Certifying Algorithms and Forbidden Induced Subgraphs

Transcription

1 /32 and P. Heggernes 1 D. Kratsch 2 1 Institutt for Informatikk Universitetet i Bergen Norway 2 Laboratoire d Informatique Théorique et Appliquée Université Paul Verlaine - Metz France Dagstuhl - Germany May 20-25, 2007

2 2/32 Outline 1 Why certifying algorithms? A Certifying Planarity Test Certificates and authentication What is a Good Certifying Algorithm? 2 Characterisation of Graph Classes Subgraphs as Certificates Authentication of a Subgraph 3 Previous Work Our Results Cographs Trivially Perfect Graphs

3 /32 Why certifying Algorithms? Why certifying algorithms? A Certifying Planarity Test Certificates and authentication What is a Good Certifying Algorithm? The LEDA Platform of Combinatorial and Geometric Computing : The Planarity Test Story Software Engineering : hard to avoid bugs in software Algorithm vs. software : Correctness of an algorithm does not imply that its implementations have no bugs. Bugs : no termination, wrong result, much too time consuming, much too space consuming, etc.

4 /32 What to do about bugs? Why certifying algorithms? A Certifying Planarity Test Certificates and authentication What is a Good Certifying Algorithm? Program Verification : Methods to find and avoid bugs in software. Algorithm Design : Methods to design algorithms such that bugs in the implementation can be avoided?? Algorithm Design : Algorithms that support an easy authentication of their results using certificates. The implementation may or may not have bugs.

5 /32 A Planarity Test : Certificates Why certifying algorithms? A Certifying Planarity Test Certificates and authentication What is a Good Certifying Algorithm? 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?

6 /32 A Planarity Test : Certificates Why certifying algorithms? A Certifying Planarity Test Certificates and authentication What is a Good Certifying Algorithm? 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?

7 /32 A Planarity Test : Certificates Why certifying algorithms? A Certifying Planarity Test Certificates and authentication What is a Good Certifying Algorithm? 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?

8 /32 A Planarity Test : Certificates Why certifying algorithms? A Certifying Planarity Test Certificates and authentication What is a Good Certifying Algorithm? 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?

9 A Planarity Test : Authentication Why certifying algorithms? A Certifying Planarity Test Certificates and authentication What is a Good Certifying Algorithm? We need to verify the correctness of each certificate!! Verify whether the planar embedding is indeed a planar embedding of the input graph. Verify whether the Kuratowski graph is indeed a subdivision of the input graph. Note that if authentication is positive then the input is certainly planar resp. non planar, no matter whether the planarity test has bugs or not. /32

10 A Planarity Test : Authentication Why certifying algorithms? A Certifying Planarity Test Certificates and authentication What is a Good Certifying Algorithm? We need to verify the correctness of each certificate!! Verify whether the planar embedding is indeed a planar embedding of the input graph. Verify whether the Kuratowski graph is indeed a subdivision of the input graph. Note that if authentication is positive then the input is certainly planar resp. non planar, no matter whether the planarity test has bugs or not. /32

11 A Planarity Test : Authentication Why certifying algorithms? A Certifying Planarity Test Certificates and authentication What is a Good Certifying Algorithm? We need to verify the correctness of each certificate!! Verify whether the planar embedding is indeed a planar embedding of the input graph. Verify whether the Kuratowski graph is indeed a subdivision of the input graph. Note that if authentication is positive then the input is certainly planar resp. non planar, no matter whether the planarity test has bugs or not. /32

12 A Planarity Test : Authentication Why certifying algorithms? A Certifying Planarity Test Certificates and authentication What is a Good Certifying Algorithm? We need to verify the correctness of each certificate!! Verify whether the planar embedding is indeed a planar embedding of the input graph. Verify whether the Kuratowski graph is indeed a subdivision of the input graph. Note that if authentication is positive then the input is certainly planar resp. non planar, no matter whether the planarity test has bugs or not. /32

13 Why certifying algorithms? A Certifying Planarity Test Certificates and authentication What is a Good Certifying Algorithm? 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 /32 Authentication input : graph G, output of recognition algorithm including certificate output : YES if the certificate has all required properties w.r.t. to the input G

