Matroid Representation of Clique Complexes Kenji Kashiwabara (Univ Tokyo) Yoshio Okamoto* (ETH Zurich) Takeaki Uno (NII Japan) July 25{28, 2003 COCOON 2003 @ Big Sky Resort, MT Supported by the Berlin-Zurich¨ * Joint Graduate Program 1 () Authors p K. Kashiwabara Y. Okamoto T. Uno Univ Tokyo ETH NII Japan Switzerland Japan 1 2 () Overview . BackgroundBackgroundp An independence system can be represented as the intersection of finitely many matroids. The clique complex of a graph is an independence system. QuestionQuestion How many matroids do we need for clique complexes? . MotivationMotivation Given later. 2 3 () Def.: independence system Vpa nonempty finite set . Def.:Def.: an independence system on V is Def.: a nonempty family of subsets of V s.t. I X = Y for all Y X. 2 I 2 I ⊆ . Example:Example: V = f1; 2; 3; 4g; ) f ; f1g; f2g; f3g; f4g; f1; 2g; f1; 3g; f2; 3g; f3; 4g; f1; 2; 3gg ; 3 3 () Def.: independence system Vpa nonempty finite set . Def.:Def.: an independence system on V is Def.: a nonempty family of subsets of V s.t. I X = Y for all Y X. 2 I 2 I ⊆ . Example:Example: V = f1; 2; 3; 4g; ) f ; f1g; f2g; f3g; f1; 2g; f1; 3g; f2; 3g; f3; 4g; f1; 2; 3gg. ; 3 3 () Def.: independence system Vpa nonempty finite set . Def.:Def.: an independence system on V is Def.: a nonempty family of subsets of V s.t. I X = Y for all Y X. 2 I 2 I ⊆ . RemaRemark:rk: ) also called an abstract simplicial complex. 3 4 () Typical independence systems from ... p From graphs the family of the cliques •• the family of the forests •• the family of the matchings •• From partially ordered sets the family of all chains •• From polytopes the family of all faces of a simplicial polytope •• From game theory the family of losing coalitions of a weighted •• majority game 4 5 () Understanding independence systems = f ; 1; 2; 3; 4; 12; 13; 23; 34; 123g. I p ; 1234 V 124 234 123 134 12 13 23 14 24 34 1 2 3 4 ; ; (known as the face poset of , when we see as an abstractI simplicial complex) I 5 6 () Def.: matroids . Def.:Def.:p An indep. system is called a matroid I Def.: if satisfies the augmentation axiom: I X; Y ; jXj > jYj = z X n Y s.t. Y fzg . 2 I 9 2 [ 2 I ) z = 9 X Y Y ) X Appearance of matroids: Combinatorial optimization,Combinatorial geometry, Combinatorial topology, Coding theory, Design theory, etc. 6 7 () Intersection of independence systems Thep intersection of indep. systems 1, 2 on V is I I 1 2 = fX V : X 1 and X 2g: I \ I ⊆ 2 I 2 I 1 2 I I 1 2 I \ I 7 7 () Intersection of independence systems Thep intersection of indep. systems 1, 2 on V is I I 1 2 = fX V : X 1 and X 2g: I \ I ⊆ 2 I 2 I The intersection of more indep. systems is defined analogously. 7 7 () Intersection of independence systems Thep intersection of indep. systems 1, 2 on V is I I 1 2 = fX V : X 1 and X 2g: I \ I ⊆ 2 I 2 I The intersection of more indep. systems is defined analogously. The intersection of indep. systems is again an indep. system. 7 7 () Intersection of independence systems Thep intersection of indep. systems 1, 2 on V is I I 1 2 = fX V : X 1 and X 2g: I \ I ⊆ 2 I 2 I The intersection of more indep. systems is defined analogously. The intersection of indep. systems is again an indep. system. The intersection of matroids is an indep. system but not necessarily a matroid. However ... 7 8 () An important fact . FFpact:act: Every indep. system is the intersection of Fact: finitely many matroids. ProProof.of. Min. sets not in are shown by . I • V V V 1 2 I I V ; ; I 3 I ; = 1 2 3 I I \ I \ I ; 8 9 () Question . Question:Question:p How many matroids do we need to represent a given indep. system as their intersection?? . The proof in the prev. page says that: this number the number of for . ≤ • I But, we might do better. F We study this number for clique complexes. 9 10 () Def.: Clique complex . Def.:Def.:p The clique complex of a graph G = (V; E) Def.: is the family of all cliques of G. Example:Example: 1234 V = f1; 2; 3; 4g 124 234 123 134 1 12 13 23 14 24 34 1 2 3 4 2 3 4 G cliq(G) ; 10 11 () Question (again) . Question:Question:p How many matroids do we need to represent a clique complex as their intersection?? . Result:Result: We investigate and characterize these numbers. However, you may ask... (1) Why do we consider such a representation?? (2) Why do we consider clique complexes?? 