
• If L ³ k answer is YES
• Select in each of the first k long clause a
  literal, whose complement is not yet selected
• Set these all to true
• k long clauses are satisfied

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Reducing to only short clauses

• If less than k long clauses
• Make new instance, with only the short clauses
  and k set to k-L
• There is a truth assignment that satisfies at
  least k-L short clauses, if and only if there is
  a truth assignment that satisfies at least k
  clauses
• gt choose for each satisfied short clause a
  variable that makes the clause true. We may
  change all other variables, and can choose for
  each long clause another variable that makes it
  true
• lt trivial

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An O(k2) kernel for Maximum Satisfiability

• If at least 2k clauses return YES
• If at least k long clauses return YES
• Else remove all L long clauses, and set kk-L

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Formal definition of kernelisation

• Let P be a parameterized problem. (Each input of
  the form (I,k).) A reduction to a problem kernel
  is an algorithm, that transforms A inputs of P to
  inputs of P, such that
• (I,k)) Î P , if and only if A(I,k) Î P for all
  (I,k)
• If A(I,k) (I,k), then k f(k), and I
  g(k) for some functions f, g
• A uses time, polynomial in I and k

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Kernelisation implies FPT and vice versa

• If P has a reduction to a problem kernel, and is
  decidable, then P is in FPT
• Use transformation
• Solve remaining instance
• Uses polynomial time, plus T(g(I)) time
• If P is in FPT, then P has a (perhaps trivial)
  reduction to a problem kernel
• Suppose we have an f(k) nc algorithm
• If I gt f(k), solve problem exactly this is
  O(nc1) time
• Formality take small yes or no-instance
  afterwards
• Otherwise, take original instance as kernel

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Improved kernelisation for Vertex Cover

• Nemhauser/Trotter kernel for VC of size at most
  2k
• ILP formulation of Vertex Cover
• Minimize SvÎ V xv
• For all v,w Î E xv xw ³ 1
• For all v Î V xv Î 0,1

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Relaxation

• Solve relaxation
• Minimize SvÎ V xv
• For all v,w Î E xv xw ³ 1
• For all v Î V 0 xv 1
• Define
• C0 v Î V xv lt 0.5
• V0 v Î V xv 0.5
• I0 v Î V xv gt 0.5

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Theorem and reduction

• Theorem There is a minimum size vertex cover S
  with C0 Í S and S Ç I0 Æ .
• Reduction rule remove all vertices in C0 and I0
  and set k k - C0 .
• Safe
• New, equivalent instance (GV0, k - C0 )
• If V0 gt 2k, relaxation, and hence also ILP has
  optimal solution gt k, answer NO.
• We have a kernel with at most 2k vertices.

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Example to explain techniques

• Convex recoloring
• Techniques
• Annotation
• Rules that either
• Decide
• Make the instance smaller
• Improve structural insight

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Connected tree coloring

• Tree
• Each vertex has a color
• Coloring is connected, if each set of vertices
  with the same color is connected

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Connected recoloring

• Given vertex-colored tree T
• Task give some vertices a different color, such
  that we obtain a connected coloring.
• With as few as possible recolored vertices

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Two recolored vertices
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A problem on evolution trees

• Tree models evolutionary ancestry of species
• Colors model values for some attributes
• Often only once evolutionary change to specific
  value is made
• Set of species with a specific value forms
  connected part of tree
• Correct data connected coloring of tree
• Incorrect data how to change as few as possible
  values to obtain consistent tree?

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Problem definition

• Convex tree recoloring
• Given Tree T(V,E), color c(v) for each vertex
  v, integer k
• Parameter k
• Question can we change the color of at most k
  vertices such that for each color, all vertices
  with this color form a subtree?

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Results

• For this problem B et al. O(n2) size kernel
• Here polynomial kernel
• Reducing number of vertices per color
• Reducing number of colors

80
Fixing colors
Fixing colors improves structural insight

• Trick annotation some vertices get a fixed
  color
• Annotation solving a different (generalized)
  problem, with some additional conditions for
  objects in problem
• At end undoing annotation
• Simple example if v is incident to k2 vertices
  with color C, then v must have color C

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Color fixing rule

• Suppose the trees of T-v have a1 a2 ar
  vertices with color c, and a1 ar-1 ³ k1.
• Then in any solution v gets color c.
• Otherwise we must recolor the vertices with color
  c in all but one of the subtrees of T-v at least
  k1 recolorings.
• Give v fixed color c possibly decrease k by 1

v
k4
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Analysis

• If there are at least 2k 3 vertices with color
  C, then we can fix a vertex with color C

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One fixed color vertex per color

• If v and w have the same color C, both fixed,
  then
• Fix the color of all vertices on the path from v
  to w to C, possibly decreasing k, or rejecting if
  there is a vertex with a different fixed color on
  the path
• If v and w are adjacent, and have the same fixed
  color C, then contract the edge v,w
• After these rules each color C has at most one
  vertex with fixed color C

84
Subtrees for a fixed color vertex

• Suppose v has a fixed color C. Look at the
  subtrees T-v
• If one of these has at least 2k3 vertices with
  color C, we can fix a vertex with color C in this
  subtree, and reduce T
• So, to bound the number of vertices with color C,
  we should make sure that v has small degree

85
Getting rid of uninteresting subtrees

• Uninteresting subtree No color in the subtree is
  broken, except perhaps C, and the vertices with
  color C are connected and incident to v, v has
  fixed color C
• Rule remove all vertices in an uninteresting
  subtree

v
Red, yellow, green do not appear elsewhere in T
86
Bounding degree of fixed color vertices

