Packing and covering balls in graphs excluding a minor
Nicolas Bousquet, Wouter Cames van Batenburg, Louis Esperet, Gwenaël Joret, William Lochet, Carole Muller, François Pirot
PPACKING AND COVERING BALLS IN GRAPHS EXCLUDING AMINOR
NICOLAS BOUSQUET, WOUTER CAMES VAN BATENBURG, LOUIS ESPERET, GWENA¨EL JORET,WILLIAM LOCHET, CAROLE MULLER, AND FRANC¸ OIS PIROT
Abstract.
We prove that for every integer t (cid:62) c t such that forevery K t -minor-free graph G , and every set S of balls in G , the minimum size of a set ofvertices of G intersecting all the balls of S is at most c t times the maximum number ofvertex-disjoint balls in S . This was conjectured by Chepoi, Estellon, and Vax`es in 2007 in thespecial case of planar graphs and of balls having the same radius. Introduction
A hypergraph H is a pair ( V, E ) where V is the vertex set and E ⊆ V is the edge set of H .A matching in a hypergraph H is a set of pairwise vertex-disjoint edges, and a transversal is aset of vertices that intersects every edge. The matching number of a hypergraph H , denoted by ν ( H ), is the maximum number of edges in a matching. The transversal number of H , denotedby τ ( H ), is the minimum size of a transversal of H . We can also consider the linear relaxationof these two parameters: we define the fractional matching number ν ∗ ( H ) and the fractionaltransversal number τ ∗ ( H ) as follows. ν ∗ ( H ) = max (cid:88) e ∈E ( H ) w e given that (cid:80) e (cid:51) v w e (cid:54) v of H w e (cid:62) e of H , and the dual of this linear program is τ ∗ ( H ) = min (cid:88) v ∈ V ( H ) w v given that (cid:80) v ∈ e w v (cid:62) e of H w v (cid:62) v of H . By the strong duality theorem, ν ( H ) (cid:54) ν ∗ ( H ) = τ ∗ ( H ) (cid:54) τ ( H ) for every hypergraph H .Given a class C of hypergraphs, a classical problem in combinatorial optimization is to decidewhether there exists a function f such that τ ( H ) (cid:54) f ( ν ( H )) for every H ∈ C . If this isthe case the class C is sometimes said to have the Erd˝os-P´osa property . Classical examplesinclude the family of all cycles of a graph [12] (i.e. given a graph G = ( V, E ) we considerthe hypergraph with vertex set V whose edges are all the cycles of G ), and the family of all N. Bousquet is supported by ANR Project DISTANCIA (
ANR-17-CE40-0015 ). W. Cames van Batenburg,G. Joret, and F. Pirot are supported by an ARC grant from the Wallonia-Brussels Federation of Belgium. L.Esperet is supported by ANR Projects GATO (
ANR-16-CE40-0009-01 ) and GrR (
ANR-18-CE40-0032 ). C.Muller is supported by the Luxembourg National Research Fund (FNR) (Grant Agreement Nr 11628910). a r X i v : . [ m a t h . C O ] J u l N. BOUSQUET, W. CVB, L. ESPERET, G. JORET, W. LOCHET, C. MULLER, AND F. PIROT directed cycles of a directed graph [23]. A desirable property is the existence of a constant c such that τ ( H ) (cid:54) c · τ ∗ ( H ) or ν ∗ ( H ) (cid:54) c · ν ( H ), or (even better) τ ( H ) (cid:54) c · ν ( H ) for every H ∈ C . These properties are often useful in the design of approximation algorithms using aprimal-dual approach (see for instance [16, 14]).Given a graph G = ( V, E ), an integer r (cid:62)
0, and a vertex v ∈ V , we denote by B r ( v ) the ball of radius r in G centered in v , that is B r ( v ) := { u ∈ V ( G ) | d G ( u, v ) (cid:54) r } , where d G ( u, v ) denotes the distance between u and v in G (we will omit the subscript G whenthe graph is clear from the context). We say that a hypergraph H is a ball hypergraph of G if H has vertex set V = V ( G ) and each edge of H is a ball B r ( v ) in G for some integer r andsome vertex v ∈ V . If all the balls forming the edges of H have the same radius r , we say that H is an r -ball hypergraph of G .Chepoi, Estellon, and Vax`es [7] proved the existence of a universal constant ρ such thatfor every r (cid:62) G of diameter at most 2 r , the vertices of G can becovered with at most ρ balls of radius r . This result was extended to graphs embeddable on afixed surface with a bounded number of apices in [1]. Note that G has diameter at most 2 r ifand only if there are no two disjoint balls of radius r in G . Also, a set of balls of radius r in G covers all of V ( G ) if and only if their centers intersect all balls of radius r in G . Thus, theseresults state equivalently the existence of a universal constant ρ such that for every r (cid:62) G , if the r -ball hypergraph H consistingof all balls of radius r satisfies ν ( H ) = 1, then τ ( H ) (cid:54) ρ . With this interpretation in mind,Chepoi, Estellon, and Vax`es [3] conjectured the following generalization in 2007 (see also [13]). Conjecture 1 (Chepoi, Estellon, and Vax`es [3]) . There exists a constant c such that forevery integer r (cid:62) , every planar graph G , and every r -ball hypergraph H of G , we have τ ( H ) (cid:54) c · ν ( H ) . If one considers all metric spaces obtained as standard graph-metrics of planar graphs, thenConjecture 1 states that these metric spaces satisfy the so-called bounded covering-packingproperty [6]. Recently, Chepoi, Estellon, and Naves [6] showed that other metric spaces dohave this property, including the important case of Busemann surfaces. (Quoting [6], the latterare roughly the geodesic metric spaces homeomorphic to R in which the distance function isconvex; they generalize Euclidean spaces, hyperbolic spaces, Riemannian manifolds of globalnonpositive sectional curvatures, and CAT(0) spaces.)Going back to Conjecture 1, let us emphasize that a key aspect of this conjecture is that theconstant c is independent of the radius r . If c is allowed to depend on r , then the conjecture isknown to be true. In fact, it holds more generally for all graph classes with bounded expansion,as shown by Dvoˇr´ak [10].Some evidence for Conjecture 1 was given by Bousquet and Thomass´e [2], who proved thatit holds with a polynomial bound instead of a linear one. More generally, they proved that forevery integer t (cid:62)