11 11 () Question (again) . Question:Question:p How many matroids do we need to represent a clique complex as their intersection?? . Result:Result: We investigate and characterize these numbers. However, you may ask... (1) Why do we consider such a representation?? (2) Why do we consider clique complexes?? 11 12 () Motivation (1) . Q.Q.p Why do we consider such a representation? . Ans.Ans. in the following proposition. Prop.:Prop.: (Jenkyns, '76; Korte{Hausmann, '78) (Translation of their prop. to clique complexes) cliq(G) is the intersection of k matroids = Greedy Alg. is a k-approx. alg. for = Max weighted clique problem in G. k )how complex G is w.r.t. clique problem. F ≈ ) 12 12 () Motivation (1) . Q.Q.p Why do we consider such a representation? . Ans.Ans. in the following proposition. Prop.:Prop.: (Jenkyns, '76; Korte{Hausmann, '78) (Translation of their prop. to clique complexes) cliq(G) is the intersection of k matroids = Greedy Alg. is a k-approx. alg. for = Max weighted clique problem in G. k )how complex G is w.r.t. clique problem. F ≈ ) 12 13 () Motivation (2) . Q.Q.p Why do we consider clique complexes? . Ans.Ans. The class of clique complexes includes Ans. some important classes like the classes of the family of the matchings of a graph G = cliq(the complement of the line graph of G) studied by Fekete{Firla{Spille '03 in the same framework as ours. the family of the chains of a poset P = cliq(the comparability graph of P) ... 13 13 () Motivation (2) . Q.Q.p Why do we consider clique complexes? . Ans.Ans. The class of clique complexes includes Ans. some important classes like the classes of the family of the matchings of a graph G = cliq(the complement of the line graph of G) studied by Fekete{Firla{Spille '03 in the same framework as ours. the family of the chains of a poset P = cliq(the comparability graph of P) ... 13 14 () The talk plan In thep rest of my talk (1) clique complex which is a matroid Key concept: partition matroid (2) clique complex which is the intersection of k matroids Key concept: stable-set partition (3) other results 14 15 () Def.: Partition matroid V apnonempty finite set = fV1; V2; : : : ; Vrg a partition of V P . Def.:Def.: A partition matroid ( ) of is I P P Def.: ( ) = fX V : jX Vij 1 for all i = 1; : : : ; rg. I P ⊆ \ ≤ . Example:Example: = fV1; V2; V3g P 3 4 ( ) = 2 fI146;P 147; 148; 156; 157; 158; 1 5 f246; 247; 248; 256; 257; 258; V2 V1 f346; 347; 348; 356; 357; 358; fand their subsetsg 6 7 8 V3 15 15 () Def.: Partition matroid V apnonempty finite set = fV1; V2; : : : ; Vrg a partition of V P . Def.:Def.: A partition matroid ( ) of is I P P Def.: ( ) = fX V : jX Vij 1 for all i = 1; : : : ; rg. I P ⊆ \ ≤ . Example:Example: = fV1; V2; V3g P 3 4 2 5 1 . Notice:Notice: In fact, V V2 ( ) is a matroid. 1 I P 6 7 8 V3 15 16 () Partition matroids are clique complexes . Lem.:Lem.:p Partition matroids are clique complexes. ProProof.of. Given a partition = fV1; : : : ; Vrg of V. P V1 V2 V1 V2 V3 V3 Construct a complete r-partite graph G s.t. P each part of G corresponds to Vi. P Then, cliq(G ) = ( ). [qed] P I P 16 17 () Consequence p . RemaRemark:rk: G a graph G is complete multipartite cliq(G) is a matroid. + 17 18 () Matroidal clique complexes p . ThmThm 1:1: G a graph G is complete multipartite cliq(G) is a matroid. m . ProProof.of. Omitted. 18 19 () The talk plan In thep rest of my talk (1) clique complex which is a matroid Key concept: partition matroid (2) clique complex which is the intersection of k matroids Key concept: stable-set partition (3) other results 19 20 () Def.: Stable-set partitions G =p(V; E) a graph . Def.:Def.: A stable-set partition of G Def. is a partition = fV1; : : : ; Vrg of V P Def. s.t. Vi is a stable set of G (i = 1; : : : ; r). Example:Example: 20 21 () Construct a graph from a stable-set partition G apgraph a stable-set partition of G P Construct a complete r-partite graph G from . P P . Observation:Observation: An edge of G is an edge of G . P This means: cliq(G) cliq(G ) = ( ). ⊆ P I P G G P 21 22 () Three stable-set partitions Generallyp , cliq(G) cliq(G 1 ) cliq(G 2 ) cliq(G 3 ). ⊆ P \ P \ P However, in this example, G 1 G 2 G 3 P P P cliq(G) = cliq(G 1 ) cliq(G 2 ) cliq(G 3 ) P \ P \ P That is because 22 22 () Three stable-set partitions Generallyp , cliq(G) cliq(G 1 ) cliq(G 2 ) cliq(G 3 ).