Improved bounds for coloring locally sparse hypergraphs
aa r X i v : . [ c s . D M ] J un Girth-reducibility and the algorithmic barrier for coloring
Fotis Iliopoulos ∗ Institute for Advanced Study [email protected]
Abstract
All known efficient algorithms for constraint satisfaction problems are stymied by random instances.For example, no efficient algorithm is known that can q -color a random graph with average degree (1 + ǫ ) q ln q , even though random graphs remain q -colorable for average degree up to (2 − o (1)) q ln q .Similar failure to find solutions at relatively low constraint densities is known for random CSPs such asrandom k -SAT and other hypergraph-based problems. The constraint density where algorithms breakdown for each CSP is known as the “algorithmic barrier” and provably corresponds to a phase transitionin the geometry of the space of solutions [Achlioptas and Coja-Oghlan 2008]. In this paper we aim toshed light on the following question: Can algorithmic success up to the barrier for each CSP be ascribedto some simple deterministic property of the inputs?We answer this question positively for graph coloring by identifying the property of girth-reducibility .We prove that every girth-reducible graph of average degree (1 − o (1)) q ln q is efficiently q -colorable andthat the threshold for girth reducibility of random graphs coincides with the algorithmic barrier. Thus,we link the tractability of graph coloring up to the algorithmic barrier to a single deterministic property.Our main theorem actually extends to coloring k -uniform hypergraphs. As such, we believe that it isan important first step towards discovering the structural properties behind the tractability of arbitrary k -CSPs for constraint densities up to the algorithmic barrier. ∗ This material is based upon work directly supported by the IAS Fund for Math and indirectly supported by the National ScienceFoundation Grant No. CCF-1900460. Any opinions, findings and conclusions or recommendations expressed in this material arethose of the author(s) and do not necessarily reflect the views of the National Science Foundation.
Introduction
Due to intense research in the past couple of decades, for many random constraint satisfaction problems(CSPs), such as random graph coloring, random k -SAT, and other hypergaph-based problems, we are nowaware of asymptotically tight [5], and in some cases even exact [16], estimates for the largest constraint den-sity for which typical instances have solutions. At the same time though, current state-of-the-art algorithmsstop finding solutions efficiently at much lower constraint densities than those for which the existence ofsolutions is guaranteed. For example, no efficient algorithm is known that is able to q -color a random graphwith average degree (1 + ǫ ) q ln q , even though random graphs remain q -colorable for average degree up to (2 − o (1)) q ln q . (Equivalently, while random graphs of bounded average degree d are known to be almostsurely ( + o (1)) d ln d -colorable [4, 14, 31], all known efficient algorithms require twice as many colors.)This is not a coincidence, as the point where all known algorithms stop corresponds precisely to a phasetransition in the geometry of the space of solutions known as the shattering threshold [1, 39], often referredto as the “algorithmic barrier” [1]. In particular, Achlioptas and Coja-Oghlan [1] proved that while the set ofsolutions for low densities forms a giant well-connected cluster, at some critical threshold this “huge ball”shatters into an exponential number pieces (clusters of solutions), where each cluster is well-connected andany two clusters are well-separated by huge “energy barriers”. This phenomenon is a large part of the “1-stepReplica Symmetry Breaking” hypothesis [29, 32] in statistical physics, and finding an efficient algorithm tosolve any CSP for constraint-densities beyond the algorithmic barrier is a major open problem. For example,it seems that local algorithms are bound to fail beyond this threshold [3, 12, 13].In this paper we aim to shed light on the question: Can algorithmic success up to the barrier for each CSP be ascribed to some simple deterministic prop-erty of the inputs?
