A note on the random greedy triangle-packing algorithm

A note on the r andom greedy triangle-pac king algorithm T om Bohman ∗ Alan F rieze † Ey al Lub etzky ‡ Abstract The random greedy algorithm for constructing a large p artial Steiner-T riple-System is defined as follo ws. W e b egin with a complete graph on n v ertices and pro ceed to remo v e the edges of triangles one at a time, wh ere eac h triangle remov e d is c hosen uniformly at rand om from the collection of all remaining triangles. This sto c hastic pro cess t erminates o nce i t a rrive s at a triangle-free graph. In this note we sho w that with high probabilit y the n um b er of edges in t he final graph is at most O  n 7 / 4 log 5 / 4 n  . 1 In tro du ction W e consider the random greedy alg orithm for triangle-packing. This sto c hastic graph pro cess b egins with the graph G (0 ), set to b e the complete graph o n ve rtex set [ n ], then pro ceeds to rep eatedly remov e the edges of r a ndomly c hosen triangles (i.e. copies of K 3 ) from the graph. Namely , letting G ( i ) denote the graph that remains after i triangles ha v e b een remo v ed, the ( i + 1)-th triangle remo v e d is c hosen uniformly at random from the set of all triangles in G ( i ). The pro cess terminates at a triangle- free graph G ( M ). In this w ork we study the random v ariable M , i.e., the nu m b er of triangles remo v e d until o bt a ining a triang le-free g raph (or equiv alen tly , ho w many edges t here a re in the final t riangle-free gra ph). This pro cess and its v ariations play an imp o rtan t role in the history of com binatorics. Note that the collection of triangles remo v ed in the course of the pro cess is a maximal collection of 3-elemen t subsets o f [ n ] with t he prop erty that a n y pair of distinct triples in the collection hav e pairwise interse ction less tha n 2. F or integers t < k < n a p artial ( n, k , t ) - Steiner system is a collection of k - elemen t subsets of an n -elemen t set with the pro p ert y that an y pairwise in t ersection of sets in the collection ha s cardinality less than t . Not e that the n umber o f sets in a par t ia l ( n, k , t )-Steiner system is at most  n t  /  k t  . Let S ( n, k , t ) b e the ∗ Department o f Mathematical Sciences, Carnegie Mellon Universit y , Pittsburg h, P A 1 5213 , USA. E ma il: tbohma n@mat h.cmu.e du . Research s upp o rted in part by NSF g rant DMS-07 0118 3. † Department o f Mathematical Sciences, Carnegie Mellon Universit y , Pittsburg h, P A 1 5213 , USA. E ma il: alan@r andom .math.c mu.edu . Res e arch supp or ted in part b y NSF grant DMS-07 2187 8. ‡ Microsoft Res earch, O ne Micr osoft W ay , Redmond, W A 9805 2, USA. Email: e yal@m icros oft.com . 1 maxim um num b er of k - sets in a partial ( n, k, t )-Steiner system. In the early 1960’s Erd˝ os and Ha nani [5] conjectured that for any in t egers t < k lim n →∞ S ( n, k , t )  k t   n t  = 1 . (1) In w ords, for any t < k there exist partial ( n, k, t )-Steiner systems that are essen tially as large as a llow ed b y the simple volume upp er b ound. This conjecture w as pro v ed b y R¨ odl [7] in the early 1980’s by w ay of a randomized construction t ha t is now kno wn a s the R ¨ odl nibble. This construction is a semi-random v ariatio n on the random greedy tria ngle-pac king pro cess defined ab ov e, and thereafter suc h semi-random constructions hav e b een successfully applied to establish v arious k ey results in Com binatorics ov er t he last