A note on the random greedy triangle-packing algorithm

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1 A note on the random greedy triangle-acking algorithm Tom Bohman Alan Frieze Eyal Lubetzky Abstract The random greedy algorithm for constructing a large artial Steiner-Trile-System is defined as follows. We begin with a comlete grah on n vertices and roceed to remove the edges of triangles one at a time, where each triangle removed is chosen uniformly at random from the collection of all remaining triangles. This stochastic rocess terminates once it arrives at a triangle-free grah. In this note we show that with high robability the number of edges in the final grah is at most O n 7/4 log 5/4 n ). 1 Introduction We consider the random greedy algorithm for triangle-acking. This stochastic grah rocess begins with the grah G0), set to be the comlete grah on vertex set [n], then roceeds to reeatedly remove the edges of randomly chosen triangles i.e. coies of K 3 ) from the grah. Namely, letting Gi) denote the grah that remains after i triangles have been removed, the i+1)-th triangle removed is chosen uniformly at random from the set of all triangles in Gi). The rocess terminates at a triangle-free grah GM). In this work we study the random variable M, i.e., the number of triangles removed until obtaining a triangle-free grah or equivalently, how many edges there are in the final triangle-free grah). This rocess and its variations lay an imortant role in the history of combinatorics. Note that the collection of triangles removed in the course of the rocess is a maximal collection of 3-element subsets of [n] with the roerty that any air of distinct triles in the collection have airwise intersection less than. For integers t < k < n a artial n, k, t)- Steiner system is a collection of k-element subsets of an n-element set with the roerty that any airwise intersection of sets in the collection has cardinality less than t. Note that the number of sets in a artial n, k, t)-steiner system is at most n t) / k t). Let Sn, k, t) be the Deartment of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 1513, USA. tbohman@math.cmu.edu. Research suorted in art by NSF grant DMS Deartment of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 1513, USA. alan@random.math.cmu.edu. Research suorted in art by NSF grant DMS Microsoft Research, One Microsoft Way, Redmond, WA 9805, USA. eyal@microsoft.com. 1

2 maximum number of k-sets in a artial n, k, t)-steiner system. In the early 1960 s Erdős and Hanani [5] conjectured that for any integers t < k ) lim n Sn, k, t) k t n = 1. 1) t) In words, for any t < k there exist artial n, k, t)-steiner systems that are essentially as large as allowed by the simle volume uer bound. This conjecture was roved by Rödl [7] in the early 1980 s by way of a randomized construction that is now known as the Rödl nibble. This construction is a semi-random variation on the random greedy triangle-acking rocess defined above, and thereafter such semi-random constructions have been successfully alied to establish various key results in Combinatorics over the last three decades see e.g. [1] for further details). Desite the success of the Rödl nibble, the limiting behavior of the random greedy acking rocess remains unknown, even in the secial case of triangle acking considered here. Recall that Gi) is the grah remaining after i triangles have been removed. Let Ei) be the edge set of Gi). Note that Ei) = n ) 3i and that EM) is the number of edges in the triangle-free grah roduced by the rocess. Observe that if we show EM) = on ) with non-vanishing robability then we will establish 1) for k = 3, t = and obtain that the random greedy triangle-acking rocess roduces an asymtotically otimal artial Steiner system. This is in fact the case: It was shown by Sencer [9] and indeendently by Rödl and Thoma [7] that EM) = on ) with high robability 1. This was extended to EM) n 11/6+o1) by Grable in [6], where the author further sketched how similar arguments using more delicate calculations should extend to a bound of n 7/4+o1) w.h.. By comarison, it is widely believed that the grah roduced by the random greedy triangle-acking rocess behaves similarly to the Erdős-Rényi random grah with the same edge density, hence the rocess should end once its number of remaining edges becomes comarable to the number of triangles in the corresonding Erdős-Rényi random grah. Conjecture Folklore). With high robability EM) = n 3/+o1). Joel Sencer has offered $00 for a resolution of this question. In this note we aly the differential-equation method to achieve an uer bound on EM). In contrast to the aforementioned nibble-aroach, whose alication in this setting involves delicate calculations, our aroach yields a short roof of the following best-known result: Theorem 1. Consider the random greedy algorithm for triangle-acking on n vertices. Let M be the number of stes it takes the algorithm to terminate and let EM) be the edges of the resulting triangle-free grah. Then with high robability, EM) = O n 7/4 log 5/4 n ). 1 Here and in what follows, with high robability w.h..) denotes a robability tending to 1 as n.