1, there exists a constant c t such that for every integer r (cid:62)
0, every K t -minorfree graph G , and every r -ball hypergraph H of G , we have τ ( H ) (cid:54) c t · ν ( H ) t +1 .The main result of this paper is that Conjecture 1 is true, and furthermore it is not necessaryto assume that all the balls have the same radius. Theorem 2 (Main result) . For every integer t (cid:62) , there is a constant c t such that τ ( H ) (cid:54) c t · ν ( H ) for every K t -minor-free graph G and every ball hypergraph H of G . ACKING AND COVERING BALLS IN GRAPHS EXCLUDING A MINOR 3
A set S of vertices of a graph G is r -dominating if each vertex of G is at distance at most r from S , and r -independent if any two vertices of S are at distance at least 2 r + 1 apart in G . Note that if we take H to be the r -ball hypergraph consisting of all balls of radius r in G ,Theorem 2 has the following interesting graph-theoretic interpretation: if G is K t -minor-free,then the minimum size of an r -dominating set is at most c t times the maximum size of an r -independent set in G .Our proof of Theorem 2 follows a bootstrapping approach. It relies on the existence ofsome function f t such that τ ( H ) (cid:54) f t ( ν ( H )), i.e. on the Erd˝os-P´osa property of the ballhypergraphs of K t -minor-free graphs, which is used in the proof when ν ( H ) is not ‘too big’.However, showing this property was an open problem. This was known for r -ball hypergraphs,by the result of Bousquet and Thomass´e [2], but their proof method does not extend to thecase of balls of arbitrary radii. For this reason, as a first step towards proving Theorem 2,we prove Theorem 3 below establishing said Erd˝os-P´osa property. We also note that, whilethe bounding function in Theorem 3 is not optimal, it is a near linear bound of the form τ ( H ) (cid:54) c t · ν ( H ) log ν ( H ) where c t is a small explicit constant polynomial in t . This is incontrast with the constant c t in our proof of Theorem 2 which is large, exponential in t . Thus,the bound in Theorem 3 is better for small values of ν ( H ). (We note that logarithms in thispaper are natural, and the base of the natural logarithm is denoted by e.) Theorem 3 (Near linear bound) . Let G be a graph with no K t -minor and such that everyminor of G has average degree at most d . Then for every ball hypergraph H of G , τ ( H ) (cid:54) t − d · ν ( H ) · log(11e d · ν ( H )) . In particular, τ ( H ) (cid:54) ct √ log t · ν ( H ) · log( t · ν ( H )) for some absolute constant c > , and if G is planar then τ ( H ) (cid:54)
48 e · ν ( H ) · log(66 e · ν ( H )) . The proof of Theorem 3 uses known results on the VC-dimension of ball hypergraphs of G when G excludes a minor, together with classical bounds relating τ ( H ) and τ ∗ ( H ) when H has bounded VC-dimension, as well as the following theorem. Theorem 4 (Fractional version) . Let G be a graph and let d be the maximum average degreeof a minor of G . Then for every ball hypergraph H of G , we have ν ∗ ( H ) (cid:54) e d · ν ( H ) .In particular, if G is planar then ν ∗ ( H ) (cid:54) · ν ( H ) and if G has no K t -minor then ν ∗ ( H ) (cid:54) c · t √ log t · ν ( H ) , for some absolute constant c > . We note that results on the VC-dimension of ball hypergraphs in graphs excluding a minorhave also been used recently to obtain improved algorithms for the computation of the diameterin sparse graphs [9, 20].The proofs of Theorems 2, 3, and 4 are constructive, and can be transformed into efficientalgorithms producing transversals (in the case of Theorems 2 and 3) or matchings (in the caseof Theorem 4) of the desired size.The paper is organized as follows. Sections 2 and 3 are devoted to technical lemmas thatwill be used in our proofs. Theorems 3 and 4 are proved in Section 4. Theorem 2 is proved inSection 5. Finally, we conclude the paper in Section 6 with a construction suggesting thatTheorem 2 does not extend way beyond proper minor-closed classes.2.
Hypergraphs, balls, and minors
We will need two technical lemmas, whose proofs are very similar to the proof of [2, Theorem4] and [7, Proposition 1]. We start with Lemma 1, which will be used in the proofs of Theorem 4and Theorem 2. We first need the following definitions.
N. BOUSQUET, W. CVB, L. ESPERET, G. JORET, W. LOCHET, C. MULLER, AND F. PIROT
We say that two balls B and B are comparable if B ⊂ B or B ⊂ B (where ⊂ denotes the strict inclusion), and otherwise they are incomparable . Note in particular that if B = B then B and B are incomparable. Consider two intersecting and incomparable balls B = B r ( v ) and B = B r ( v ) in a graph G , and let d := d G ( v , v ). A median vertex of B and B is any vertex u lying on a shortest path between v and v , at distance (cid:98) r − r + d (cid:99) from v and at distance (cid:100) r − r + d (cid:101) from v , or symmetrically at distance (cid:100) r − r + d (cid:101) from v and at distance (cid:98) r − r + d (cid:99) from v . Since B and B intersect, we have r + r (cid:62) d andsince B and B are incomparable, we have r (cid:54) r + d and r (cid:54) r + d , and in particular (cid:98) r − r + d (cid:99) (cid:62) (cid:100) r − r + d (cid:101) (cid:62) (cid:98) r − r + d (cid:99) = (cid:98) r − r − r + d (cid:99) (cid:54) r and (cid:100) r − r + d (cid:101) = (cid:100) r − r − r + d (cid:101) (cid:54) r , so any median vertex of B and B lies in B ∩ B . Finally, note that by the definition of a median vertex u of B and B , • for every { i, j } = { , } we have r j − d ( v j , u ) (cid:54) r i − d ( v i , u ) + 1, and • if v = v (which implies B = B since the balls are incomparable), then u = v = v . Lemma 1.