three decades (see e.g. [1] for further details). Despite the succ ess of the R¨ odl nibble, the limiting b eha vior of the random greedy pac king pro cess r emains unkno wn, eve n in the sp ecial case of triangle pac king considered here. Recall that G ( i ) is the graph remaining a fter i triangles ha v e b een remo v ed. Let E ( i ) b e the edge set of G ( i ). Note that | E ( i ) | =  n 2  − 3 i and that E ( M ) is the num b er of edges in the triangle-free graph pro duced b y the pro cess. Observ e that if we sho w | E ( M ) | = o ( n 2 ) with non-v anishing probabilit y then w e will establish (1) for k = 3 , t = 2 and obtain that the random greedy triangle-pac king pro cess pro duces an asymptotically optimal partia l Steiner system. This is in fa ct the case: It w as sho wn by Sp encer [9] a nd indep enden tly by R¨ odl and Thoma [7] that | E ( M ) | = o ( n 2 ) with high probability 1 . This was extended to | E ( M ) | ≤ n 11 / 6+ o (1) b y Grable in [6], where the author f ur t her sk etc hed how similar a r g umen ts using mo r e delicate calculations should extend to a b ound of n 7 / 4+ o (1) w.h.p. By comparison, it is widely b eliev ed that the g r a ph pro duced b y the random gr eedy triangle-pac king pro cess b ehav es similarly to the Erd˝ os-R´ en yi random graph with the same edge densit y , hence the pro cess should end once its n um b er of remaining edges b ecomes comparable to the n umber o f triangles in the corresp onding Erd˝ os-R ´ en yi random gra ph. Conjecture (F olklore) . With h igh pr ob ability | E ( M ) | = n 3 / 2+ o (1) . Jo el Sp encer ha s offered $200 for a resolution o f this question. In this note we apply the differential-equation metho d to ac hiev e an upp er b ound on E ( M ). In contrast to the aforemen tioned nibble-approac h, whose application in t his setting in volv es delicate calculations, our approac h yields a short pro o f of the following b est-known result: Theorem 1. Conside r the r andom gr e e dy alg orithm f or triangle-p acking on n vertic es . L et M b e the numb er of steps it take s the algorithm to terminate and let E ( M ) b e the e dges o f the r esulting triangle-fr e e gr aph. T hen with high p r ob abi lity, | E ( M ) | = O  n 7 / 4 log 5 / 4 n  . 1 Here and in what follows, “with high probability” ( w.h.p.) denotes a pro ba bility tending to 1 as n → ∞ . 2 W ormald [11] also applied the differen t ial-equation metho d to this problem, deriving an upp er b ound of n 2 − ǫ on E ( M ) for any ǫ < ǫ 0 = 1 / 57 while stating that “some no n-trivial mo dification would b e required to equal o r b etter Grable’s result.” Indeed, in a companio n pap er we com bine the metho ds in tro duced here with some other ideas (and a significan tly more in volv ed analysis) to improv e the exp onent o f the upp er b ound on E ( M ) to a b out 1 . 65. This f ollo w-up w ork will app ear in [3]. 2 Ev oluti on of the pro cess in detail As is usual for applications of the differen tial equations metho d, w e b egin by sp ecifying the random v ariables that w e tra ck. Of course, our main in terest is in the v ariable Q ( i ) △ = # o f triangles in G ( i ) . In or der to trac k Q ( i ) w e also consider the co-degrees in the graph G ( i ): Y u,v ( i ) △ = |{ x ∈ [ n ] : xu, xv ∈ E ( i ) }| for all { u , v } ∈  [ n ] 2  . Our in terest in Y u,v is motiv ated by the follow ing observ ation: If the ( i + 1)-th triangle tak en is abc then Q ( i + 1) − Q ( i ) = Y a,b ( i ) + Y b,c ( i ) + Y a,c ( i ) − 2 . Th us, b o unds on Y u,v yield imp ortant information ab out the underlying pro cess. No w that we ha ve identifie d our v ar iables, w e determine the con tin uous tra jectories that they should follo w. W e establish a corresp ondence with con t inuous time by introducing a con tinuous v ariable t and setting t = i/n 2 (this is our time scaling). W e exp ect the graph G ( i ) to resem ble a unifor mly c hosen graph with n v ertices and  n 2  − 3 i edges, whic h in turn resem bles the Erd˝ os-R ´ en yi graph G n,p with p = 1 − 6 i/n 2 = p ( t ) = 1 − 6 t . (Note that w e can view p as either a contin uo us function of t or as a f unction of the discrete v ariable i . W e pass b etw een these in terpretations o f p without comment.) F ollowing this in tuition, we exp ect to ha ve Y u,v ( i ) ≈ p 2 n and Q ( i ) ≈ p 3 n 3 / 6. F or ease o f notation define y ( t ) = p 2 ( t ) , q ( t ) = p 3 ( t ) / 6 . W e state our main result in terms of an error function that slow ly grows as the pro cess ev olv es. D efine f ( t ) = 5 − 3 0 log(1 − 6 t ) = 5 − 30 lo g p ( t ) . Our main result is the f o llo wing: 3 Theorem 2. With high p r ob ab ility we have Q ( i ) ≥ q ( t ) n 3 − f 2 ( t ) n 2 log n p ( t ) and (2) | Y u,v ( i ) − y ( t ) n | ≤ f ( t ) p n log n for al l { u, v } ∈  [ n ] 2  , (3) holding fo r every i ≤ i 0 = 1 6 n 2 − 5 3 n 7 / 4 log 5 / 4 n. F urthermor e , for al l i = 1 , . . . , M we have Q ( i ) ≤ q ( t ) n 3 + 1 3 n 2 p ( t ) . (4) Note t ha t the error term in the upp er b ound (4) de cr e ases a s the pro cess ev olv es. This is not a common feat ure of applications of t he differential equations metho d for random graph pro cess; indeed, the usual approac h requires an error b ound that gro ws as the pro cess evolv es. While nov el tec hniques are in tro duced here to get this ‘self-correcting’ upp er b ound, tw o v ersions of ‘self-correcting’ estimates ha v e app eared to date in a pplicatio ns of the differen tia l equations metho d in the literature (see [4] and [10]). The stronger upp er bo und o n the n umber of edges in the graph pro duced b y the random greedy triangle-pack ing pro cess giv en in the companion pap er [3] is prov ed b y establishing self-correcting estimates for a large collection of v ariables (including the v aria ble Y u,v in tro duced here). Observ e that (2) (with i = i 0 ) establishes Theorem 1. W e conclude this section with a discussion of the implications of (4) for the end of the pro cess, the part of the pro cess where there ar e few er then n 3 / 2 edges remaining. Our first observ ation is that at any step i w e can deduce a low er b ound on the n umber o f edges in the final graph; in particular, for an y i w e ha ve E ( M ) ≥ E ( i ) − 3 Q ( i ). W e might hop e to establish a low er b ound o n the n um b er of edges remaining a t the end o f the pro cess b y sho wing that there is a step i where E ( i ) − 3 Q ( i ) is lar ge. The b ound (4) is (j ust barely) to o w eak for this argumen t to b e useful. But we can deduce the follo wing. Consider i = n 2 / 6 − Θ( n 3 / 2 ); that is, consider p = cn − 1 / 2 . Once c is small enough the upp er b ound (4) is dominated b y the ‘error’ term n 2 p/ 3. If Q remains close to this upp er bo und then for the rest of the pro cess w e are usually just choosing triangles in which eve ry edge is in exactly o ne triangle; in other w ords, the remaining graph is an appro ximate partia l Steiner triple system. If Q drops significan tly b elo w this b ound then the pro cess will so o n terminate. 