3 Wormald [11] also alied the differential-equation method to this roblem, deriving an uer bound of n ɛ on EM) for any ɛ < ɛ 0 = 1/57 while stating that some non-trivial modification would be required to equal or better Grable s result. Indeed, in a comanion aer we combine the methods introduced here with some other ideas and a significantly more involved analysis) to imrove the exonent of the uer bound on EM) to about This follow-u work will aear in [3]. Evolution of the rocess in detail As is usual for alications of the differential equations method, we begin by secifying the random variables that we track. Of course, our main interest is in the variable Qi) = # of triangles in Gi). In order to track Qi) we also consider the co-degrees in the grah Gi): Y u,v i) = {x [n] : xu, xv Ei)} for all {u, v} ) [n]. Our interest in Yu,v is motivated by the following observation: If the i + 1)-th triangle taken is abc then Qi + 1) Qi) = Y a,b i) + Y b,c i) + Y a,c i). Thus, bounds on Y u,v yield imortant information about the underlying rocess. Now that we have identified our variables, we determine the continuous trajectories that they should follow. We establish a corresondence with continuous time by introducing a continuous variable t and setting t = i/n this is our time scaling). We exect the grah Gi) to resemble a uniformly chosen grah with n vertices and n ) 3i edges, which in turn resembles the Erdős-Rényi grah Gn, with = 1 6i/n = t) = 1 6t. Note that we can view as either a continuous function of t or as a function of the discrete variable i. We ass between these interretations of without comment.) Following this intuition, we exect to have Y u,v i) n and Qi) 3 n 3 /6. For ease of notation define yt) = t), qt) = 3 t)/6. We state our main result in terms of an error function that slowly grows as the rocess evolves. Define ft) = 5 30 log1 6t) = 5 30 log t). Our main result is the following: 3

4 Theorem. With high robability we have holding for every Qi) qt)n 3 f t)n log n t) Y u,v i) yt)n ft) n log n for all {u, v} [n] Furthermore, for all i = 1,..., M we have i i 0 = 1 6 n 5 3 n7/4 log 5/4 n. and ) ), 3) Qi) qt)n n t). 4) Note that the error term in the uer bound 4) decreases as the rocess evolves. This is not a common feature of alications of the differential equations method for random grah rocess; indeed, the usual aroach requires an error bound that grows as the rocess evolves. While novel techniques are introduced here to get this self-correcting uer bound, two versions of self-correcting estimates have aeared to date in alications of the differential equations method in the literature see [4] and [10]). The stronger uer bound on the number of edges in the grah roduced by the random greedy triangle-acking rocess given in the comanion aer [3] is roved by establishing self-correcting estimates for a large collection of variables including the variable Y u,v introduced here). Observe that ) with i = i 0 ) establishes Theorem 1. We conclude this section with a discussion of the imlications of 4) for the end of the rocess, the art of the rocess where there are fewer then n 3/ edges remaining. Our first observation is that at any ste i we can deduce a lower bound on the number of edges in the final grah; in articular, for any i we have EM) Ei) 3Qi). We might hoe to establish a lower bound on the number of edges remaining at the end of the rocess by showing that there is a ste i where Ei) 3Qi) is large. The bound 4) is just barely) too weak for this argument to be useful. But we can deduce the following. Consider i = n /6 Θn 3/ ); that is, consider = cn 1/. Once c is small enough the uer bound 4) is dominated by the error term n /3. If Q remains close to this uer bound then for the rest of the rocess we are usually just choosing triangles in which every edge is in exactly one triangle; in other words, the remaining grah is an aroximate artial Steiner trile system. If Q dros significantly below this bound then the rocess will soon terminate. 3 Proof of Theorem The structure of the roof is as follows. For each variable of interest and each bound meaning both uer and lower) we introduce a critical interval that has one extreme at the bound we are trying to maintain and the other extreme slightly closer to the exected 4