In other words, given a CSP of interest, is there a structural property P such that (i) the family of instancesthat admit property P are tractable; and (ii) almost all instances of constraint density up to the algorithmicbarrier admit P ? Besides its theoretical importance, an answer to this question would allow the design ofalgorithms that are robust enough to apply to deterministic instances, while at the same time matching theperformance of the best known algorithms for random models.In this paper we study this question for the problem of coloring graphs and, more generally, k -uniformhypergraphs. To simplify the exposition, we first focus on the case of graphs.Our contribution is to identify a family of graphs which we call girth-reducible and prove the followingtheorem regarding their chromatic number and their relation to sparse random graphs. Theorem 1.1 (Informal statement) . There exists an efficient deterministic algorithm that properly colorsany girth-reducible graph of average degree d using (1 + o (1)) d/ ln d colors. Additionally, almost everygraph of bounded average degree is girth-reducible. The bound on the chromatic number in Theorem 1.1 matches the algorithmic barrier [1, 39] for coloringsparse random graphs. Thus, Theorem 1.1 links the tractability of graph q -coloring up to the algorithmicbarrier to a single deterministic property. (To see this, say that a graph has the q -girth-reducibility propertyif it is of average degree (1 − o (1)) q ln q and is girth-reducible. Theorem 1.1 implies that any such graphis efficiently q -colorable, and that the threshold for q -girth-reducibility of random graphs coincides with thealgorithmic barrier.) Note also that Theorem 1.1 extends to the more general list-coloring problem and, asa consequence, we obtain the first efficient deterministic algorithm for list-coloring sparse random graphsthat works up to the algorithmic barrier. To the best of our knowledge, the currently best list-coloringalgorithm [3] for sparse random graphs is randomized.Roughly speaking, a graph of average degree d is girth-reducible if it can be treated as a graph of girth and of maximum degree d for the purposes of coloring. This means that its vertex set can be seen as1he union of two parts: A “low-degeneracy” part, which contains all vertices of degree more than d , and a“high-girth” part, which induces a graph of maximum degree roughly d and girth . (Recall that a graph is κ -degenerate if its vertices can be ordered so that every vertex has at most κ neighbors greater than itself.Thus, any such graph can be greedily colored with κ + 1 colors.) Definition 1.2 (Informal definition) . We say that graph G ( V, E ) of average degree d is girth-reducible if itsvertex set can be partitioned in two sets, U and V \ U , such that:(a) subgraph G [ U ] is d ln d -degenerate;(b) subgraph G [ V \ U ] has maximum degree at most (1 + o (1)) d and is of girth at least ;(c) every vertex in V \ U has o (cid:0) d ln d (cid:1) neighbors in U . As we will see, there exists a simple and efficient procedure that decides whether a given graph isgirth-reducible, which also outputs the promised partition in case it is. Furthermore, the definition of girth-reducibility naturally extends to k -uniform hypergraphs (see Section 1.1, Definition 1.5). In fact, Theo-rem 1.1 is a special case of the following more general result, which gives a bound on the chromatic numberof k -uniform hypergraphs that is within a factor of ( k − of the algorithmic barrier [9, 21]. Theorem 1.3 (Informal statement) . There exists an efficient deterministic algorithm that properly colors anygirth-reducible k -uniform hypergraph of average degree d using (1 + o (1))( k −
1) ( d/ ln d ) / ( k − colors.Additionally, almost every k -uniform hypergraph of bounded average degree is girth-reducible. As we discuss in Section 1.2, dealing with constraints of large arity is highly non-trivial when the input isdeterministic. Thus, we believe that Theorem 1.3 is an important first step towards discovering the structuralproperties behind the tractability of arbitrary k -CSPs for constraint densities up to the algorithmic barrier.It is also worth noting that the definition of girth-reducibility emerges quite naturally in the light of asimple observation regarding the ways in which we can properly color the neighborhood of any fixed vertexin a high-girth k -uniform hypergraph. To describe the observation we will again focus for simplicity on thecase of graphs, but an analogous phenomenon takes place for any k ≥ .Given a triangle-free graph fix one of its vertices, say v , and let d v be its degree. Consider now all