certifying algorithms and forbidden induced subgraphskratsch/talks/talkdagstuhl2007.pdf · the leda...

Certifying AlgorithmsForbidden Induced Subgraphs

Previous Work and Our Results

Certifying Algorithms and

Forbidden Induced Subgraphs

P. Heggernes1 D. Kratsch2

1Institutt for InformatikkUniversitetet i Bergen

Norway

2Laboratoire d’Informatique Theorique et AppliqueeUniversite Paul Verlaine - Metz

France

Dagstuhl - GermanyMay 20-25, 2007

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Outline

1 Certifying AlgorithmsWhy certifying algorithms ?A Certifying Planarity TestCertificates and authenticationWhat is a Good Certifying Algorithm ?

2 Forbidden Induced SubgraphsCharacterisation of Graph ClassesSubgraphs as CertificatesAuthentication of a Subgraph

3 Previous Work and Our ResultsPrevious WorkOur ResultsCographsTrivially Perfect Graphs

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Why certifying algorithms ?A Certifying Planarity TestCertificates and authenticationWhat is a Good Certifying Algorithm ?

Why certifying Algorithms ?

The LEDA Platform of Combinatorial and GeometricComputing : The Planarity Test Story

Software Engineering : hard to avoid bugs in software

Algorithm vs. software : Correctness of an algorithm does notimply that its implementations have no bugs.

Bugs : no termination, wrong result, much too timeconsuming, much too space consuming, etc.

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What to do about bugs ?

Program Verification : Methods to find and avoid bugs insoftware.

Algorithm Design : Methods to design algorithms such thatbugs in the implementation can be avoided ? ?

Algorithm Design :

Algorithms that support an easy authentication of their resultsusing certificates.The implementation may or may not have bugs.

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A Planarity Test : Certificates

Linear-time planarity test (e.g. [Hopcroft, Tarjan]) outputs :

YES or NO

planar embedding (certificate) if input planar

Kuratowski graph (certificate) if input non planar

Is this all we need ?

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A Planarity Test : Authentication

We need to verify the correctness of each certificate ! !

Verify whether the planar embedding is indeed a planarembedding of the input graph.

Verify whether the Kuratowski graph is indeed a subdivision ofthe input graph.

Note that if authentication is positive then the input is certainlyplanar resp. non planar, no matter whether the planarity test hasbugs or not.

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General framework of a Certifying Recognition Algorithm

Recognition algorithm

input : graph G

output : YES and certificate for membership

output : NO and certificate for non-membership

Authentication

input : graph G , output of recognition algorithmincluding certificate

output : YES if the certificatehas all required properties w.r.t. to the input G

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Time and space

Resources needed by a certifying algorithm :

running time of recognition algorithm

running time of authentication algorithm(membership/non-membership)

space needed by certificate

Certifying planarity test :

running time of recognition algorithm : O(n + m)

running time of authentication algorithm for membership :O(n + m)

running time of authentication algorithm fornon-membership : O(n)

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Sublinear, Linear and Weak Certificates

sublinear certificates

A certificate is sublinear if the running time of its authenticationalgorithm is tighter than a linear one.

linear certificates

A certificate is linear if the running time of its authenticationalgorithm linear.

weak certificates

A certificate is weak if the running time of its authenticationalgorithm is the same (or even longer) as the one of therecognition algorithm.

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A Good Certifying Recognition Algorithm

A certifying algorithm to recognize a graph class should preferablyhave the following properties :

Good certifying algorithm

Recognition algorithm has linear running time.

Membership certificates are linear.

Non-membership certificates are sublinear.

Informal and Important :An authentication algorithm should be simple and easy toimplement. It should not redo the computation from scratch and itshould by no means rely on the recognition algorithm.

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Characterisation of Graph ClassesSubgraphs as CertificatesAuthentication of a Subgraph

Forbidden Induced Subgraphs




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Characterisation of Graph Classes

Certifying recognition algorithms for a graph class G often rely oncharacterisations of G.