Let G be a graph, let S = { B i = B r i ( s i ) } i ∈ [ n ] be a set of n pairwise incomparableballs in G , with pairwise distinct centers, and let E S ⊆ (cid:0) S (cid:1) be a subset of pairs of intersectingballs { B i , B j } ⊆ S , each of which is associated with a median vertex x { i,j } of B i and B j , andsuch that the only balls of S containing x { i,j } are B i and B j . Then the graph H = ( S, E S ) isa minor of G .Proof. Let us fix a total ordering ≺ on the vertices of G . In the proof, all distances are in thegraph G , so we write d ( u, v ) instead of d G ( u, v ) for the sake of readability. For every pair ofballs { B i , B j } ∈ E S , we write x ij or x ji instead of x { i,j } , for the sake of readability ( x ij , x ji ,and x { i,j } all correspond to the same median vertex of B i and B j ). We also let P ( s i , x ij ) be ashortest path from s i to x ij , and we assume that the sequence of vertices from s i to x ij on thepath is minimum with respect to the lexicographic order induced by ≺ (among all shortestpaths from s i to x ij ). By the assumptions, we know that P ij := P ( s i , x ij ) ∪ P ( s j , x ij ) is ashortest path from s i to s j .For every i ∈ [ n ], we define T i := (cid:91) j : { B i ,B j }∈ E S P ( s i , x ij ) . Claim 1.
For every i ∈ [ n ], T i is a tree.Assume for the sake of contradiction that there is a cycle C in T i . Observe that, by construction,if uv is an edge of T i then | d ( s i , u ) − d ( s i , v ) | = 1. Let y be a vertex of C maximizing d ( s i , y ),and let z , z denote its two neighbors in C . Then d ( s i , z ) = d ( s i , z ) = d ( s i , y ) −
1, and thereexist j , j such that z y is an edge of P ( s i , x ij ) and z y is an edge of P ( s i , x ij ). Let P and P be the subpaths from s i to y of P ( s i , x ij ) and P ( s i , x ij ), respectively. Then P and P are two different paths from s i to y , and one of them is not minimum either in terms of length,or with respect to the lexicographic order induced by ≺ . This contradicts the definition of P ( s i , x ij ) and P ( s i , x ij ). Claim 2.
For every two pairs of balls { B i , B k } , { B j , B (cid:96) } ∈ E S with i (cid:54) = j , if P ( s i , x ik ) and P ( s j , x j(cid:96) ) intersect in some vertex y such that d ( y, x ik ) (cid:54) d ( y, x j(cid:96) ), then j = k and y = x ij .Note that d ( s j , x ik ) (cid:54) d ( s j , y ) + d ( y, x ik ) (cid:54) d ( s j , y ) + d ( y, x j(cid:96) ) = d ( s j , x j(cid:96) ). Since x j(cid:96) is amedian vertex of B j and B (cid:96) , we have d ( s j , x j(cid:96) ) (cid:54) r j , which implies that d ( s j , x ik ) (cid:54) r j and ACKING AND COVERING BALLS IN GRAPHS EXCLUDING A MINOR 5 thus x ik ∈ B j . By definition, x ik is only contained in the balls B i and B k of S and thus j = k .If we also have i = (cid:96) , then necessarily y = x ij .From now on, we assume that i (cid:54) = (cid:96) . Since P ij = P ( s i , x ij ) ∪ P ( s j , x ij ) is a shortest pathcontaining the vertex y , the s j – y section of that path (which contains x ij ) has the same lengthas the s j – y section of P ( s j , x j(cid:96) ). Replacing the latter section by the former, we obtain ashortest path from s j to x j(cid:96) containing x ij , which we denote Q ( s j , x j(cid:96) ). As a consequence, d ( x j(cid:96) , x ij ) = d ( x j(cid:96) , s j ) − d ( s j , x ij ) (cid:54) r j − d ( s j , x ij ) (cid:54) r i − d ( s i , x ij ) + 1 , where the last inequality follows from the definition of x ij . We now use the fact that y appearson the path P ( s i , x ij ) and on the x ij – x j(cid:96) section of Q ( s j , x j(cid:96) ), and obtain d ( s i , x j(cid:96) ) (cid:54) d ( s i , y ) + d ( y, x j(cid:96) ) = d ( s i , x ij ) + d ( x ij , x j(cid:96) ) − d ( y, x ij ) (cid:54) r i + 1 − d ( y, x ij ) . Since x j(cid:96) / ∈ B i by definition (and so d ( s i , x j(cid:96) ) > r i ), this implies that y = x ij , as desired.This claim immediately implies that for every i, j ∈ [ n ] with i (cid:54) = j , we have V ( T i ) ∩ V ( T j ) = { x ij } if { B i , B j } ∈ E S , and V ( T i ) ∩ V ( T j ) = ∅ otherwise. Another consequence is that forevery { B i , B j } ∈ E S , the vertex x ij is a leaf in at least one of the two trees T i and T j (sinceotherwise there exist k (cid:54) = j and (cid:96) (cid:54) = i such that x ij ∈ P ( s i , x ik ) and x ij ∈ P ( s j , x j(cid:96) ), whichreadily contradicts Claim 2 above).In the subgraph (cid:83) i ∈ [ n ] T i of G , for each i ∈ [ n ] we contract each edge of T i except the onesincident to a leaf of T i . It follows from the paragraph above that the resulting graph is preciselya graph obtained from H = ( S, E S ) by subdividing each edge at most once, and thus H is aminor of G . (cid:3) The next result has a very similar proof , but the setting is slightly different. It will be usedin the proof of Theorem 2. Lemma 2.