14 Why certifying algorithms? A Certifying Planarity Test Certificates and authentication What is a Good Certifying Algorithm? 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 /32 Authentication input : graph G, output of recognition algorithm including certificate output : YES if the certificate has all required properties w.r.t. to the input G

15 Why certifying algorithms? A Certifying Planarity Test Certificates and authentication What is a Good Certifying Algorithm? 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 /32 Authentication input : graph G, output of recognition algorithm including certificate output : YES if the certificate has all required properties w.r.t. to the input G

16 Why certifying algorithms? A Certifying Planarity Test Certificates and authentication What is a Good Certifying Algorithm? 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 /32 Authentication input : graph G, output of recognition algorithm including certificate output : YES if the certificate has all required properties w.r.t. to the input G

17 Why certifying algorithms? A Certifying Planarity Test Certificates and authentication What is a Good Certifying Algorithm? 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 /32 Authentication input : graph G, output of recognition algorithm including certificate output : YES if the certificate has all required properties w.r.t. to the input G

18 8/32 Time and space Why certifying algorithms? A Certifying Planarity Test Certificates and authentication What is a Good Certifying Algorithm? 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 for non-membership : O(n)

19 8/32 Time and space Why certifying algorithms? A Certifying Planarity Test Certificates and authentication What is a Good Certifying Algorithm? 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 for non-membership : O(n)

20 9/32 Why certifying algorithms? A Certifying Planarity Test Certificates and authentication What is a Good Certifying Algorithm? Sublinear, Linear and Weak Certificates sublinear certificates A certificate is sublinear if the running time of its authentication algorithm is tighter than a linear one. linear certificates A certificate is linear if the running time of its authentication algorithm linear. weak certificates A certificate is weak if the running time of its authentication algorithm is the same (or even longer) as the one of the recognition algorithm.

21 10/32 Why certifying algorithms? A Certifying Planarity Test Certificates and authentication What is a Good Certifying Algorithm? A Good Certifying Recognition Algorithm A certifying algorithm to recognize a graph class should preferably have 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 to implement. It should not redo the computation from scratch and it should by no means rely on the recognition algorithm.

22 11/32 1 Why certifying algorithms? A Certifying Planarity Test Certificates and authentication What is a Good Certifying Algorithm? 2 Characterisation of Graph Classes Subgraphs as Certificates Authentication of a Subgraph 3 Previous Work Our Results Cographs Trivially Perfect Graphs Characterisation of Graph Classes Subgraphs as Certificates Authentication of a Subgraph

23 12/32 Characterisation of Graph Classes Characterisation of Graph Classes Subgraphs as Certificates Authentication of a Subgraph Certifying recognition algorithms for a graph class G often rely on characterisations of G. Characterizations by forbidden (induced) subgraphs are of particular interest when designing certifying algorithms : highly regarded in graph theory any hereditary graph class can be characterized by their minimal forbidden induced subgraphs corresponding certificates often sublinear corresponding certificates very easy to authenticate

24 13/32 Characterisation of Graph Classes Subgraphs as Certificates Authentication of a Subgraph Split Graphs, Cographs and Trivially Perfect Graphs Split Graphs [Földes & Hammer 77] A graph is split if and only if it contains no vertex set that induces 2K 2, C 4, or C 5. Cographs A graph is a cograph if and only if it contains no vertex set that induces P 4. Trivially Perfect Graphs [Golumbic 78] A graph is trivially perfect if and only if it contains no vertex subset that induces P 4 or C 4.

25 14/32 Subgraphs as Certificates Characterisation of Graph Classes Subgraphs as Certificates Authentication of a Subgraph 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 convincing certificates for the user of a corresponding software package. E.g. they can be highlighted in a graphical presentation of the input graph.

26 5/32 Authentication in O(n) Time Characterisation of Graph Classes Subgraphs as Certificates Authentication of a Subgraph authentication algorithm The input to the authentication algorithm (to check whether the certificate induces a fixed subgraph H of G) is the input graph G = (V, E) represented by adjacency lists and a certificate A V of constant size. It can be authenticated in time O(n) whether vertex set A induces a subgraph H in the input graph G. Preferably for some fixed labeling of the vertices of H, the recognition algorithm assigns (by pointers) the vertices of H to the set A, indicating an isomorphism between H and G[A].

