On density of subgraphs of halved cubes Victor Chepoi, Arnaud Labourel, Sébastien Ratel To cite this version: Victor Chepoi, Arnaud Labourel, Sébastien Ratel. On density of subgraphs of halved cubes. European Journal of Combinatorics, Elsevier, 2019, 80, pp.57-70. 10.1016/j.ejc.2018.02.039. hal-02268756 HAL Id: hal-02268756 https://hal.archives-ouvertes.fr/hal-02268756 Submitted on 20 May 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. On density of subgraphs of halved cubes In memory of Michel Deza Victor Chepoi, Arnaud Labourel, and S´ebastienRatel Laboratoire d'Informatique Fondamentale, Aix-Marseille Universit´eand CNRS, Facult´edes Sciences de Luminy, F-13288 Marseille Cedex 9, France victor.chepoi, arnaud.labourel, sebastien.ratel @lif.univ-mrs.fr f g Abstract. Let S be a family of subsets of a set X of cardinality m and VC-dim(S) be the Vapnik-Chervonenkis dimension of S. Haussler, Littlestone, and Warmuth (Inf. Comput., 1994) proved that if G1(S) = (V; E) is the jEj subgraph of the hypercube Qm induced by S (called the 1-inclusion graph of S), then jV j ≤ VC-dim(S). Haussler (J. Combin. Th. A, 1995) presented an elegant proof of this inequality using the shifting operation. In this note, we adapt the shifting technique to prove that if S is an arbitrary set family and G1;2(S) = (V; E) 2 is the 1,2-inclusion graph of S (i.e., the subgraph of the square Qm of the hypercube Qm induced by S), then jEj d ∗ jV j ≤ 2 , where d := cVC-dim (S) is the clique-VC-dimension of S (which we introduce in this paper). The 1,2-inclusion graphs are exactly the subgraphs of halved cubes and comprise subgraphs of Johnson graphs as a subclass. 1. Introduction Let be a family of subsets of a set X of cardinality m and VC-dim( ) be the Vapnik- ChervonenkisS dimension of . Haussler, Littlestone, and Warmuth [19, LemmaS 2.4] proved that S if G1( ) = (V; E) is the subgraph of the hypercube Qm induced by (called the 1-inclusion S S graph of ), then the following fundamental inequality holds: jEj VC-dim( ). They used this S jV j ≤ S inequality to bound the worst-case expected risk of a prediction model of learning of concept classes based on the bounded degeneracy of their 1-inclusion graphs. Haussler [18] presented an elegantS proof of this inequality using the shifting (push-down) operation. 1-Inclusion graphs have many other applications in computational learning theory, for example, in sample compression schemes [21]. They are exactly the induced subgraphs of hypercubes and in graph theory they have been studied under the name of cubical graphs [14]. Finding a densest n-vertex subgraph of the hypercube Qm (i.e., an n-vertex subgraph G of Qm with the maximum number of edges) is equivalent to finding an n-vertex subgraph G of Qm with the smallest edge-boundary (the number of edges of Qm running between V and its complement in Qm). This is the classical arXiv:1711.11414v2 [cs.DM] 6 Dec 2017 edge-isoperimetric problem for hypercubes [3,17]. Harper [16] nicely characterized the solutions of this problem: for any n, this is the subgraph of the hypercube induced by the initial segment of length n of the lexicographic numbering of the vertices of the hypercube. One elegant way of proving this result is using compression [17]. Generalizing the density inequality jEj VC-dim( ) of [18, 19] to more general classes of jV j ≤ S graphs is an interesting and important problem. In the current paper, we present a density result for 1,2-inclusion graphs G1;2( ) of arbitrary set families . The 1,2-inclusion graphs are 2 S S the subgraphs of the square Qm of the hypercube Qm and they are exactly the subgraphs of the 1 halved cube 2 Qm+1 (Johnson graphs and their subgraphs constitute an important subclass). Since 1,2-inclusion graphs may contain arbitrary large cliques for constant VC-dimension, we have to adapt the definition of classical VC-dimension to capture this phenomenon. For this purpose, we introduce the notion of clique-VC-dimension cVC-dim∗( ) of any set family . Here is the main result