Let G be a graph and S = { B i = B r i ( s i ) } i ∈ [ n ] be a set of n pairwise vertex-disjointballs in G , and let E S ⊆ (cid:0) S (cid:1) be a subset of pairs of balls { B i , B j } ⊆ S , each of which isassociated with a ball B { i,j } / ∈ S of G which intersects only B i and B j in S . Then the graph H = ( S, E S ) is a minor of G .Proof. Let us fix a total ordering ≺ on the vertices of G . As before, all distances are in thegraph G , and we write d ( u, v ) instead of d G ( u, v ). For every { B i , B j } ∈ E S we write B ij or B ji interchangeably for B { i,j } , and we denote by x ij the center of the ball B ij , and by r ij itsradius ( x ij = x ji and r ij = r ji ). We can assume that the centers x ij are chosen so that theradii r ij are minimal (among all balls of G not in S that intersect only B i and B j in S ).We let P ( s i , x ij ) be the shortest path from s i to x ij which minimizes the sequence of verticesfrom s i to x ij with respect to the lexicographic ordering induced by ≺ (among all shortestpaths from s i to x ij ). Observe that P ( s i , x ij ) and P ( s j , x ij ) only intersect in x ij (if not, wecould replace x ij by a vertex that is closer to s i and s j and reduce the radius r ij accordingly– the new ball B ij would still intersect B i and B j , and no other ball of S , and this wouldcontradict the minimality of r ij ). We may also assume that r i + r ij − (cid:54) d ( s i , x ij ) (cid:54) r i + r ij (otherwise we could replace x ij by its neighbor on P ( s j , x ij ) and decrease r ij by 1).For every i ∈ [ n ], we define T i := (cid:91) j : { B i ,B j }∈ E S P ( s i , x ij ) . Despite our best effort, we have not been able to prove the two results at once in a satisfactory way, i.e.with a proof that would be both readable and shorter than the concatenation of the two existing proofs.
N. BOUSQUET, W. CVB, L. ESPERET, G. JORET, W. LOCHET, C. MULLER, AND F. PIROT
Claim 1.
For every i ∈ [ n ], T i is a tree.The proof is exactly the same as that of Claim 1 in the proof of Lemma 1 (we do not repeat ithere).On the path P ( s i , x ij ), we let z i,ij be the vertex at distance r i from s i (and since x ij = x ji weuse z i,ij and z i,ji interchangeably). As we assumed above that r i + r ij − (cid:54) d ( s i , x ij ) (cid:54) r i + r ij ,we also have r ij − (cid:54) d ( x ij , z i,ij ) (cid:54) r ij . In particular, d ( x ij , z j,ij ) − (cid:54) d ( x ij , z i,ij ) (cid:54) d ( x ij , z j,ij ) + 1. Claim 2.
For every two pairs of balls { B i , B k } , { B j , B (cid:96) } ∈ E S , with i (cid:54) = j , if P ( s i , x ik ) and P ( s j , x j(cid:96) ) intersect in some vertex y such that d ( y, z i,ik ) (cid:54) d ( y, z j,j(cid:96) ), then i = (cid:96) and y = x ij .We first argue that y appears after z j,j(cid:96) when traversing P ( s j , x j(cid:96) ) from s j to x j(cid:96) . Indeed,otherwise we would have d ( s j , z i,ik ) (cid:54) d ( s j , y ) + d ( y, z i,ik ) (cid:54) d ( s j , y ) + d ( y, z j,j(cid:96) ) = d ( s j , z j,j(cid:96) ) = r j , which means that B i and B j intersect, contradicting the assumptions that i (cid:54) = j and all ballsin S are vertex-disjoint. So y lies on the z j,j(cid:96) – x j(cid:96) section of P ( s j , x j(cid:96) ), and we infer that d ( x j(cid:96) , z i,ik ) (cid:54) d ( x j(cid:96) , y ) + d ( y, z i,ik ) (cid:54) d ( x j(cid:96) , y ) + d ( y, z j,j(cid:96) ) = d ( x j(cid:96) , z j,j(cid:96) ) (cid:54) r j(cid:96) . It follows that the ball B j(cid:96) intersects the ball B i . By the assumption, this means that i = (cid:96) ,and thus s (cid:96) = s i and z j,j(cid:96) = z j,ij . We now argue that y lies in the z i,ik – x ik section of P ( s i , x ik ).Suppose for a contradiction that y appears strictly before z i,ik when traversing P ( s i , x ik ) from s i to x ik . By definition of z i,ik , it then follows that d ( s i , y ) (cid:54) r i −
1. On the other hand d ( s j , y ) = d ( s j , x ij ) − d ( y, x ij ) = d ( s j , z j,ij ) + d ( z j,ij , x ij ) − d ( y, x ij ) . Note that r i + r ij − (cid:54) d ( s i , x ij ) (cid:54) d ( s i , y ) + d ( y, x ij ) (cid:54) r i − d ( y, x ij ), and thus d ( y, x ij ) (cid:62) r ij (cid:62) d ( z i,ij , x ij ). We obtain that d ( z j,ij , x ij ) (cid:54) d ( z i,ij , x ij ) + 1 (cid:54) d ( y, x ij ) + 1, andit follows that d ( s j , y ) (cid:54) d ( s j , z j,ij ) + 1 = r j + 1. Hence d ( s i , s j ) (cid:54) d ( s i , y ) + d ( y, s j ) (cid:54) r i + r j ,so B i and B j intersect, a contradiction. We conclude that y lies in the z i,ik – x ik section of P ( s i , x ik ), and thus d ( x ik , y ) + d ( y, z i,ik ) = d ( x ik , z i,ik ).Recall that by the initial assumption of the claim, combined with i = (cid:96) , we have d ( y, z i,ik ) (cid:54) d ( y, z j,ij ). Assume first that d ( y, z i,ik ) = d ( y, z j,ij ). Then d ( x ik , z j,ij ) (cid:54) d ( x ik , y ) + d ( y, z j,ij ) = d ( x ik , y ) + d ( y, z i,ik ) (cid:54) r ik , which implies that B j intersects B ik . Thus j = k , P ( s i , x ik ) = P ( s i , x ij ), and P ( s j , x j(cid:96) ) = P ( s j , x ij ). Since these two paths have only x ij in common, in this case we conclude that y = x ij . We can now assume that d ( y, z i,ik ) (cid:54) d ( y, z j,ij ) −
1. Recall that by definition of x ij ,we have d ( x ij , z i,ij ) (cid:62) d ( x ij , z j,ij ) −
1, which implies that d ( y, z i,ik ) + d ( y, x ij ) (cid:54) d ( y, z j,ij ) − d ( y, x ij ) = d ( z j,ij , x ij ) − (cid:54) d ( z i,ij , x ij ) . Since z i,ik and z i,ij are both at distance r i from s i and P ( s i , x ij ) is a shortest path from s i to x ij , it follows that the concatenation of the s i – y section of P ( s i , x ik ) and the y – x ij section of P ( s j , x ij ) is a shortest path from s i to x ij (containing y ). As y is also on a shortest path from s j to x ij , if we had d ( y, x ij ) >