• If v has degree at least 2k1, a fixed color, and
  each subtree of T-v is interesting, then return
  NO
• Each color change can help at most 2 subtrees
  of T-v
• Bounds number of vertices per color to O(k2)
  after reductions
• If more than 2k2 vertices, we can fix a vertex
  with color C
• One fixed color C vertex
• Each subtree has at most 2k2 vertices with color
  C
• At most 2k subtrees in T-v

87
Types of colors

• Broken colors the vertices with color C are not
  connected
• Unbroken colors
• Rule If there are more than 2k broken colors,
  return NO
• Each color change can fix at most 2 colors the
  old and new color of the recolored vertex
• Consequence there are at most 2k broken colors

88
Broken colors

• A color is broken, if it is not connected in the
  given coloring.
• If there are more than 2k broken colors, there is
  no solution.
• Each recoloring can correct at most two colors
  (the old and the new color of the vertex) but not
  more.
• So Assume there are at most 2k broken colors.

v
v
89
Bounding number of colors

• A color is interesting, if it is
• Broken, or
• Appears on a path of length at most k between two
  vertices with the same broken color
• We never need to recolor vertices that are not
  interesting
• Rule remove all vertices whose color is not
  interesting
• Gives a forest

90
Rough analysis

• O(k) broken colors, each with O(k2) vertices
• O(k3) possible paths in tree, so O(k4)
  interesting colors
• O(k6) vertices in reduced instance
• Working harder gives better bound (quadratic)
• We are not yet finished going back to original
  problem
• From forest to tree
• Without fixed colors

91
From forest to tree

• Take one new color, and one vertex v with this
  color, fixed.
• Make v incident to a vertex with this color at
  each subtree

Like this (click)
92
From forest to tree

• Take one new color, and one vertex v with this
  color, fixed.
• Make v incident to a vertex with this color at
  each subtree

93
Getting rid of annotations

• If v with color C is annotated (fixed color),
  then add k2 leaves with color C, incident to v
  and remove the annotation
• Still O(k6) size

94
Lessons

• Reduction rules that
• Make the graph smaller and transform
  Yes-instances to Yes-instances and No-instances
  to No-instances
• Or improve structural map insight in instance
• Different rules ensure that different aspects of
  the resulting instance are small
• Annotation
• Analysis of problem and sizes

95
6

• Iterative compression

96
Idea of Iterative Compression

• Take small subset of input
• Repeat until we have a solution on entire input
• Solve problem on small subset
• We have a solution from the previous round, and
  this often helps!
• If solution gt k, return NO
• Add tiny part of input to subset of input, e.g.,
  add one additional vertex

97
Example

• Feedback vertex set problem
• Recent (Chen et al. 2007) result, combining
  different techniques

98
Feedback Vertex Set

• Instance graph G(V,E)
• Parameter integer k
• Question Is there a set of at most k vertices W,
  such that G-W is a forest?
• Known in FPT
• Here recent algorithm O(5k p(n)) time algorithm
• Can be done in O(5k kn) or less with kernelisation

99
Iterative compression technique

• Number vertices v1, v2, , vn
• Let X v1, v2, , vk
• for i k1 to n do
• Add vi to X
• Note X is a FVS of size at most k1 of v1, v2,
  , vi
• Call a subroutine that either
• Finds (with help of X) a feedback vertex set Y of
  size at most k in v1, v2, , vi set X Y OR
• Determines that Y does not exist stop, return NO

100
Compression subroutine

• Given graph G, FVS X of size k 1
• Question find if existing FVS of size k
• Is subroutine of main algorithm
• for all subsets S of X do
• Determine if there is a FVS of size at most k
  that contains all vertices in S and no vertex in
  X S

101
Yet a deeper subroutine

• Given Graph G, FVS X of size k1, set S
• Question find if existing a FVS of size k
  containing all vertices in S and no vertex from X
  S
• Remove all vertices in S from G
• Mark all vertices in X S
• If marked cycles contain a cycle, then return NO
• While marked vertices are adjacent, contract them
• Set k k - S. If k lt 0, then return NO
• If G is a forest, then return YES S

102
Subroutine continued

• If an unmarked vertex v has at least two edges to
  marked vertices
• If these edges are parallel, i.e., to the same
  neighbor, then v must be in a FVS (we have a
  cycle with v the only unmarked vertex)
• Put v in S, set k k 1 and recurse
• Else recurse twice
• Put v in S, set k k 1 and recurse
• Mark v, contract v with all marked neighbors and
  recurse
• The number of marked vertices is one smaller

103
Other case

• Choose an unmarked vertex v that has at most one
  unmarked neighbor (a leaf in GV-X)
• By step 7, it also has at most one marked
  neighbor
• If v is a leaf in G, then remove v
• If v has degree 2, then remove v and connect its
  neighbors

104
Analysis

• Precise analysis gives O(5k) subproblems in
  total
• Imprecise 2k subsets S
• Only branching step
• k is decreased by one, or
• Number of marked vertices is decreased by one
• Initially number of marked vertices k is at
  most 2k
• Bounded by 2k.22k 8k

105
7

• Conclusions

106
Conclusions

• Fixed parameter tractability gives interesting
  new insights on combinatorial problems
• Here examples from (mainly) graphs and logic
• Can be applied to different fields