# Dynamic Parameterized Problems - Algorithms and Complexity

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• Dynamic Parameterized Problems

R. Krithika The Institute of Mathematical Sciences, Chennai, India

NMI Workshop on Complexity TheoryIIT Gandhinagar

Nov 2016

(Joint work with Abhishek Sahu and Prafullkumar Tale)

• Dynamic Graphs

• Dynamic Graphs

• Dynamic Graphs

• Dynamic Graphs

• Dynamic Graphs

• Dynamic Graphs

• Dynamic Graphs

• Dynamic Graphs

• Dynamic Graphs

• Dynamic Graphs

Current graph is at a distance 8 from the initial graph

• Dynamic Graphs

• Dynamic GraphsG

• Dynamic Graphs

vertex cover of G

G

• Dynamic Graphs

vertex cover of G

G

• Dynamic Graphs

vertex cover of G

G H

• Dynamic Graphs

edit distance 8

vertex cover of G

G H

• Dynamic Graphs

edit distance 8

vertex cover of G

G H

vertex cover of H

• Dynamic Graphs

edit distance 8

Hamming distance 1

vertex cover of G

G H

vertex cover of H

• Dynamic Graphs

edit distance 8

G H

• Dynamic Graphs

edit distance 8

G H

vertex cover of G

• Dynamic Graphs

edit distance 8

G H

Hamming distance 2

vertex cover of G

• Dynamic Graphs

edit distance 8

G H

Hamming distance 2

vertex cover of G vertex cover of H

• Dynamic Problem Template

• Dynamic Problem TemplateInstance:

Graphs G, H on same vertex set s.t de(G,H) k

A solution S of G

An integer r

Question: Does H have a solution T s.t dv(S,T) r?

Parameter(s): k, r

• Dynamic Problem TemplateInstance:

Graphs G, H on same vertex set s.t de(G,H) k

A solution S of G

An integer r

Question: Does H have a solution T s.t dv(S,T) r?

Parameter(s): k, r

k- edit parameter

r- distance parameter

• Dynamic Dominating Set

• Dynamic Dominating Set

D

• Dynamic Dominating Set

D D

• Dynamic Dominating Set

D D

edit distance kG H

• Dynamic Dominating Set

• Dynamic Dominating SetS

G

S is a DS for G

• Dynamic Dominating SetS

G

S is a DS for G

TH

T is a DS for H

• Dynamic Dominating Set

SS

G

S is a DS for G

TH

T is a DS for H

• Dynamic Dominating Set

S

(S T) is a DS for H

SG

S is a DS for G

TH

T is a DS for H

• Dynamic Dominating Set

S

d(S,S T) = |T-S| |T-S| + |S-T| = d(S,T)

(S T) is a DS for H

SG

S is a DS for G

TH

T is a DS for H

• Dynamic Dominating Set

S

d(S,S T) = |T-S| |T-S| + |S-T| = d(S,T)

(S T) is a DS for H

SG

S is a DS for G

TH

T is a DS for H

Suffices to search for a DS of H that contains S

• Dynamic Dominating Set

• Dynamic Dominating SetD

• Dynamic Dominating SetD D

C

• Dynamic Dominating SetD D

C

Addition of edges do not affect the solution

• Dynamic Dominating Set

• Dynamic Dominating Set

D

• Dynamic Dominating Set

D D

C

• Dynamic Dominating Set

D D

C

Deletion of edges within C or within D do not affect the solution

• Dynamic Dominating Set

• Dynamic Dominating SetD

• Dynamic Dominating SetD D

C

• Dynamic Dominating SetD D

C

Deletion of edges between C and D could result in undominated vertices!

• Dynamic Dominating Set

• Dynamic Dominating Set

D

• Dynamic Dominating Set

D D

C

• Dynamic Dominating Set

D

At most k vertices are not dominated by D

D

C

• Dynamic Dominating Set

• Dynamic Dominating SetD

• Dynamic Dominating SetD D

C

• Dynamic Dominating SetD D

C

A set cover instance on k-element universe

• Dynamic Dominating SetD D

C

A set cover instance on k-element universe

O*(2k) algorithm

• Dynamic Connected Dominating Set

• Dynamic Connected Dominating Set

D

• Dynamic Connected Dominating Set

D D

• Dynamic Connected Dominating Set

D D

edit distance kG H

k1 del k2 add

• Dynamic Connected Dominating Set

• Dynamic Connected Dominating Set

SG

S is a CDS for G

• Dynamic Connected Dominating Set

SG

S is a CDS for G

TH

T is a CDS for H

• Dynamic Connected Dominating Set

SS

G

S is a CDS for G

TH

T is a CDS for H

• Dynamic Connected Dominating Set

S

(S T) is a DS for H

SG

S is a CDS for G

TH

T is a CDS for H

• Dynamic Connected Dominating Set

S

d(S,S T) = |T-S| |T-S| + |S-T| = d(S,T)

