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. ∈ I ∈ I ⊆ . Example:Example: V = {1, 2, 3, 4}; ⇒ { , {1}, {2}, {3}, {4}, {1, 2}, {1, 3}, {2, 3}, {3, 4}, {1, 2, 3}} ∅

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. ∈ I ∈ I ⊆ . Example:Example: V = {1, 2, 3, 4}; ⇒ { , {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {3, 4}, {1, 2, 3}}. ∅

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. ∈ I ∈ 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 = { , 1, 2, 3, 4, 12, 13, 23, 34, 123}. 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 , |X| > |Y| = z X \ Y s.t. Y {z} . ∈ I ∃ ∈ ∪ ∈ I

⇒ z = ∃ 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 = {X V : X 1 and X 2}. I ∩ I ⊆ ∈ I ∈ 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 = {X V : X 1 and X 2}. I ∩ I ⊆ ∈ I ∈ 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 = {X V : X 1 and X 2}. I ∩ I ⊆ ∈ I ∈ 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

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 = {1, 2, 3, 4} 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 = {V1, V2, . . . , Vr} a partition of V P . Def.:Def.: A partition matroid ( ) of is I P P Def.: ( ) = {X V : |X Vi| 1 for all i = 1, . . . , r}. I P ⊆ ∩ ≤ . Example:Example: = {V1, V2, V3} P 3 4 ( ) = 2 I{146,P 147, 148, 156, 157, 158, 1 5 {246, 247, 248, 256, 257, 258, V2 V1 {346, 347, 348, 356, 357, 358, {and their subsets} 6 7 8 V3 15 15 () Def.: Partition matroid V apnonempty ﬁnite set = {V1, V2, . . . , Vr} a partition of V P . Def.:Def.: A partition matroid ( ) of is I P P Def.: ( ) = {X V : |X Vi| 1 for all i = 1, . . . , r}. I P ⊆ ∩ ≤ . Example:Example: = {V1, V2, V3} 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 = {V1, . . . , Vr} 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 = {V1, . . . , Vr} 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 ). ⊆ 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 No red edge is contained in all of these graphs. 22 23 () To summarize G =p(V, E) a graph 1, 2, . . . , k stable-set partitions of G P P P cliq(G) = { ( i) : i = 1, . . . , k} I P T

No red edge is contained in all G i ’s. P m

23 24 () In other words... . RemaRemap rk:rk: G = (V, E) a graph cliq(G) is the intersection of k matroids

k stable-set partitions 1, 2, . . . , k of G s.t. ∃ P P P cliq(G) = { ( i) : i =m1, . . . , k} I P T k stable-set partitions 1, 2, . . . , k of G s.t. ∃ P P P no red edge is containedmin all G i ’s. P . ProProof.of. Omitted. 24 25 () Moreover... . ThmThmp 2:2: G = (V, E) a graph cliq(G) is the intersection of k matroids

k stable-set partitions 1, 2, . . . , k of G s.t. ∃ P P P cliq(G) = { ( i) : i =m1, . . . , k} I P T k stable-set partitions 1, 2, . . . , k of G s.t. ∃ P P P no red edge is containedmin all G i ’s. P . ProProof.of. Omitted. 25 26 () What does this theorem mean? (1) Givenp a graph G. Partition matroids arising from stable-set partitions of G

Partition matroids

Matroids When we want a representation of cliq(G) by k matroids, just search k partition matroids from stable-set partitions of G. 26 27 () What does this theorem mean? (2) Considerp the following decision problem. Instance: a graph G and a positive integer k Question: ?? k matroids 1, . . . , k ∃ I I s.t. cliq(G) = 1 k ?? I ∩ · · · ∩ I Thm 2 implies: This problem . ∈ N P

27 28 () 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 (ﬁve more pages!!!)

28 29 () Clique complexes represented by two matroids . ThmThmp 3:3: G a graph cliq(G) is the intersection of two matroids

The stable-set graph of G is bipartite. Theom rem 3 and . Note: Combining Note: Protti–Szwarcﬁter (’02), we can tell, in polynomial time, whether the clique complex of a given graph is the intersection of two matroids or not. 29 29 () Clique complexes represented by two matroids . ThmThmp 3:3: G a graph cliq(G) is the intersection of two matroids

The stable-set graph of G is bipartite. Theom rem 3 and . Note: Combining Note: Protti–Szwarcﬁter (’02), we can tell, in polynomial time, whether the clique complex of a given graph is the intersection of two matroids or not. 29 30 () Complexity issue Considerp the following decision problem. Fix: a positive integer k Instance: a graph G

Question: ? k matroids 1, . . . , k ∃ I I s.t. cliq(G) = 1 k ? I ∩ · · · ∩ I What is the complexity of this problem for each k? k 1 2 3 4 ? ? · · · P P · · ·

30 31 () Graphs as independence systems A graphp can be seen as an independence system: or X G X = {∅v} or ∈  {u, v} if it’s an edge. ⇔ . ThmThm 4:4: G a graph  G is the intersection of k matroids

cliq(G) is the intersection of k matroids. m 31 32 () Number of matroids we need Letp k(G) = min{k : cliq(G) is the intersection of k matroids }. k(n) = max{k(G) : G has n vertices }.

. ThmThm 5:5: k(n) = n − 1.

32 33 () Summary . QuestionQuestionp

How many matroids do we need to represent a given clique complex as their intersection??

. ResultResult

cliq(G) is the intersection of k matroids

cliq(G) is the intersection of k partition matroids from stable-setm partitions of G . [End of the talk] 33