Characterizations by forbidden (induced) subgraphs are ofparticular interest when designing certifying algorithms :

highly regarded in graph theory

any hereditary graph class can be characterized by theirminimal forbidden induced subgraphs

corresponding certificates often sublinear

corresponding certificates very easy to authenticate

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Split Graphs, Cographs and Trivially Perfect Graphs

Split Graphs [Foldes & Hammer 77]

A graph is split if and only if it contains no vertex set that induces2K2, C4, or C5.

Cographs

A graph is a cograph if and only if it contains no vertex set thatinduces P4.

Trivially Perfect Graphs [Golumbic 78]

A graph is trivially perfect if and only if it contains no vertexsubset that induces P4 or C4.

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Subgraphs as Certificates

Small forbidden (induced) subgraphs are ...

natural certificates (typically of non-membership)

sublinear certificates

often not provided by classical recognition algorithms

Convincing certificates

Small forbidden induced subgraphs are particularly convincingcertificates for the user of a corresponding software package. E.g.they can be highlighted in a graphical presentation of the inputgraph.

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Authentication in O(n) Time

authentication algorithm

The input to the authentication algorithm (to check whetherthe certificate induces a fixed subgraph H of G ) is the inputgraph G = (V ,E ) represented by adjacency lists and acertificate A ⊆ V of constant size.

It can be authenticated in time O(n) whether vertex set Ainduces a subgraph H in the input graph G .

Preferably for some fixed labeling of the vertices of H, therecognition algorithm assigns (by pointers) the vertices of H to theset A, indicating an isomorphism between H and G [A].

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Authentication in O(1) Time

authentication in O(1) time

The input to the authentication algorithm (to check whetherthe certificate induces a fixed subgraph H of G ) is the inputgraph G = (V ,E ) represented by ordered adjacency lists anda certificate A ⊆ V of constant size.

It can be authenticated in time O(1) whether vertex set Ainduces a subgraph H of the input graph G .

Use ordered adjacency lists to represent the input graph G .The recognition algorithm passes the set A to theauthentication algorithm by adding pointers to all the edgesand non edges of G [A] in the ordered adjacency lists.

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Previous WorkOur ResultsCographsTrivially Perfect Graphs





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Previous Work

Linear-time certifying algorithms to recognize ...

planar graphs [LEDA99]

chordal graphs [Tarjan & Yannakakis 84/85]

cographs [Corneil et al. 85]

interval and permutation graphs [Kratsch et al. 06]

proper interval graphs [Hell & Huang 04, Meister 05]

proper interval bigraphs [Hell & Huang 04]

proper circular-arc graphs [Kaplan & Nussbaum 06]

unit circular-arc graphs [Kaplan & Nussbaum 06]

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Our Results I

Linear-time certifying algorithms to recognize ...

split graphs {2K2,C4,C5}-freethreshold graphs {2K2,C4,P4}-freebipartite chain graphs {2K2,C3,C5}-freecobipartite chain graphs {2K2,C3,C5}-freetrivially perfect graphs {C4,P4}-free

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Our Results II

All our certifying algorithms are such that ...

recognition algorithm has linear running time

membership certificate is model of the class

membership certificate is linear

non-membership certificate is a small forbidden inducedsubgraph of the class

non-membership certificate is sublinear

authentication of non-membership certificates in O(1)

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Cographs I

definition by graph operations

Cographs are defined as follows :

A graph consisting of a single vertex is a cograph.

Let G1 and G2 be cographs. Then the join of G1 and G2 isagain a cograph.

Let G1 and G2 be cographs. Then the union of G1 and G2 isagain a cograph.

There are no other cographs.

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Cographs II

P4-free graphs

A graph G is a cograph iff it has no vertex subset that induces aP4.

cotree

A graph G is a cograph iff it has a cotree representation.

The cotree of a cograph is uniquely determined.

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Linear time recognition algorithms

There are various linear-time recognition algorithms for cographs.

[Corneil et al. 85]

linear running time

cotree as membership certificate

vertex set inducing a P4 as non-membership certificate

sublinear non-membership certificate

Does this imply immediately a good certifyingalgorithm ?