0, then we could replace x ij by y and reduce r ij to r ij − d ( y, x ij )( B ij would still intersect B i and B j and only these balls of S ), which would contradict theminimality of r ij . It follows that y = x ij , as desired.As in the proof of Lemma 1, the claim implies that for i (cid:54) = j ∈ [ n ], T i ∩ T j = { x ij } if { B i , B j } ∈ E S , and otherwise the trees T i and T j are vertex-disjoint. Another directconsequence is that for every { B i , B j } ∈ E S , the vertex x ij is a leaf in at least one of the twotrees T i and T j . As before, we can contract the edges of each tree T i not incident to a leaf of ACKING AND COVERING BALLS IN GRAPHS EXCLUDING A MINOR 7 T i , and the resulting graph is precisely a graph obtained from H = ( S, E S ) by subdividingeach edge at most once, and thus H is a minor of G . (cid:3) Hypergraphs and density A partial hypergraph of H is a hypergraph obtained from H by removing a (possibly empty)subset of the edges. In addition to hypergraphs, it will also be convenient to consider multi-hypergraphs , i.e. hypergraphs H = ( V, E ) where E is a multiset of edges. The rank of ahypergraph or multi-hypergraph H is the maximum cardinality of an edge of H .We start with a useful tool, inspired by [15] (see also [4]), itself inspired by the Crossinglemma. Given a graph G = ( V, E ), we denote by ad( G ) the average degree of G , that isad( G ) = 2 | E | / | V | . Lemma 3.
Let H = ( V, E ) be a multi-hypergraph of rank at most k (cid:62) on n vertices, andlet E ⊆ (cid:0) V (cid:1) be a set of pairs of vertices { u, v } of V such that there exists an edge e uv of H containing u and v . (Note that we allow that e uv = e xy for two different pairs { u, v } and { x, y } .) Then the graph ( V, E ) contains a subgraph H such that ad( H ) (cid:62) | E | n e k and for everyedge uv of H , the corresponding edge e uv of H contains no vertex from V ( H ) − { u, v } .Proof. Let H be the (random) graph obtained by selecting each vertex of H independentlywith probability 1 /k , and keeping a single edge (of cardinality 2) between u and v wheneverthe only selected vertices of e uv are u and v . Then we have E ( | V ( H ) | ) = nk , and E ( | E ( H ) | ) (cid:62) | E | · k (cid:18) − k (cid:19) k − > | E | e k , since k (cid:62)
2. It follows that E (cid:16) | E ( H ) | − | E | n e k | V ( H ) | (cid:17) >
0. In particular, there exists asubgraph H of ( V, E ) such that ad( H ) (cid:62) | E | n e k and for every edge uv of H , the edge e uv of H contains no vertex from V ( H ) − { u, v } , as desired. (cid:3) The proof actually gives a randomized algorithm producing the graph H . This algorithm caneasily be derandomized using the method of conditional expectations, giving a deterministicalgorithm running in time O ( | E | + n ).Given a hypergraph H and a matching B in H , we define the packing-hypergraph P ( H , B )as the hypergraph with vertex set B , in which a subset B (cid:48) ⊆ B is an edge if some edge of H intersects all the edges in B (cid:48) and no other edge of B . Lemma 4.
Let G be a graph such that each minor of G has average degree at most d , let H be a ball hypergraph of G , and let B be a matching of size n in H . For every integer k (cid:62) , thenumber of edges of cardinality at most k in the packing-hypergraph P ( H , B ) is at most (1 + d e k ) k − · n. Proof.
Let P (cid:48) be the partial hypergraph of P ( H , B ) induced by the edges of cardinality atmost k . Let H be the graph with vertex set B in which two distinct vertices are adjacent ifthey are contained in an edge of P (cid:48) (i.e. an edge of P ( H , B ) of cardinality at most k ). Let m be the number of edges of H . Applying Lemma 3 to P (cid:48) , we obtain a subgraph H (cid:48) of H ofaverage degree at least mn e k , and such that for any pair x, y of adjacent vertices in H (cid:48) , there isan edge of P (cid:48) that contains x and y and no other vertex of H (cid:48) . The vertices of H (cid:48) correspond N. BOUSQUET, W. CVB, L. ESPERET, G. JORET, W. LOCHET, C. MULLER, AND F. PIROT to a subset S of pairwise disjoint balls of G (since B is a matching), and each edge of H (cid:48) corresponds to a ball of G that intersects some pair of balls of S (and does not intersect anyother ball of S ).By Lemma 2, H (cid:48) is a minor of G , so in particular mn e k (cid:54) ad( H (cid:48) ) (cid:54) d , and hence m (cid:54) d e kn .It follows that H contains a vertex of degree at most d e k , and the same is true for every inducedsubgraph of H (since we can replace B in the proof by any subset of B ). As a consequence, H is (cid:98) d e k (cid:99) -degenerate. It is a folklore result that (cid:96) -degenerate graphs on n vertices have at most (cid:0) (cid:96)t − (cid:1) n cliques of size t (see for instance [28, Lemma 18], where the proof gives a linear timealgorithm to enumerate all the cliques of size t when t and (cid:96) are fixed), and hence there are atmost n · k (cid:88) i =1 (cid:18) (cid:98) d e k (cid:99) i − (cid:19) (cid:54) n · (1 + d e k ) k − cliques of size at most k in H , which is an upper bound on the number of edges of cardinalityat most k in P ( H , B ). (cid:3) Note that the proof gives an O ( n ) time algorithm enumerating all edges of cardinality atmost k in the packing-hypergraph P ( H , B ), when k and d are fixed (note that since H is (cid:98) d e k (cid:99) -degenerate, it contains a linear number of edges, and thus the application of Lemma 3takes time O ( n )). 4. Fractional packings of balls
We now prove Theorem 4. The proof is inspired by ideas from [22].