5 trajectory relative to the magnitude of the error bound in question). The length of this interval is generally a function of t. If a articular bound is violated then sometime in the rocess the variable would have to cross this critical interval. To show that this event has low robability we introduce a collection of sequences of random variables, a sequence starting at each ste j of the rocess. This sequence stos as soon as the variable leaves the critical interval which in many cases would be immediately), and the sequence forms either a submartingale or suermartinagle deending on the tye of bound in question). The event that the bound in question is violated is contained in the event that there is an index j for which the corresonding sub/suer-martingale has a large deviation. Each of these large deviation events has very low robability, even in comarison with the number of such events. Theorem then follows from the union bound. For ease of notation we set i 0 = 1 6 n 5 3 n7/4 log 5/4 n, 0 = 10n 1/4 log 5/4 n. Let the stoing time T be the minimum of M and the first ste i < i 0 at which ) or 3) fail and the first ste i at which 4) fails. Note that, since Y u,v decreases as the rocess evolves, if i 0 i T then we have Y u,v i) = O n 1/ log 5/ n ) for all {u, v} ) [n]. We begin with the bounds on Qi). The first observation is that we can write the exected one-ste change in Q as a function of Q. To do this, we note that we have E[ Q] = Y xy + Y xz + Y yz = 1 Yxy 5) Q Q and xyz Q 3Q = xy E Y xy. And, of course, E = n / n/.) Observe that if Q grows too large relative to its exected trajectory then the exected change will be become more negative, introducing a drift to Q that brings it back toward the mean. A similar henomena occurs if Q gets too small. Restricting our attention to a critical interval that is some distance from the exected trajectory allows us to take full advantage of this effect. This is the main idea in this analysis. For the uer bound on Qi) our critical interval is qt)n n, qt)n n ). 6) xy E Suose Qi) falls in this interval. Since Cauchy-Schwartz gives xy E Y xy xy E Y xy) E 9Q n /, 5

6 in this situation we have E[Qi + 1) Qi) Gi)] 18Q n < 3n 9 = 3n 5. Now we consider a fixed index j. We are interested in those indices j where Qj) has just entered the critical window from below, but our analysis will formally aly to any j.) We define the sequences of random variables Xj), Xj + 1),..., XT j ) where Xi) = Qi) qn 3 n 3 and the stoing time T j is the minimum of max{j, T } and the smallest index i j such that Qi) is not in the critical interval 6). Note that if Qj) is not in the critical interval then we have T j = j.) In the event j i < T j we have E[Xi + 1) Xi) Gi)] = E[Qi + 1) Qi) Gi)] qt + 1/n ) qt) ) n 3 t + 1/n ) t) ) n 3n 5 + 3n + + O 1/n) 0. So, our sequence of random variables is a suermartingale. Note that if Qi) crosses the uer boundary in 4) at i = T then, since the one ste change in Qi) is at most 3n, there exists a ste j such that Xj) n tj)) + On) 4 while T = T j and XT ) 0. We aly Hoeffding-Azuma to bound the robability of such an event: the number of stes is at most n tj))/6 and the maximum 1-ste difference is On 1/ log 5/ n) as i < T imlies bounds on the co-degrees). Thus the robability of such a large deviation beginning at ste j is at most { )} n tj))) { )} ntj)) ex Ω n tj))) n 1/ log 5/ n ) = ex Ω log 5. n As there are at most n ossible values of j, we have the desired bound. Now we turn to the lower bound on Q, namely ). Here we work with the critical interval qt)n 3 ft) n log n, qt)n 3 ft) ) 1)ft)n log n. 7) Suose Qi) falls in this interval for some i < T. Note that our desired inequality is in the wrong direction for an alication of Cauchy Schwartz to 5). In its lace we use the control imosed on Y u,v i) by the condition i < T. For a fixed 3Q = uv E Y u,v, the sum 6 3

7 uv E Y u,v is maximized when we make as many terms as large as ossible. Suose this allows α terms in the sum xy E Y xy equal to n + f n log n and α + β terms equal to n f n log n. For ease of notation we view α, β as rationals, thereby allowing the terms in the maximum sum to slit comletely into these two tyes. Then we have Therefore, we have xy E Y xy α βf n log n = E n 3Q = 3qn 3 3Q n. n + f ) n log n + α + β) n f ) n log n n = n ) n n 4 + n ) f n log n βf n 3/ log 1/ n = n4 5 + f n 3 log n 1 n f n log n ) ) n 3qn 3 3Q n 6n Q n4 5 + f n 3 log n + n3 4. Now, for j < i 0 define T j to be the minimum of i 0, max{j, T } and the smallest index i j such that Qi) is not in the critical interval 7). Set For j i < T j we have the bound Xi) = Qi) qt)n 3 + ft) n log n t) E[Xi + 1) Xi) Gi)] = E[Qi + 1) Qi) Gi)] n 3 qt + 1/n ) qt)) f t + 1/n ) + t + 1/n ) f ) t) n log n t) 6n + n4 5 Q f n 3 log n + O) + 3 n + O1/n) Q f f + f 1)fn log n [ 18f f ) log n + O. log 3 ) n n 3 n 4 5 qn 3 ) f n 3 log n Q f f 18f 1 + o1))3f + f f + + 6f + 6f ) log n )] log n If the rocess violates the bound ) at ste T = i then there exists a j < i such that T = T j, XT ) = Xi) < 0 and Xj) > ftj))n log n On). tj)) 7