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Missing

Find a linear time algorithm that given a graph G = (V ,E ) and atree T , decides whether T is a cotree of G .

EXERCISE

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Trivially Perfect

Definition

A graph G is trivially perfect if for each induced subgraph H of G ,the number of maximal cliques of H is equal to the maximum sizeof an independent set of H [Golumbic 78].

[Golumbic 78]

A graph is trivially perfect if and only if it contains no vertexsubset that induces P4 or C4.

[Brandstadt et al.]

Trivially perfect graphs are exactly the chordal cographs.

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How to certify ?

Using known recognition algorithms

Both chordal graphs and cographs have linear time certifyingalgorithms [Tarjan & Yannakakis 84/85, Corneil et al. 85,Habib & Paul 05].

Obtaining a forbidden induced subgraph as a certificate ofnon-membership can be done by using those algorithms.

However

The challenge is to give a certificate of membership that can beauthenticated in O(n + m) time.

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Our membership certificates

universal-in-a-component ordering (uco)

A vertex ordering α = (v1, v2, ..., vn) of a graph G is auniversal-in-a-component ordering (uco) if for 1 ≤ i ≤ n, thevertex vi is universal in the connected component ofG [{vi , vi+1, ..., vn}] that vi belongs to.

A graph is trivially perfect if and only if it has a uco.

special type of cotree

A cograph G is a trivially perfect graph if and only if, in the cotreeT of G , every 1-node has at most one child that is a 0-node.

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Linear membership certificates

authentication

For both membership certificates we provide a simpleauthentication algorithm with running time O(n + m).

good certifying algorithm

Thus we obtain two linear time certifying algorithms to recognizetrivially perfect graphs and each has linear membership andsublinear non-membership certificates.

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Merci a tous !




For Further Reading I

A. Brandstadt, V. B. Le, and J. P. Spinrad,Graph classes : A survey.Philadelphia, SIAM, 1999.

D. G. Corneil, H. Lerchs, and L. Stewart-Burlingham,Complement reducible graphs.Discrete Applied Mathematics 3 :163–174,1981.

D. G. Corneil, Y. Perl, and L. K. Stewart,A linear recognition algorithm for cographs.SIAM J. Comput., 14 :926–934, 1985.

S. Foldes and P. L. Hammer,Split graphs.Congressus Numerantium, 19 :311–315, 1977.

M.C. Golumbic,Trivially perfect graphs.Discrete Math. 24 :105–107, 1978.




For Further Reading II

M. C. Golumbic,Algorithmic Graph Theory and Perfect Graphs.Second edition, Annals of Discrete Mathematics 57. Elsevier, 2004.

M. Habib and C. Paul,A simple linear time algorithm for cograph recognition.Discrete Applied Mathematics, 145 :183–197, 2005.

P. Hell and J. Huang,Certifying LexBFS recognition algorithms for proper interval graphs and properinterval bigraphs.SIAM J. Discrete Math., 18 :554–570, 2004.

H. Kaplan and Y. Nussbaum,Certifying algorithms for recognizing proper circular-arc graphs and unitcircular-arc graphs.Proc. of WG 2006 , LNCS 4271, (2006), pp. 289–300.

D. Kratsch, R. M. McConnell, K. Mehlhorn and J. P. Spinrad,Certifying algorithms to recognize interval and permutation graphs.SIAM J. Computing, 36 :326-353, 2006.




For Further Reading III

K. Mehlhorn and S. Naher,LEDA : A Platform for Combinatorial and Geometric Computing,Cambridge University Press, 1999.

D. Meister,Recognition and computation of minimal triangulations for AT-free claw-free andco-comparability graphs.Discrete Appl. Math., 146 :193–218, 2005.

R. E. Tarjan and M. Yannakakis,Simple linear-time algorithms to test chordality of graphs, test acyclicity ofhypergraphs, and selectively reduce acyclic hypergraphs.SIAM J. Comput., 13 :566–579, 1984.Addendum : SIAM J. Computing, 14 :254–255, 1985.

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