of the paper: S S Theorem 1. Let be an arbitrary set family of 2X with X = m, let d = cVC-dim∗( ) be the S j j jSEj d clique-VC-dimension of and G1;2( ) = (V; E) be the 1,2-inclusion graph of . Then : S S S jV j ≤ 2 2. Related work 2.1. Haussler's proof of the inequality jEj VC-dim( ). We briefly review the notion of jV j ≤ S VC-dimension and the shifting method of [18] of proving the inequality jEj VC-dim( ) (the jV j ≤ S original proof of [19] was by induction on the number of sets). In the same vein, see Harper's proof [17, Chapter 3] of the isoperimetric inequality via compression. We will use the shifting method in the proof of Theorem 1. Let be a family of subsets of a set X = e1; : : : ; em ; can be viewed as a subset of vertices S f g S of the m-dimensional hypercube Qm. Denote by G1( ) the subgraph of Qm induced by the S vertices of Qm corresponding to the sets of ; G1( ) is called the 1-inclusion graph of [18,19]. S S X S Vice-versa, for any subgraph G of Qm there exists a family of subsets of 2 such that G is the 1-inclusion graph of . A subset Y of X is shattered by if for allS Y 0 Y there exists S such that S Y = SY 0. The Vapnik-Chervonenkis's dimensionS [28] VC-dim(⊆ ) of is the cardinality2 S of the largest\ subset of X shattered by . S S S X Theorem 2 ([18, 19]). If G := G1( ) = (V; E) is the 1-inclusion graph of a set family 2 S S ⊆ with VC-dimension VC-dim( ) = d, then jEj d: S jV j ≤ X For a set family 2 ; the shifting (push down or stabilization) operation 'e with respect to an element e XSreplaces ⊆ every set S of such that S e = by the set S e . Denote 2 S0 n f g 20 S 0 n f g by 'e( ) the resulting set family and by G = G1('e( )) = (V ;E ) the 1-inclusion graph of S S 'e( ). Haussler [18] proved that the shifting map 'e has the following properties: S 0 (1) 'e is bijective on the vertex-sets: V = V , j j j j 0 (2) 'e is increasing the number of edges: E E , j j ≤ j j (3) 'e is decreasing the VC-dimension: VC-dim( ) VC-dim('e( )). S ≥ S Harper [17, p.28] called Steiner operations the set-maps ' : 2X 2X satisfying (1), (2), and the following condition: ! (4) S T implies '(S) '(T ). ⊆ ⊆ He proved that the compression operation defined in [17, Subsection 3.3] is a Steiner operation. Note that 'e satisfies (4) (but is defined only on ). After a finite sequence of shiftings, any set familyS can be transformed into a set family ∗, ∗ ∗ S ∗ S such that 'e( ) = holds for any e X. The resulting set family , a complete shifting of , is downwardS closedS (i.e., is a simplicial2 complex). Consequently,S the 1-inclusion graph S ∗ ∗ G1( ) of is a bouquet of cubes, i.e., a union of subcubes of Qm with a common origin ?. Let ∗ S S∗ ∗ ∗ ∗ ∗ G = G1( ) = (V ;E ) and d = VC-dim( ). Since all shiftings satisfy the conditions (1)-(3), S S we conclude that V ∗ = V , E∗ E , and d∗ d. Therefore, to prove the inequality jEj d j j j j j j ≥ j j ≤ jV j ≤ jE∗j ∗ it suffices to show that jV ∗j d . Haussler deduced it from Sauer's lemma [26], however it is ≤ ∗ easy to prove this inequality directly, by bounding the degeneracy of G . Indeed, let v0 be the ∗ ∗ vertex of G corresponding to the origin ? and let v be a furthest from v0 vertex of G . Then ∗ ∗ v0 and v span a maximal cube of G (of dimension d ) and v belongs only to this maximal cube of G∗. Therefore, if we remove v from G∗, we≤ will also remove at most d∗ edges of G∗ and the resulting graph will be again a bouquet of cubes G− = (V −;E−) with one less vertex and dimension d∗. Therefore, we can apply the induction hypothesis to this bouquet G− and deduce that E−≤ V − d∗. Consequently, E∗ d∗ + E− d∗ + ( V ∗ 1)d∗ = V ∗ d∗. j j ≤ j j j j ≤ j j ≤ j j − j j 2 To extend Haussler's proof to subgraphs of halved cubes (and, equivalently, to subgraphs of squares of cubes), we need to appropriately define the shifting operation and the notion of VC- dimension, that satisfy the conditions (1)-(3). Additionally, the degeneracy of the 1,2-inclusion graph of the final shifted family must be bounded by a function of the VC-dimension.