(S T) is a DS for H

SG

S is a CDS for G

TH

T is a CDS for H

• Dynamic Connected Dominating Set

S

d(S,S T) = |T-S| |T-S| + |S-T| = d(S,T)

(S T) is a DS for H

SG

S is a CDS for G

TH

T is a CDS for H

S T

• Dynamic Connected Dominating Set

S

d(S,S T) = |T-S| |T-S| + |S-T| = d(S,T)

(S T) is a DS for H

SG

S is a CDS for G

TH

T is a CDS for H

S T

• Dynamic Connected Dominating Set

S

d(S,S T) = |T-S| |T-S| + |S-T| = d(S,T)

(S T) is a DS for H

SG

S is a CDS for G

TH

T is a CDS for H

S T

• Dynamic Connected Dominating Set

S

d(S,S T) = |T-S| |T-S| + |S-T| = d(S,T)

(S T) is a DS for H

SG

S is a CDS for G

TH

T is a CDS for H

S T

• Dynamic Connected Dominating Set

S

d(S,S T) = |T-S| |T-S| + |S-T| = d(S,T)

(S T) is a DS for H

SG

S is a CDS for G

TH

T is a CDS for H

S T

• Dynamic Connected Dominating Set

S

d(S,S T) = |T-S| |T-S| + |S-T| = d(S,T)

(S T) is a DS for H

SG

S is a CDS for G

TH

T is a CDS for H

S T (S T) is a CDS for H

• Dynamic Connected Dominating Set

S

d(S,S T) = |T-S| |T-S| + |S-T| = d(S,T)

(S T) is a DS for H

SG

S is a CDS for G

TH

T is a CDS for H

Suffices to search for a CDS of H that contains S

S T (S T) is a CDS for H

• Dynamic Connected Dominating Set

• Dynamic Connected Dominating Set

D

• Dynamic Connected Dominating Set

D D

• Dynamic Connected Dominating Set

D D

At most k1 vertices are not dominated by D D has at most k2 components

• Dynamic Connected Dominating Set

• Dynamic Connected Dominating Setcomponents of D

• Dynamic Connected Dominating SetD1 D2 D3 . . . Dk2 components of D

• Dynamic Connected Dominating Set

undominated vertices

D1 D2 D3 . . . Dk2 components of D

• Dynamic Connected Dominating Set

v1 v2 v3 . . . vk1 undominated vertices

D1 D2 D3 . . . Dk2 components of D

• Dynamic Connected Dominating Set

v1 v2 v3 . . . vk1 undominated vertices

N(v1) N(v2) N(v3) . . . N(vk1)

D1 D2 D3 . . . Dk2 components of D

• Dynamic Connected Dominating Set

v1 v2 v3 . . . vk1 undominated vertices

N(v1) N(v2) N(v3) . . . N(vk1)

D1 D2 D3 . . . Dk2 components of D

D1 D2 D3 . . . Dk2

• Dynamic Connected Dominating Set

v1 v2 v3 . . . vk1 undominated vertices

N(v1) N(v2) N(v3) . . . N(vk1)

D1 D2 D3 . . . Dk2 components of D

D1 D2 D3 . . . Dk2

A group Steiner tree problem with parameter k1+k2 = k

• Dynamic Connected Dominating Set

v1 v2 v3 . . . vk1 undominated vertices

N(v1) N(v2) N(v3) . . . N(vk1)

D1 D2 D3 . . . Dk2 components of D

D1 D2 D3 . . . Dk2

O*(2k) algorithm

A group Steiner tree problem with parameter k1+k2 = k

• Set Cover Conjecture

• Set Cover ConjectureSet Cover

• Set Cover ConjectureSet Cover

U = { u1, u2, u3, . . . un }

• Set Cover ConjectureSet Cover

U = { u1, u2, u3, . . . un }

F = { F1, F2, F3, . . . Fm }

• Set Cover ConjectureSet Cover

U = { u1, u2, u3, . . . un }

F = { F1, F2, F3, . . . Fm }

O*(2n) algorithm

• Set Cover ConjectureSet Cover

U = { u1, u2, u3, . . . un }

F = { F1, F2, F3, . . . Fm }

O*(2n) algorithm No faster algorithm known

• Set Cover ConjectureSet Cover

U = { u1, u2, u3, . . . un }

F = { F1, F2, F3, . . . Fm }

O*(2n) algorithm No faster algorithm knownNo lb under SETH known

• Set Cover ConjectureSet Cover

U = { u1, u2, u3, . . . un }

F = { F1, F2, F3, . . . Fm }

O*(2n) algorithm

SCC: No O*((2-c)n) algorithm for any c > 0

No faster algorithm knownNo lb under SETH known

• Set Cover ConjectureSet Cover

U = { u1, u2, u3, . . . un }

F = { F1, F2, F3, . . . Fm }

O*(2n) algorithm

SCC: No O*((2-c)n) algorithm for any c > 0

No faster algorithm knownNo lb under SETH known

Cygan, Dell, Lokshtanov, Marx, Nederlof, Okamoto, Paturi, Saurabh, Wahlstrom (2012)