3 Pro of o f The orem 2 The structure of the pro of is as f ollo ws. F o r eac h v ariable of intere st and eac h b o und (meaning b oth upp er and low er) we introduce a critic al interva l that has one extreme at the b ound w e are t r ying to main tain and the other extreme sligh t ly closer to the exp ected 4 tra jectory (relativ e to the magnitude of the error b ound in question). The length o f this in terv al is generally a f unction of t . If a particular b ound is violated then sometime in the pro cess the v ariable w ould hav e to ‘cross’ this critical in t erv al. T o show that t his ev en t has low probability w e in tro duce a collection o f sequ ences of random v ariables, a sequenc e starting at eac h step j of the pro cess. This sequence stops as so o n as the v ariable leav es the critical in terv al (whic h in man y cases w ould b e immediately), and the sequence forms either a submartingale or sup ermartinagle (dep ending on the ty p e of b ound in question). The eve n t that the b ound in question is violated is con tained in the ev ent tha t there is a n index j for whic h the corresp onding sub/sup er-marting a le has a large deviation. Each of these large deviation ev ents has v ery lo w probability , ev en in comparison with the n um b er of suc h ev ents . Theorem 2 then f o llo ws f rom the union b ound. F o r ease of notation we set i 0 = 1 6 n 2 − 5 3 n 7 / 4 log 5 / 4 n , p 0 = 10 n − 1 / 4 log 5 / 4 n . Let the stopping time T b e the minim um o f M and the first step i < i 0 at whic h (2) or (3) fail and the first step i at whic h (4) fails. Note that, since Y u,v decreases as the pro cess ev olv es, if i 0 ≤ i ≤ T then w e hav e Y u,v ( i ) = O  n 1 / 2 log 5 / 2 n  for all { u, v } ∈  [ n ] 2  . W e b egin with the b ounds on Q ( i ). The first observ ation is that w e can write the exp ected one-step change in Q as a function of Q . T o do this, w e not e that w e ha v e E [∆ Q ] = − X xy z ∈ Q Y xy + Y xz + Y y z − 2 Q = 2 − 1 Q X xy ∈ E Y 2 xy (5) and 3 Q = X xy ∈ E Y xy . (And, o f course, | E | = n 2 p/ 2 − n/ 2.) Observ e that if Q g r o ws to o la r ge relative to its exp ected tra jectory then the exp ected change will b e b ecome more negativ e, in t ro ducing a drift to Q that brings it back to w ard the mean. A similar phenomena o ccurs if Q gets to o small. Restricting our a t t ention to a critical in terv al that is some distance from the exp ected tra jectory allo ws us to ta ke full adv antage of t his effect. This is the main idea in this analysis. F o r the upp er b ound on Q ( i ) our critical inte rv a l is  q ( t ) n 3 + 1 4 n 2 p , q ( t ) n 3 + 1 3 n 2 p  . (6) Supp ose Q ( i ) falls in this in terv al. Since Cauc hy -Sc hw artz giv es X xy ∈ E Y 2 xy ≥  P xy ∈ E Y xy  2 | E | ≥ 9 Q 2 n 2 p/ 2 , 5 in this situation w e ha v e E [ Q ( i + 1) − Q ( i ) | G ( i )] ≤ 2 − 18 Q n 2 p < 2 − 3 np 2 − 9 2 = − 3 np 2 − 5 2 . No w w e consider a fixed index j . ( W e a re in terested in those indices j where Q ( j ) has just en tered the critical windo w from b elow, but our analysis will formally a pply to any j .) W e define t he sequences of random v ariables X ( j ) , X ( j + 1) , . . . , X ( T j ) where X ( i ) = Q ( i ) − q n 3 − n 2 p 3 and the stopping time T j is the minim um of max { j, T } and the smallest index i ≥ j suc h that Q ( i ) is not in the critical in terv al (6). (Note that if Q ( j ) is not