27 15/32 Authentication in O(n) Time Characterisation of Graph Classes Subgraphs as Certificates Authentication of a Subgraph authentication algorithm The input to the authentication algorithm (to check whether the certificate induces a fixed subgraph H of G) is the input graph G = (V, E) represented by adjacency lists and a certificate A V of constant size. It can be authenticated in time O(n) whether vertex set A induces a subgraph H in the input graph G. Preferably for some fixed labeling of the vertices of H, the recognition algorithm assigns (by pointers) the vertices of H to the set A, indicating an isomorphism between H and G[A].

28 6/32 Authentication in O(1) Time Characterisation of Graph Classes Subgraphs as Certificates Authentication of a Subgraph authentication in O(1) time The input to the authentication algorithm (to check whether the certificate induces a fixed subgraph H of G) is the input graph G = (V, E) represented by ordered adjacency lists and a certificate A V of constant size. It can be authenticated in time O(1) whether vertex set A induces 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 the authentication algorithm by adding pointers to all the edges and non edges of G[A] in the ordered adjacency lists.

29 6/32 Authentication in O(1) Time Characterisation of Graph Classes Subgraphs as Certificates Authentication of a Subgraph authentication in O(1) time The input to the authentication algorithm (to check whether the certificate induces a fixed subgraph H of G) is the input graph G = (V, E) represented by ordered adjacency lists and a certificate A V of constant size. It can be authenticated in time O(1) whether vertex set A induces 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 the authentication algorithm by adding pointers to all the edges and non edges of G[A] in the ordered adjacency lists.

30 16/32 Authentication in O(1) Time Characterisation of Graph Classes Subgraphs as Certificates Authentication of a Subgraph authentication in O(1) time The input to the authentication algorithm (to check whether the certificate induces a fixed subgraph H of G) is the input graph G = (V, E) represented by ordered adjacency lists and a certificate A V of constant size. It can be authenticated in time O(1) whether vertex set A induces 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 the authentication algorithm by adding pointers to all the edges and non edges of G[A] in the ordered adjacency lists.

31 Previous Work Our Results Cographs Trivially Perfect Graphs 1 Why certifying algorithms? A Certifying Planarity Test Certificates and authentication What is a Good Certifying Algorithm? 2 Characterisation of Graph Classes Subgraphs as Certificates Authentication of a Subgraph 3 Previous Work Our Results Cographs Trivially Perfect Graphs 7/32

32 18/32 Previous Work Previous Work Our Results Cographs Trivially Perfect Graphs 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]

33 19/32 Our Results I Previous Work Our Results Cographs Trivially Perfect Graphs Linear-time certifying algorithms to recognize... split graphs threshold graphs bipartite chain graphs cobipartite chain graphs trivially perfect graphs {2K 2, C 4, C 5 }-free {2K 2, C 4, P 4 }-free {2K 2, C 3, C 5 }-free {2K 2, C 3, C 5 }-free {C 4, P 4 }-free

34 20/32 Our Results II Previous Work Our Results Cographs Trivially Perfect Graphs 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 induced subgraph of the class non-membership certificate is sublinear authentication of non-membership certificates in O(1)

35 1/32 Cographs I Previous Work Our Results Cographs Trivially Perfect Graphs definition by graph operations Cographs are defined as follows : A graph consisting of a single vertex is a cograph. Let G 1 and G 2 be cographs. Then the join of G 1 and G 2 is again a cograph. Let G 1 and G 2 be cographs. Then the union of G 1 and G 2 is again a cograph. There are no other cographs.

36 2/32 Cographs II Previous Work Our Results Cographs Trivially Perfect Graphs P 4 -free graphs A graph G is a cograph iff it has no vertex subset that induces a P 4. cotree A graph G is a cograph iff it has a cotree representation. The cotree of a cograph is uniquely determined.

