Graphs and Orders Cours MPRI

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1 Graphs and Orders Cours MPRI Michel Habib Chevaleret novembre 2012

2 Table des Matières Introduction Definitions Dilworth theorem Applications Comparability invariants Permutation graphs Transitive Orientation Some conjectures

3 Introduction Modelisations using partial orders are very important in applications : distributed systems, logics, physics, biology...

4 Introduction Examples

5 Introduction Examples

6 Introduction Examples

7 Introduction Examples

8 Introduction If the data is not acurate enough, the natural generalization is to consider paths on a rooted directed tree, which leads to directed path graphs. But we could also deal with subtrees but then we found another subclass of chordal graphs. In some particular cases we can only use the size of the intersection of intervals this leads to tolerance graphs and orders.

9 Introduction 3 simple paradigms to solve an algorithmic problem Play with the definition of the problem (ex : compute factoring permutation instead of modular decomposition tree) Play with the representations (ex : use various geometric representations of particular classes of graphs) Play with dualities, if any (ex : G versus G, or in case of planarity use the dual) Of course this often is clear, only afterwards!

10 Introduction Playing with the representations : common intervals Let τ and σ be two permutations on [1, n]. wlog τ is supposed to be the identity 2 permutations τ= σ= Definition Two intervals I, J [1, n] are common intervals of τ and σ if τ(i ) = σ(j) as subsets of [1, n].

11 Introduction example τ= σ= The ordering of the elements may differ. [3, 4, 5, 6] and [4, 6, 3, 5] are the unique non trivial maximal common intervals of τ and σ.

12 Introduction Problems The size of the data is in O(n). 1. Propose an algorithm which computes all non trivial maximal common intervals 2. Propose an algorithm which computes all common intervals 3. Same problems for k permutations 4. Same problems with some fixed number of errors

13 Definitions Graphs and orders Definitions P = (X, ) is an order if the relation is reflexive, transitive and antisymmetric. Q = (X, ) is a quasi order if the relation is reflexive and transitive. If either x P y or y P x, then x, y are said to be comparable, else they are incomparable (denoted by x y). We say that y covers x denoted by x < y, if there no z X, z x, y such that x P z P y.

14 Definitions Definitions An antichain (resp. chain) is a set of pairwise incomparable (resp. comparable) elements. N.B. a chain is a directed path A linear extension τ of P is a total order compatible with P, i.e. x P y implies x τ y.

15 Definitions A partial order can be represented by an acyclic directed graph G = (X, U) satisfying x P y iff a path from x to y in G. If xy U and it exists a path of length 2 from x to y in G, then xy is called a transitivity arc, else it is a covering arc. Among the lattice of these acyclic graphs, they are two extremal ones : G t = (X, U t ) the transitive closure of P, for which x P y iff x, y U t. G r = (X, U r ) the transitive reduction of P, having no transivity arcs. Other names : Hasse diagram, directed covering graph. G t has all possible transitivity arcs, while G r has none.

16 Definitions Unfortunately these two representations of an order have not the same complexity or size. Since for a chain order, G t = (X, U t ) can be quadratic in size of G r = (X, U r ). The transitive closure-reduction gap It is not known if it is possible to extract G t = (X, U t ) or G r = (X, U r ) from a given representation G of P in linear time.

17 Definitions Best known algorithms require either O(n.m) or O(n α ), with α > 2. Best complexity for boolean matrix multiplication. Graal for algorithms on partial orders A linear-time algorithm accepting any given representation G of P.

18 Definitions Other representations : planar diagrams.

19 Definitions Undirected graphs associated with an order Comparability graphs An undirected graph G = (X, E) is a comparability graph if it can be directed as a partial order (more precisely as the graph of the transitive closure) Covering graph An undirected graph G = (X, E) is a covering graph if it can be directed as the directed covering graph of a partial order (more precisely as the graph of the transitive reduction of the partial order)

20 Definitions Examples

21 Definitions These two notions are very different It is polynomial to recognize a comparability graph, while it is NP-hard for a covering graph (Nesestril, Rödl 89). There exists a nice characterization of comparability graphs, while there exist only necessary conditions for covering graphs.

22 Definitions Examples of comparability graphs Tree are comparability graphs, but C 5 a cycle with 5 vertices is not a comparability graph. Interval graphs are complement of comparability graphs (cocomparability graphs). Chordal are not always comparability graphs (cf. the 3-sun or its complement)

23 Definitions Counterexamples a b c f d e Figure: A chordal graph which is not a comparability graph

24 Definitions The duality between graphs and orders Co-comparability graphs The complement of a comparability graph is called a co-comparability graph. There is a kind of duality between a co-comparability graph and the partial orders associated with its complement. Examples cographs and series-parallel orders Interval graphs and interval orders Permutation graphs and permutation orders also called 2-dimensional orders. Trapezoid graphs and trapezoid orders...

25 Definitions Using this duality can help to solve a problem (many examples of that). Two examples follow

26 Dilworth theorem Dilworth theorem Dilworth 1950 For every finite order P, The maximum size of an antichain (denoted by width(p)) is equal to the minimum size of a chain partition of P (denoted by θ(p)). Consequences Comparability and co-comparability graphs are perfect

27 Dilworth theorem Applications Another min-max polynomial theorem similar to max flow min cut. Computation of width(p) can be done in O(n 5/2 ) using a maximum matching algorithm.

28 Dilworth theorem Applications Erdös, Szekeres 1935 From every sequence of pq + 1 integers one can always extract a decreasing subsequence of size p + 1 or an increasing one of size q + 1.