in the critical in terv al then we hav e T j = j .) In the ev en t j ≤ i < T j w e ha v e E [ X ( i + 1) − X ( i ) | G ( i )] = E [ Q ( i + 1) − Q ( i ) | G ( i )] −  q ( t + 1 /n 2 ) − q ( t )  n 3 −  p ( t + 1 /n 2 ) − p ( t )  n 2 3 ≤ − 3 np 2 − 5 2 + 3 np 2 + 2 + O (1 /n ) ≤ 0 . So, our sequence o f ra ndom v ariables is a sup ermartingale. Note tha t if Q ( i ) crosses the upp er b oundary in (4) at i = T then, since the one step change in Q ( i ) is at most 3 n , there exists a step j suc h that X ( j ) ≤ − n 2 p ( t ( j )) 4 + O ( n ) while T = T j and X ( T ) ≥ 0. W e apply Ho effding-Azuma to b ound t he probability of suc h an ev ent: the n umber of steps is at most n 2 p ( t ( j )) / 6 and the maximu m 1-step difference is O ( n 1 / 2 log 5 / 2 n ) (as i < T implies b ounds on the co-degrees). Th us the pro ba bilit y of suc h a large deviation b eginning at step j is at most exp ( − Ω ( n 2 p ( t ( j ))) 2 ( n 2 p ( t ( j ))) ·  n 1 / 2 log 5 / 2 n  2 !) = exp  − Ω  np ( t ( j )) log 5 n  . As there are at most n 2 p ossible v alues o f j , we hav e t he desired b o und. No w w e turn to the low er b ound on Q , namely (2). Here w e w ork with the critical interv al  q ( t ) n 3 − f ( t ) 2 n 2 log n p , q ( t ) n 3 − ( f ( t ) − 1) f ( t ) n 2 log n p  . (7) Supp ose Q ( i ) falls in this in terv al for some i < T . Note that our desired inequality is in the wrong direction for an application of Cauch y Sc hw artz to (5). In its place w e use the con trol imp osed on Y u,v ( i ) b y t he condition i < T . F or a fixed 3 Q = P uv ∈ E Y u,v , t he sum 6 P uv ∈ E Y 2 u,v is maximized when w e mak e as man y terms as large a s p ossible. Supp ose this allo ws α terms in the sum P xy ∈ E Y xy equal to np 2 + f √ n log n and α + β terms equal to np 2 − f √ n log n . F or ease of notation we view α , β as rationals, thereb y allow ing the terms in the maxim um sum to split completely into these tw o t yp es. Then w e hav e β f p n log n = | E | · np 2 − 3 Q = 3 q n 3 − 3 Q − n 2 p 2 2 . Therefore, w e ha v e X xy ∈ E Y 2 xy ≤ α  np 2 + f p n log n  2 + ( α + β )  np 2 − f p n log n  2 =  n 2 p 2 − n 2  · n 2 p 4 +  n 2 p 2 − n 2  · f 2 n log n − 2 β f p 2 n 3 / 2 log 1 / 2 n = n 4 p 5 2 + f 2 pn 3 log n 2 − 1 2  n 3 p 4 + f 2 n 2 log n  − 2 p 2 n  3 q n 3 − 3 Q − n 2 p 2 2  ≤ 6 np 2 Q − n 4 p 5 2 + f 2 pn 3 log n 2 + n 3 p 4 2 . No w, for j < i 0 define T j to b e the minim um of i 0 , max { j, T } and the smallest index i ≥ j suc h that Q ( i ) is not in the critical in terv al (7). Set X ( i ) = Q ( i ) − q ( t ) n 3 + f ( t ) 2 n 2 log n p ( t ) . F o r j ≤ i < T j w e hav e the b ound E [ X ( i + 1) − X ( i ) | G ( i )] = E [ Q ( i + 1) − Q ( i ) | G ( i )] − n 3 ( q ( t + 1 /n 2 ) − q ( t )) +  f 2 ( t + 1 /n 2 ) p ( t + 1 /n 2 ) − f 2 ( t ) p ( t )  n 2 log n ≥ 2 − 6 np 2 + n 4 p 5 2 Q − f 2 pn 3 log n 2 Q + O ( p ) + 3 p 2 n + O (1 / n ) +  2 f ′ f p + 6 f 2 p 2  log n + O  log 3 n n 2 p 3  ≥ ( f − 1 ) f n 2 log n p · n 4 p 5 2( q n 3 ) 2 − f 2 pn 3 log n 2 Q +  2 f ′ f p + 6 f 2 p 2  log n ≥  18 f 2 p 2 − 18 f p 2 − (1 + o ( 1)) 3 f 2 p 2 +  2 f ′ f p + 6 f 2 p 2  log n ≥ 0 . If the pro cess violat es the b ound ( 2) a t step T = i then there exists a j < i suc h that T = T j , X ( T ) = X ( i ) < 0 and X ( j ) > f ( t ( j )) n 2 log n p ( t ( j )) − O ( n ) . 