• Tightness Under Set Cover Conjecture

• Tightness Under Set Cover Conjecture

Set Cover (l)

• Tightness Under Set Cover Conjecture

Set Cover (l)

U = { u1, u2, u3, . . . un }

• Tightness Under Set Cover Conjecture

Set Cover (l)

U = { u1, u2, u3, . . . un }

F = { F1, F2, F3, . . . Fm }

• Tightness Under Set Cover Conjecture

Set Cover (l)

U = { u1, u2, u3, . . . un }

F = { F1, F2, F3, . . . Fm }

uiU

• Tightness Under Set Cover Conjecture

Set Cover (l)

U = { u1, u2, u3, . . . un }

F = { F1, F2, F3, . . . Fm }

uiU

Fj

F

• Tightness Under Set Cover Conjecture

Set Cover (l)

U = { u1, u2, u3, . . . un }

F = { F1, F2, F3, . . . Fm }

uiU

Fj

F

• Tightness Under Set Cover Conjecture

Set Cover (l)

U = { u1, u2, u3, . . . un }

F = { F1, F2, F3, . . . Fm }

x

uiU

Fj

F

• Tightness Under Set Cover Conjecture

Set Cover (l)

U = { u1, u2, u3, . . . un }

F = { F1, F2, F3, . . . Fm }

x

uiU

Fj

F

• Tightness Under Set Cover Conjecture

Set Cover (l)

U = { u1, u2, u3, . . . un }

F = { F1, F2, F3, . . . Fm }

x

uiU

Fj

F

G

CDS

• Tightness Under Set Cover Conjecture

Set Cover (l)

U = { u1, u2, u3, . . . un }

F = { F1, F2, F3, . . . Fm }

x

uiU

Fj

F

uiFj

x

H

G

CDS

• Tightness Under Set Cover Conjecture

Set Cover (l)

U = { u1, u2, u3, . . . un }

F = { F1, F2, F3, . . . Fm }

x

uiU

Fj

F

uiFj

x

H

k=nr=l

G

CDS

• Tightness Under Set Cover Conjecture

Set Cover (l)

U = { u1, u2, u3, . . . un }

F = { F1, F2, F3, . . . Fm }

x

uiU

Fj

F

uiFj

x

H

k=nr=l

(2-c)k algo DDS

(2-c)n algo SC

G

CDS

• Concluding RemarksDynamic Problem Parameterized Complexity

k r

Dominating Set 2k2 [DEFRS14], 2k (tight) W[2]-hard [DEFRS14]

Connected Dominating Set

4k [AEFRS15], 2k (tight) W[2]-hard [AEFRS15]

Vertex Cover1.174k, 1.1277k (expo space), O(k) kernel

1.2738r, O(r2) kernel

Connected Vertex Cover 4k [AEFRS15], 2k W[2]-hard [AEFRS15]

Feedback Vertex Set1.6667k (randomized),

O(k) kernel3.592r, O(r2) kernel

-DeletionFixed-parameter (in)tractability related to that

of non-dynamic version

• Concluding Remarks

Maintain k-size solution?

Other interesting parameters

=k1 edge additions and =k2 edge deletions

treewidth, vertex cover

Relation to reconfiguration problems and online problems

Interesting data structures

• Thank you :)

Questions?

• ReferencesDynamic Problems in Parameterized Complexity Framework F. N. Abu-Khzam, J. Egan, M. R. Fellows, F. A. Rosamond, and P. Shaw. On the parameterized complexity

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S. Thomass. A quadratic kernel for feedback vertex set, chapter 13, pages 115119. 2009.

Set Cover and Group Steiner Tree M. Cygan, H. Dell, D. Lokshtanov, D. Marx, J. Nederlof, Y. Okamoto, R. Paturi, S. Saurabh, and M.

Wahlstrom. On problems as hard as CNF-SAT. ACM Transactions on Algorithms, 12(3), Article No. 41, 2016.

F. V. Fomin, D. Kratsch, and G. J. Woeginger. Exact (exponential) algorithms for the dominating set problem. In Graph-Theoretic Concepts in Computer Science, pages 245256. Springer, 2004.

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