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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

of dynamic problems. Theoretical Computer Science, 607, Part 3:426434, 2015. R.G. Downey, J. Egan, M.R. Fellows, F.A. Rosamond, and P. Shaw. Dynamic dominating set and turbo-

charging greedy heuristics. J. Tsinghua Sci. Technol, 19(4):329337, 2014. S. Hartung and R. Niedermeier. Incremental list coloring of graphs, parameterized by conservation.

Theoretical Computer Science, 494:8698, 2013.

Vertex Cover J. Chen, I. A. Kanj, and W. Jia. Vertex cover: further observations and further improvements. Journal

of Algorithms, 41(2):280301, 2001. J. Chen, I. A. Kanj, and G. Xia. Improved upper bounds for vertex cover. Theoretical Computer Science,

411(40-42):37363756, 2010.

Treewidth Bounds F. V. Fomin, S. Gaspers, S. Saurabh, and A. A. Stepanov. On two techniques of combining branching and

treewidth. Algorithmica, 54(2):181207, 2009. J. Kneis, D. Mlle, S. Richter, and P. Rossmanith. A bound on the pathwidth of sparse graphs with

applications to exact algorithms. SIAM Journal on Discrete Mathematics,23(1):407427, 2009.

ReferencesFeedback Vertex Set F. V. Fomin, S. Gaspers, D. Lokshtanov, and S. Saurabh. Exact algorithms via monotone local

search. In Proceedings of the 48th Annual ACM Symposium on Theory of Computing, STOC 16. ACM, 2016.

T. Kociumaka and M. Pilipczuk. Faster deterministic feedback vertex set. Information Processing Letters, 114(10):556560, 2014.

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.

N. Misra, G. Philip, V. Raman, S. Saurabh, and S. Sikdar. FPT algorithms for connected feedback vertex set. Journal of combinatorial optimization, 24(2):131146, 2012.