37 Linear time recognition algorithms Previous Work Our Results Cographs Trivially Perfect Graphs There are various linear-time recognition algorithms for cographs. [Corneil et al. 85] linear running time cotree as membership certificate vertex set inducing a P 4 as non-membership certificate sublinear non-membership certificate Does this imply immediately a good certifying algorithm? 3/32

38 Linear time recognition algorithms Previous Work Our Results Cographs Trivially Perfect Graphs There are various linear-time recognition algorithms for cographs. [Corneil et al. 85] linear running time cotree as membership certificate vertex set inducing a P 4 as non-membership certificate sublinear non-membership certificate Does this imply immediately a good certifying algorithm? 3/32

39 4/32 Missing Previous Work Our Results Cographs Trivially Perfect Graphs Find a linear time algorithm that given a graph G = (V, E) and a tree T, decides whether T is a cotree of G. EXERCISE

40 4/32 Missing Previous Work Our Results Cographs Trivially Perfect Graphs Find a linear time algorithm that given a graph G = (V, E) and a tree T, decides whether T is a cotree of G. EXERCISE

41 5/32 Trivially Perfect Previous Work Our Results Cographs Trivially Perfect Graphs 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 size of an independent set of H [Golumbic 78]. [Golumbic 78] A graph is trivially perfect if and only if it contains no vertex subset that induces P 4 or C 4. [Brandstädt et al.] Trivially perfect graphs are exactly the chordal cographs.

42 6/32 How to certify? Previous Work Our Results Cographs Trivially Perfect Graphs Using known recognition algorithms Both chordal graphs and cographs have linear time certifying algorithms [Tarjan & Yannakakis 84/85, Corneil et al. 85, Habib & Paul 05]. Obtaining a forbidden induced subgraph as a certificate of non-membership can be done by using those algorithms. However The challenge is to give a certificate of membership that can be authenticated in O(n + m) time.

43 7/32 Our membership certificates Previous Work Our Results Cographs Trivially Perfect Graphs universal-in-a-component ordering (uco) A vertex ordering α = (v 1, v 2,..., v n ) of a graph G is a universal-in-a-component ordering (uco) if for 1 i n, the vertex v i is universal in the connected component of G[{v i, v i+1,..., v n }] that v i 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 cotree T of G, every 1-node has at most one child that is a 0-node.

44 Linear membership certificates Previous Work Our Results Cographs Trivially Perfect Graphs authentication For both membership certificates we provide a simple authentication algorithm with running time O(n + m). good certifying algorithm Thus we obtain two linear time certifying algorithms to recognize trivially perfect graphs and each has linear membership and sublinear non-membership certificates. 8/32

45 Previous Work Our Results Cographs Trivially Perfect Graphs Merci à tous!

46 For Further Reading I Previous Work Our Results Cographs Trivially Perfect Graphs A. Brandstädt, V. B. Le, and J. P. Spinrad, Graph classes : A survey. Philadelphia, SIAM, D. G. Corneil, H. Lerchs, and L. Stewart-Burlingham, Complement reducible graphs. Discrete Applied Mathematics 3 : ,1981. D. G. Corneil, Y. Perl, and L. K. Stewart, A linear recognition algorithm for cographs. SIAM J. Comput., 14 : , S. Földes and P. L. Hammer, Split graphs. Congressus Numerantium, 19 : , M.C. Golumbic, Trivially perfect graphs. Discrete Math. 24 : , 1978.

47 For Further Reading II Previous Work Our Results Cographs Trivially Perfect Graphs M. C. Golumbic, Algorithmic Graph Theory and Perfect Graphs. Second edition, Annals of Discrete Mathematics 57. Elsevier, M. Habib and C. Paul, A simple linear time algorithm for cograph recognition. Discrete Applied Mathematics, 145 : , P. Hell and J. Huang, Certifying LexBFS recognition algorithms for proper interval graphs and proper interval bigraphs. SIAM J. Discrete Math., 18 : , H. Kaplan and Y. Nussbaum, Certifying algorithms for recognizing proper circular-arc graphs and unit circular-arc graphs. Proc. of WG 2006, LNCS 4271, (2006), pp D. Kratsch, R. M. McConnell, K. Mehlhorn and J. P. Spinrad, Certifying algorithms to recognize interval and permutation graphs. SIAM J. Computing, 36 : , 2006.

48 For Further Reading III Previous Work Our Results Cographs Trivially Perfect Graphs K. Mehlhorn and S. Näher, LEDA : A Platform for Combinatorial and Geometric Computing, Cambridge University Press, D. Meister, Recognition and computation of minimal triangulations for AT-free claw-free and co-comparability graphs. Discrete Appl. Math., 146 : , R. E. Tarjan and M. Yannakakis, Simple linear-time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs. SIAM J. Comput., 13 : , Addendum : SIAM J. Computing, 14 : , 1985.