7 The submartingale X ( j ) , X ( j + 1) , . . . X ( T j ) has length at most n 2 p ( t ( j )) / 6 a nd maxim um one-step c hange O ( n 1 / 2 log 5 / 2 n ). The probability that w e violate the low er b ound (2) is at most n 2 · exp  − Ω  f 2 ( t ( j )) n 4 log 2 n/p 2 ( t ( j )) n 2 p ( t ( j )) · n log 5 n  = n 2 · exp  − Ω  f 2 ( t ( j )) n log 3 n  = o (1 ) . Finally , w e turn to the co-degree estimate Y u,v . L et u , v b e fixed. W e b egin with the upp er b ound. Our critical in t erv al here is  y ( t ) n + ( f ( t ) − 5) p n log n , y ( t ) n + f ( t ) p n log n  . (8) F o r a fixed j < i 0 w e consider t he sequence of random v ariables Z u,v ( j ) , Z u,v ( j +1) , . . . , Z u,v ( T j ) where Z u,v ( i ) = Y u,v ( i ) − y ( t ) n − f ( t ) p n log n and T j is defined to b e the minim um o f i 0 , max { j, T } and the smallest index i ≥ j suc h that Y u,v ( i ) is not in the critical in terv al (8) . T o see tha t this sequence forms a sup ermartingale, w e note that i < T giv es   Q ( i ) − q ( t ) n 3   ≤ f ( t ) 2 n 2 log n p ( t ) , and t herefore E [ Z u,v ( i + 1) − Z u,v ( i )] ≤ − X x ∈ N ( u ) ∩ N ( v ) Y u,x + Y v,x − 1 uv ∈ E ( i ) Q − n  y ( t + 1 /n 2 ) − y ( t )  − p n log n  f ( t + 1 /n 2 ) − f ( t )  ≤ − 2( y n + ( f − 5) √ n log n )( y n − f √ n log n ) Q + O  1 n 2 p  − y ′ ( t ) n − f ′ ( t ) log 1 / 2 n n 3 / 2 + O  1 n 3 p 2  ≤ − 2( y n + ( f − 5) √ n log n )( y n − f √ n log n ) q n 3 + 2 · f 2 n 2 log n p · ( y n ) 2 ( q n 3 ) 2 − y ′ ( t ) n − f ′ ( t ) log 1 / 2 n n 3 / 2 + O  1 n 2 p  ≤ 10 y n 3 / 2 log 1 / 2 n q n 3 + 14 f 2 n log n q n 3 + O  1 n 2 p  − f ′ ( t ) log 1 / 2 n n 3 / 2 T o get the sup ermartingale condition w e consider eac h po sitiv e term here separately . The follo wing b ounds would suffice 60 p ≤ f ′ 3 , 84 f 2 √ log n p 3 n 1 / 2 ≤ f ′ 3 , 1 n 1 / 2 p = o  f ′ p log n  . 8 The first term requires f ′ ( t ) ≥ 180 p ( t ) = 180 1 − 6 t . W e see that this requiremen t, together with the initial condition f (0) ≥ 5, imp oses f ( t ) ≥ 5 − 30 log(1 − 6 t ) = 5 − 30 log p ( t ) . But t his v alue for f also suffices to handle the remaining terms a s w e restrict our attention to p ≥ p 0 = 10 n − 1 / 4 log 5 / 4 n . Th us, w e hav e established that Z u,v ( i ) is a sup ermartingale. T o b ound the pro babilit y o f a lar g e deviation w e recall a Lemma fro m [2]. A sequence of random v ariables X 0 , X 1 , . . . is ( η , N ) -b ounde d if for all i we ha v e − η < X i +1 − X i < N . Lemma 3. Supp ose 0 ≡ X 0 , X 1 , . . . is an ( η , N ) -b ounde d submartingale for som e η < N / 10 . Then for any a < η m we have P ( X m < − a ) < exp  − a 2 / (3 η N m )  . As − Z u,v ( j ) , − Z u,v ( j + 1) , . . . is a (6 /n, 2)-b ounded submartingale, the probability that we ha ve T = T j with Y u,v ( T ) > y n + f √ n log n is at most exp  − 25 n log n 3 · (6 /n ) · 2 · ( p ( t ( j )) n 2 / 6)  = exp  − 25 log n 6  . Note that there are at most n 4 c hoices fo r j and the pair u, v . As the argumen t for the low er b ound in (3) is the symmetric analogue of the reasoning w e hav e just completed, Theorem 2 follo ws.  References [1] N. Alon and J.H. Spencer, The Pr ob a bilistic Metho d (3r d Ed.), Wiley-In terscience, 2 008. [2] T. Bohman, The triangle-free pro cess, A dv anc es in Mathema tics 221 (2009) 165 3 -1677. [3] T. Bohman, A. F rieze, E. Lub etzky , Impro ved analysis of the random greedy triangle- pac king algorithm: b eating the 7/4 exp onen t . [4] T. Bohman, M. Picollelli, Ev olutio n of SIR epidemics on rando m graphs with a fixed degree sequence, man uscript. [5] P . Erd˝ os, H. 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