Proof of Theorem 4.
Let H be a ball hypergraph of G . Since ν ∗ ( H ) is attained and is a rationalnumber (recall that ν ∗ ( H ) is the solution of a linear program with integer coefficients), thereexists a multiset B of p balls of G , such that every vertex v ∈ V ( G ) is contained in at most q balls of B , and ν ∗ ( H ) = p/q (see for instance [22], where the same argument is applied tofractional cycle packings). We may assume that q is arbitrarily large (by taking k copies ofeach ball of B , with multiplicities, for some arbitrarily large constant k ), so in particular wemay assume that q (cid:62)
2. We may also assume that G contains at least one edge (i.e. d (cid:62) B as B , B , . . . , B p (and recallthat since B is a multiset, some balls B i and B j might coincide). We may assume that thereis no pair of balls B i , B j such that B i ⊂ B j (otherwise we can replace B j by B i in B , and westill have a fractional matching). It follows that the balls of B are pairwise incomparable (asdefined at the beginning of Section 2). For any two intersecting balls B i and B j we define x ij as a median vertex of B i and B j (also defined at the beginning of Section 2). Recall thatit implies in particular that whenever B i and B j intersect, x ij ∈ B i ∩ B j , and if B i and B j coincide then x ij is the center of B i and B j .We let G be the intersection graph of the balls in B , that is V ( G ) = B and two vertices B i , B j ∈ B = V ( G ) with i (cid:54) = j are adjacent in G if and only if B i ∩ B j (cid:54) = ∅ . (In particular,there is an edge linking B i and B j when B i and B j are two copies of the same ball.) Let m bethe number of edges of G . Let B ∗ denote the multi-hypergraph with vertex set B , where forevery vertex of G of the form x ij there is a corresponding edge consisting of the balls in B thatcontain x ij . Note that two distinct such vertices could possibly define the same edge, which iswhy edges in B ∗ could have multiplicities greater than 1. The multi-hypergraph B ∗ has rankat most q and contains p vertices. Note moreover that the number of pairs of vertices B i , B j of B ∗ with i (cid:54) = j such that there exists an edge of B ∗ containing B i and B j is precisely m . ACKING AND COVERING BALLS IN GRAPHS EXCLUDING A MINOR 9
By Lemma 3 applied to the multi-hypergraph B ∗ , we obtain a graph H = ( S, E S ) satisfyingthe following properties: • S ⊆ B ; • for each edge { B i , B j } ∈ E S , x ij is contained in B i and B j but in no other ball from S , and • ad( H ) (cid:62) mp e q .We would like to apply Lemma 1 to H but this is not immediately possible, since someballs of S might coincide (recall that B is a multiset), and therefore the centers of the ballsof S might not be pairwise distinct. However, observe that if two balls of S coincide, thenby definition the two corresponding vertices of H have degree either 0 or 1 in H (and in thelatter case the two vertices are adjacent in H ). Indeed, if two balls B i , B j of S coincide and B i is adjacent to B k in H with k (cid:54) = j , then the only balls of S containing x ik are B i and B k ,contradicting the fact that x ik is also in B j .Let S ⊆ S be the subset of balls of S having multiplicity 1 in S . Since no ball of B is astrict subset of another ball of B , the centers of the balls of S are pairwise distinct. As aconsequence of the previous paragraph, if we consider the subgraph H of H induced by S ,then ad( H ) (cid:54) max(1 , ad( H )).By Lemma 1 applied to the set of balls S in G , we obtain that H is a minor of G and thusad( H ) (cid:54) d . It follows that mp e q (cid:54) ad( H ) (cid:54) max(1 , d ) (cid:54) d (since d (cid:62) m/p of G is at most e dq . By the Caro-Wei inequality [5, 27] (or Tur´an’stheorem [25]), it follows that G contains an independent set of size at least | V ( G ) | ad( G ) + 1 (cid:62) p e dq + 1 = ν ∗ ( H )e d + 1 /q . An independent set in G is precisely a matching in H , and thus ν ( H ) (cid:62) d +1 /q · ν ∗ ( H ) and ν ∗ ( H ) (cid:54) (e d + 1 /q ) · ν ( H ). Since we can assume that q is arbitrarily large, it follows that ν ∗ ( H ) (cid:54) e d · ν ( H ), as desired.The rest of the result follows from well known results on the average degree of graphs. Onthe one hand, an easy consequence of Euler’s formula is that planar graphs have average degreeat most 6. On the other hand, it was proved by Kostochka [18] and Thomason [24] that every K t -minor-free graph has average degree O ( t √ log t ). (cid:3) The linear program for ν ∗ has coefficients in { , } , and can thus be solved in time O ( n ),since we can assume that the balls have pairwise distinct centers (and so the number of variablesand inequalities is linear in the number of vertices). The associated rational coefficients w e can thus be found in time O ( n ). It is then convenient to define w (cid:48) e as the largest (cid:96)n (cid:54) w e with (cid:96) ∈ N . Note that the coefficients ( w (cid:48) e ) still satisfy the inequalities of the linear programfor ν ∗ , and their sum is at least ν ∗ − n balls(since there centers are pairwise distinct). There is a small loss on the multiplicative constant(compared to the statement of Theorem 4), but we can now assume that in the proof we have q (cid:54) n and thus p (cid:54) n and m = O ( n ). It follows that the application of Lemma 3 can bedone in time O ( m ) = O ( n ), and the construction of a stable set of suitable size in G canalso be done in time O ( m ) = O ( n ). Therefore, the proof of Theorem 4 gives an O ( n ) timealgorithm constructing a matching of size Ω( ν ∗ ( H )) in H .The VC-dimension of a hypergraph H is the cardinality of a largest subset X of verticessuch that for every X (cid:48) ⊆ X , there is an edge e in H such that e ∩ X = X (cid:48) . Bousquet andThomass´e [2] proved the following result. Theorem 5. If G has no K t -minor, then every ball hypergraph H of G has VC-dimension atmost t − . A classical result is that for hypergraphs of bounded VC-dimension, τ = O ( τ ∗ log τ ∗ ). Wewill use the following precise bound of Ding, Seymour, and Winkler [8]. Theorem 6.