29 Dilworth theorem Applications Application for computing a maximum independent set in a comparability graph G 1. Transitively orient G as a partial order P 2. Compute a minimal path partition of P via a matching algorithm on a bipartite graph 3. Extract an independent set form this set of paths

30 Dilworth theorem Comparability invariants Comparability invariants For an invariant f defined on partial orders : f : P f (P) N f is a comparability invariant if for every partial orders P, Q having the same comparability graphs : f (P) = f (Q).

31 Dilworth theorem Comparability invariants Examples of comparability invariants f = being an interval order (resp. a permutation ordering) The minimum size of a chain decomposition of P (using Dilworth s theorem). The dimension of a partial order dim(p) (i.e. the minimum number of linear extensions whose intersection is P) The total number of linear extensions of P, denoted by Ext(P). #P-complete to compute Ext(P) Brightwell, Winkler

32 Dilworth theorem Comparability invariants As the duality operation on planar graphs, we can use this duality between undirected graphs (co-comparability graphs) and orders to solve some optimization problems which can be easier on the dual, if the optimization problem corresponds to a comparability invariant. This can produce efficient algorithms in particular when using partition refinement tools which can be applied within the same complexity on the graph or its complement. Examples : Recognition of permutation graphs and generalization to trapezoid graphs. Computation of a minimum path covering of a comparability graph (Corneil, Dalton, Habib 2011)...

33 Dilworth theorem Comparability invariants Characterization theorem : MH 84, Dresen, Pogunke, Winkler 85 For an invariant f defined on partial orders : f : P f (P) N f is a comparability invariant iff it satisfies : P, f(p)=f (P ) P, Q partial orders and x P, f (P Q x ) = f (P Q x )

34 Dilworth theorem Comparability invariants Partial order dimension Spiljzran Lemma Let P = (X, ) be a partial order, and a, b X with a b. Then Q = (P + {a < b}) t is a partial order extension of P. Using repeatedly this lemma with will reach a linear extension of P, moreover any linear extension of P can be obtained that way.

35 Dilworth theorem Comparability invariants Definition dim(p) = min k s.t. P = 1 i k L i where L i is a linear extension of P dim(p) n dim(p Q x ) = max(2, dim(p), dim(q)) (Consequences permutation graphs are closed under substitution)

36 Dilworth theorem Comparability invariants Geometric representation Every co-comparability graph can be represented as the intersection graph of k-polylines Application : a way to randomly generate co-comparability graphs.

37 Permutation graphs Permutation graphs

38 Permutation graphs

39 Permutation graphs

40 Permutation graphs

41 Permutation graphs Characterization theorem The 3 following statements are equivalent : 1. G is a permutation graph 2. G and G are comparability graphs

42 Permutation graphs Being a permutation graph is a comparability invariant. Dushnik-Miller s theorem (1940) 1. P is a two dimensional order 2. P admits a non-separating linear extension Nice lemma Let P (resp. Q) be a transitive orientation of a comparability graph G (resp. G). Then P + Q is a total ordering.

43 Transitive Orientation Original problem Find for each edge in G a unique orientation, such that the resulting graph is acyclic and transitive. Little Variation Find a linear extension of P which is an acyclic orientation of G.

44 Transitive Orientation Transitive orientation Strong relationship with modular decomposition Lemma MacConnell, Spinrad The last vertex of a LBFS on G can be taken as a source in G. Lemma Let M be a module, the restriction of a LBFS to M is a legitimate LBFS. Consequence An O(n + mlogn) algorithm for transitive orientation using LBFS.

45 Transitive Orientation Transitive Orientation Algorithm 1. Compute σ a LBFS ordering of G 2. Take x = σ(n) as a source and propagate using generic partition refinement Initial partition : {x, N(x), N(x)} Insertion rule : N(x) close to pivot.

46 Transitive Orientation Main Invariant It exists a transitive orientation of G compatible with the current ordered partition. Checking Transitivity Checking the result requires boolean matrix multiplication. Not yet linear? Computing is easier than checking!

47 Transitive Orientation Application to recognize permutation graphs 1. Compute a LBFS on G 2. Compute a linear extension τ of P 3. Compute an orientation H G of G 4. Replay the 3 previous steps for G 5. Compute : σ = H G + H G σ = H G + H G 6. Check if σ and σ provide a good representation of G

48 Graphs and Orders Cours MPRI Transitive Orientation

49 Transitive Orientation

50 Transitive Orientation

51 Transitive Orientation Why this algorithm is Fantastic? 1. The whole complexity is linear (a little tricky for the step 5, since it seems necessary to pay O(n 2 )) 2. Apply dually on G and G using partition refinement 3. Avoids the checking of transitivity.

52 Some conjectures To finish with two nice and old conjectures The 1/3-2/3 Conjecture This conjecture asserts that every finite partially ordered set that is not a chain contains an incomparable pair (x,y) so that 1/3 Prob[x < y] 2/ shows that the bound is tight. Kahn and Saks proved that there is always a pair for which 3/11 Prob[x < y] 8/11

53 Some conjectures Seymour s second neighbourhood conjecture Directed graphs (digraphs) are orientations of graphs, so they do not contain loops, parallel arcs, or digons (directed cycles of length 2). N + (x) = successors of x (at distance exactly one). N ++ (x) =vertices at distance exactly 2 from x Conjecture For every directed graph G there exists at least one vertex x such that : N ++ (x) N + (x)

Chordal graphs MPRI

Chordal graphs MPRI 2017 2018 Michel Habib habib@irif.fr http://www.irif.fr/~habib Sophie Germain, septembre 2017 Schedule Chordal graphs Representation of chordal graphs LBFS and chordal graphs More structural