8 The submartingale Xj), Xj + 1),... XT j ) has length at most n tj))/6 and maximum one-ste change On 1/ log 5/ n). The robability that we violate the lower bound ) is at most n ex { Ω f tj))n 4 log )} n/ tj)) n tj)) n log 5 n { = n ex Ω )} f tj))n log 3 = o1). n Finally, we turn to the co-degree estimate Y u,v. Let u, v be fixed. We begin with the uer bound. Our critical interval here is yt)n + ft) 5) n log n, yt)n + ft) ) n log n. 8) For a fixed j < i 0 we consider the sequence of random variables Z u,v j), Z u,v j+1),..., Z u,v T j ) where Z u,v i) = Y u,v i) yt)n ft) n log n and T j is defined to be the minimum of i 0, max{j, T } and the smallest index i j such that Y u,v i) is not in the critical interval 8). To see that this sequence forms a suermartingale, we note that i < T gives Qi) qt)n 3 ft) n log n, t) and therefore E[Z u,v i + 1) Z u,v i)] x Nu) Nv) Y u,x + Y v,x 1 uv Ei) Q n yt + 1/n ) yt) ) n log n ft + 1/n ) ft) ) yn + f 5) n log n)yn f n log n) + O Q ) y t) n f t) log1/ n 1 + O n 3/ n 3 yn + f 5) n log n)yn f n log n) qn 3 + f n log n 10yn3/ log 1/ n qn 3 yn) qn 3 ) y t) n f t) log1/ n n 3/ + 14f n log n qn 3 + O ) 1 n ) 1 + O n ) 1 f t) log1/ n n n 3/ To get the suermartingale condition we consider each ositive term here searately. The following bounds would suffice 60 f 3, 84f log n f 3 n 1/ 3, 1 n 1/ = o f ) log n. 8

9 The first term requires f t) 180 t) = t. We see that this requirement, together with the initial condition f0) 5, imoses ft) 5 30 log1 6t) = 5 30 log t). But this value for f also suffices to handle the remaining terms as we restrict our attention to 0 = 10n 1/4 log 5/4 n. Thus, we have established that Z u,v i) is a suermartingale. To bound the robability of a large deviation we recall a Lemma from []. A sequence of random variables X 0, X 1,... is η, N)-bounded if for all i we have η < X i+1 X i < N. Lemma 3. Suose 0 X 0, X 1,... is an η, N)-bounded submartingale for some η < N/10. Then for any a < ηm we have PX m < a) < ex a /3ηNm) ). As Z u,v j), Z u,v j + 1),... is a 6/n, )-bounded submartingale, the robability that we have T = T j with Y u,v T ) > yn + f n log n is at most { } { } 5n log n 5 log n ex = ex. 3 6/n) tj))n /6) 6 Note that there are at most n 4 choices for j and the air u, v. As the argument for the lower bound in 3) is the symmetric analogue of the reasoning we have just comleted, Theorem follows. References [1] N. Alon and J.H. Sencer, The Probabilistic Method 3rd Ed.), Wiley-Interscience, 008. [] T. Bohman, The triangle-free rocess, Advances in Mathematics 1 009) [3] T. Bohman, A. Frieze, E. Lubetzky, Imroved analysis of the random greedy triangleacking algorithm: beating the 7/4 exonent. [4] T. Bohman, M. Picollelli, Evolution of SIR eidemics on random grahs with a fixed degree sequence, manuscrit. [5] P. Erdős, H. Hanani, On a limit theorem in combinatorial analysis. Publicationes Mathematicae Debrecen ), [6] D. Grable, On random greedy triangle acking. Electronic Journal of Combinatorics ), R11, 19. 9

10 [7] V. Rödl, On a acking and covering roblem. Euroean Journal of Combinatorics ) [8] V. Rödl, L. Thoma, Asymtotic acking and the random greedy algorithm. Random Structures and Algorithms ) [9] J.H. Sencer, Asymtotic acking via a branching rocess. Random Structures and Algorithms ) [10] A. Telcs, N.C. Wormald, S. Zhou, Hamiltonicity of random grahs roduced by - rocesses. Random Structures and Algorithms ) [11] N.C. Wormald, The differential equation method for random grah rocesses and greedy algorithms, in Lectures on Aroximation and Randomized Algorithms M. Karonski and H.J. Prömel, eds), PWN, Warsaw,