If a hypergraph H has VC-dimension at most δ , then τ ( H ) (cid:54) δτ ∗ ( H ) log(11 τ ∗ ( H )) . Combining Theorems 4, 5, and 6, and using that ν ∗ ( H ) = τ ∗ ( H ), we obtain Theorem 3 as adirect consequence.As before, the linear program for τ ∗ has coefficients in { , } , and can thus be solved in time O ( n ), since we can assume that the balls have pairwise distinct centers (and so the numberof variables and inequalities is linear in the number of vertices). The associated rationalcoefficients w v can thus be found in time O ( n ). Using algorithmic versions of Theorem 6(see [17, 21]) and the coefficients ( w v ), a transversal of H of size O ( τ ∗ log τ ∗ ) = O ( ν log ν )can be found by a randomized algorithm sampling O ( τ ∗ log τ ∗ ) vertices according to thedistribution given by ( w v ), or a deterministic algorithm running in time O ( n ( τ ∗ log τ ∗ ) t ). Sothe overall complexity of obtaining a transversal of the desired size is O ( n ) (randomized) and O ( n + n ( τ ∗ log τ ∗ ) t ) (deterministic). In the remainder of the paper, the result will be usedwhen τ ∗ is a fixed constant, in which case the complexity of the deterministic algorithm is also O ( n ). 5. Linear bound
In this section we prove Theorem 2. Recall that by Theorem 3, there is a (monotone)function f t such that τ ( H ) (cid:54) f t ( ν ( H )) for every ball hypergraph H of a K t -minor-free graph.In the proof, we write d t for the supremum of the average degree of G taken over all graphs G excluding K t as a minor. Recall that d t = O ( t √ log t ) [18, 24].Let t (cid:62) c t := 2 · (1 + d t e) d t / · f t ( d t ). We will prove that everyball hypergraph H of a K t -minor-free graph satisfies τ ( H ) (cid:54) c t · ν ( H ). Proof of Theorem 2.
We prove the result by induction on k := ν ( H ). The result clearlyholds if k = 0 so we may assume that k (cid:62)
1. If k (cid:54) d t then by the definition of f t we have τ ( H ) (cid:54) f t ( d t ) (cid:54) c t (cid:54) c t · k , as desired.Assume now that k (cid:62) d t and for every ball hypergraph H (cid:48) of a K t -minor-free graph with ν ( H (cid:48) ) < k , we have τ ( H (cid:48) ) (cid:54) c t · ν ( H (cid:48) ). Let G be a K t -minor-free graph and H be a ballhypergraph of G with ν ( H ) = k . Our goal is to show that τ ( H ) (cid:54) c t · k . Note that we canassume that H is minimal , in the sense that no edge of H is contained in another edge of H (otherwise we can remove the larger of the two from H , this does not change the matchingnumber nor the transversal number).Consider a maximum matching B (of cardinality k ) in H . Let E be the set consisting of allthe edges of H that intersect at most d t edges of B . Note that each edge of B lies in E , andtherefore E is non-empty. By Lemma 4, the packing-hypergraph P ( H , B ) contains at most(1 + d t e) d t / · k edges of cardinality at most d t . For each such edge e of P ( H , B ), considerthe corresponding subset B e of at most d t edges of B , and the subset E e of edges of H thatintersect each ball of B e , and no other ball of B . Denoting by H e the partial hypergraph of H with edge set E e , observe that by the maximality of the matching B we have ν ( H e ) (cid:54) d t (since in B , replacing the edges of B e by a matching of E e again gives a matching of H ). It ACKING AND COVERING BALLS IN GRAPHS EXCLUDING A MINOR 11 follows that τ ( H e ) (cid:54) f ( d t ). And thus, if we denote by H the partial hypergraph of H withedge set E , we have τ ( H ) (cid:54) (1 + d t e) d t / · f ( d t ) · k = c t · k. Consider now the subset E consisting of all the edges of H that intersect more than d t edges of B , and let H be the partial hypergraph of H with edge set E . Note that E and E partition the edge set of H and thus τ ( H ) (cid:54) τ ( H ) + τ ( H ). Let B be a maximum matchingin H , and let (cid:96) = ν ( H ) = |B | . Let H be the (bipartite) intersection graph of the edges of B ∪ B , i.e. each vertex of H corresponds to an edge of B ∪ B , and two vertices are adjacent ifthe corresponding edges intersect. (The graph is bipartite because B and B are matchings.)Note that since H is bipartite, for every two distinct edges { B, B (cid:48) } and { C, C (cid:48) } of H , thesets B ∩ B (cid:48) and C ∩ C (cid:48) are disjoint. Moreover, no ball of B ∪ B is a subset of another ball of B ∪ B , and thus the balls of B ∪ B are pairwise incomparable (as defined at the beginning ofSection 2). So, enumerating the balls in B ∪ B as B , B , . . . , B n , we can choose, for each edge { B i , B j } of H , a median vertex x ij of B i and B j (also defined at the beginning of Section 2).Recall that x ij ∈ B i ∩ B j , and thus it follows from the property above that the only balls of B ∪ B containing x ij are B i and B j . By Lemma 1, H is a minor of G and thus has averagedegree at most d t . On the other hand, the vertices of H corresponding to the edges of B havedegree at least d t in H , and thus d t · (cid:96) (cid:54) ad( H )( k + (cid:96) ) (cid:54) d t · ( k + (cid:96) ) , where the central term counts the number of edges of H . It follows that ν ( H ) = (cid:96) (cid:54) k , andthus by the induction hypothesis we have τ ( H ) (cid:54) c t · ν ( H ) (cid:54) c t · k . As a consequence, τ ( H ) (cid:54) τ ( H ) + τ ( H ) (cid:54) c t · k + c t · k = c t · k, which concludes the proof of Theorem 2. (cid:3) The first part of the proof of Theorem 2 uses Theorem 3 when ν (and thus τ ∗ , by Theorem 4)is bounded by a function of the constant t , and in this case, by the discussion after the proofof Theorem 3, a transversal of the desired size can be found deterministically in time O ( n ).The second part of the proof of Theorem 2 can be made constructive by performing thefollowing small modification. We observe that we have not quite used the fact that B is a maximum matching of H , simply that it has the property that, for any edge e in the packing-hypergraph P ( H , B ) of cardinality at most d t , the matching number of H e is bounded. Aswe have explained after Lemma 4, such edges can be enumerated in linear time when t isfixed. We can then compute each τ ∗ ( H e ) = ν ∗ ( H e ) in time O ( n ) and if this value is morethan e d t · | e | , then we can find a matching of size more than | e | = |B e | in H e in time O ( n ) byTheorem 4, and replace B e by this larger matching in B , thus increasing the size of B (thiscan be done at most ν ( H ) times). On the other hand, if for all the (linearly many) edges e asabove, we have τ ∗ ( H e ) (cid:54) e d t · | e | = O ( d t ), then by Theorem 3, we can find a transversal ofsize O ( d t log d t ) in each hypergraph H e in time O ( n ). So overall we find a matching B thathas the desired property, and a transversal of the partial hypergraph of H with edge set E ofthe desired size in time O ( ν ( H ) · n ). Taking the induction step into account (which divides ν by at least 2), we obtain a deterministic algorithm constructing a transversal of size O ( ν ( H ))in H , in time O ( (cid:80) i (cid:62) i · ν ( H ) · n ) = O ( ν ( H ) · n ), when t is a fixed constant. Conclusion
The proof of Theorem 2 gives a bound of the order of exp( t log / t ) for the constant c t . Itwould be interesting to improve this bound to a polynomial in t .It is also natural to wonder whether Theorem 2 remains true in a setting broader than properminor-closed classes. Natural candidates are graphs of bounded maximum degree, graphsexcluding a topological minor, k -planar graphs, classes with polynomial growth (meaning thatthe size of each ball is bounded by a polynomial function of its radius, see e.g. [19]), and classeswith strongly sublinear separators (or equivalently, classes with polynomial expansion [11]).We now observe that in all these cases, the associated ball hypergraphs do not satisfy theErd˝os-P´osa property, even if all the balls have the same radius. That is, we can find r -ballhypergraphs in these classes with bounded ν and unbounded τ . Our construction shows thatthis is true even in the seemingly simple case of subgraphs of a grid with all diagonals (i.e.strong products of two paths).Fix two integers k, (cid:96) with k (cid:62)
3, and (cid:96) sufficiently large compared to k and divisible by2( (cid:0) k (cid:1) − k vertices v , v , . . . v k − , an (cid:96) - broom with root v and leaves v , . . . , v k − isa tree T of maximum degree 3 with root v and leaves v , . . . , v k − such that(1) each leaf is at distance (cid:96) from the root v ,(2) the ball of radius (cid:96)/ v in T is a path (called the handle of the broom),and(3) the distance between every two vertices of degree 3 in T is sufficiently large comparedto k .We now construct a graph G k,(cid:96) as follows. We start with a set X of k vertices x , . . . , x k ,and a path of (cid:0) k (cid:1) vertices with vertex set Y = { y { i,j } | (cid:54) i < j (cid:54) k } , disjoint from X . Wethen subdivide each edge of the latter path (cid:96) ( k ) − − (cid:96)/
2. Finally, for each 1 (cid:54) i (cid:54) k , we add an (cid:96) -broom T i with root x i and leaves Y i = { y { i,j } | j (cid:54) = i } . x x x x y { , } y { , } y { , } y { , } y { , } y { , } (cid:96)/ (cid:96)/ (cid:96)/ Figure 1.
An embedding of the graph G ,(cid:96) in the 2-dimensional grid with alldiagonals (the grid itself is not depicted for the sake of clarity).We first claim that G k,(cid:96) is a subgraph of the 2-dimensional grid with all diagonals (i.e. thestrong products of two paths). To see this, place X on a single column on the left, and Y on another column on the right (in the sequence given by the path), at distance (cid:96) from the ACKING AND COVERING BALLS IN GRAPHS EXCLUDING A MINOR 13 column of X , then draw each of the brooms in the plane (with crossings allowed). Since thedistance between two vertices of degree 3 in a broom is sufficiently large compared to k , wecan safely embed each topological crossing in the strong product of two edges (see Figure 1 foran example).Let H k,(cid:96) be the (cid:96) -ball hypergraph of G k,(cid:96) obtained by considering all the balls of radius (cid:96) in G k,(cid:96) . We first observe that ν ( H k,(cid:96) ) = 1: this follows from the fact that each ball of radius (cid:96) centered in a vertex that does not belong to the handle of a broom contains all the vertices of Y , while every two vertices on the handles of two brooms T i and T j are at distance at most (cid:96) from y { i,j } . Finally, for every two vertices x i and x j of X , note that y { i,j } is the unique vertexof G k,(cid:96) lying at distance at most (cid:96) from x i and x j , and thus τ ( H k,(cid:96) ) (cid:62) k . It follows that thereis no function f such that τ ( H ) (cid:54) f ( ν ( H )) for every ball hypergraph of a subgraph of thestrong product of two paths (even when all the balls in the ball hypergraph have the sameradius). Acknowledgments.
We thank the two anonymous reviewers for their detailed commentsand suggestions.
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Laboratoire G-SCOP, CNRS, Univ. Grenoble Alpes, Grenoble, France.
E-mail address : [email protected] D´epartement d’Informatique, Universit´e Libre de Bruxelles, Brussels, Belgium.
E-mail address : [email protected] Laboratoire G-SCOP, CNRS, Univ. Grenoble Alpes, Grenoble, France.
E-mail address : [email protected] D´epartement d’Informatique, Universit´e Libre de Bruxelles, Brussels, Belgium.
E-mail address : [email protected] Algorithms Research Group, University of Bergen, Bergen, Norway.
E-mail address : [email protected] D´epartement de Math´ematique, Universit´e Libre de Bruxelles, Brussels, Belgium.
E-mail address : [email protected] D´epartement d’Informatique, Universit´e Libre de Bruxelles, Brussels, Belgium, and Labora-toire G-SCOP, CNRS, Univ. Grenoble Alpes, Grenoble, France.
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