thepossible ways to color the neighborhood of v using q colors. Say that a color is “available” for v in sucha coloring if assigning it to v does not create any monochromatic edge. A simple argument reveals thatif q ≤ (1 − ǫ ) d v / ln d v , then the majority of ways to properly color the neighbors of v leaves it with noavailable colors while, if q ≥ (1 + ǫ ) d v / ln d v , then the number of available colors for v is non-vanishingas d v grows. (We give the details in Section 1.2.) Our key insight is that this local phenomenon suffices toexplain the tractability of coloring up to the algorithm barrier. To get a feeling for why this is the case, at firstrecall that our best algorithms are not able to efficiently color a random graph of bounded average degree d using (1 − ǫ ) d/ ln d colors, i.e., this point corresponds to the algorithmic barrier for coloring. Further, it iswell-known that sparse random graphs are “locally tree-like”, namely a random vertex does not participatein any cycle of constant length and, in particular, in any triangles. Therefore, our initial observation impliesthat the algorithmic barrier coincides with the point at which a typical vertex of a random graph is most likelyleft with no available colors after a random coloring of its neighborhood. (“Typical” here means a vertexof degree at most d that is not contained in short cycles.) This is too striking to be a mere coincidence, andindeed we show that it is not. At the very least, it suggests that in order to efficiently color any deterministicgraph of average degree d using (1 + o (1)) d/ ln d colors, we should be able to color the vertices of degreehigher than d before the rest of the graph. For otherwise, the vast majority of ways to color the neighborhoodof any such vertex leaves it with no available colors. Now the most straightforward way to color thesevertices is via a greedy algorithm, and this is going to work on a deterministic instance only if the subgraphinduced by these vertices is of low degeneracy. 2ndeed, this is precisely the strategy we follow to color a girth-reducible hypergraph. That is, we firstcolor its low-degeneracy part using the greedy algorithm. The remaining vertices form a high-girth hyper-graph that can be efficiently colored by a generalization of a classical result of Kim [26] which we developin this paper. Theorem 1.4 (Informal statement) . Every k -uniform hypergaph of degree ∆ and girth at least is efficiently (1 + o (1))( k − / ln ∆) / ( k − -list colorable via a deterministic algorithm. Theorem 1.4 is of independent interest as it implies the classical theorem of Ajtai-Koml´os-Pintz-Spencer-Szemer´edi [6] regarding the independence number of k -uniform hypergraphs of degree ∆ and girth . Thelatter is a seminal result in combinatorics, with applications in geometry and coding theory [27, 28, 30].Further, Theorem 1.4 is tight up to a constant [10]. Note also that, without the girth assumption, the bestpossible bound [17] on the chromatic number of k -uniform hypergraphs is O (∆ / ( k − ) , i.e., it is asymp-totically worse than the one of Theorem 1.4. For example, there exist graphs of degree ∆ whose chromaticnumber is exactly ∆ + 1 . We further discuss the relation of Theorem 1.4 to past results in Section 1.1.We remark that our work is inspired by a recent paper of Molloy [33], who showed that triangle-freegraphs of maximum degree ∆ can be efficiently properly colored using at most (1 + o (1)) ∆ln ∆ colors, andpointed out that this bound matches the algorithmic barrier for coloring random regular graphs of boundeddegree [39]. Molloy’s result is an extension of the one of Kim [26], who showed the same bound for graphsof degree ∆ and girth at least . Remarkably, the results of Kim and Molloy imply that the tractabilityof coloring sparse regular graphs for densities up to the algorithmic barrier boils down to a very simpleproperty, namely the absence of short cycles. (Random regular graphs of bounded degree are essentiallyhigh-girth graphs: We can almost surely remove a matching containing only a few edges to get a graph ofgirth . This modification changes the chromatic number of the graph by at most one.) Here we show thatthis phenomenon extends to general, i.e., not necessarily regular, sparse k -uniform hypergraphs and that thecrucial structural property is girth-reducibility.To conclude, we stress that the technique of Molloy does not seem to easily extend to hypergraphs eventhough it is significantly simpler than the one of Kim. Indeed, our main technical contribution is dealingwith constraints of large arity and our approach is based on the so-called semi-random method . We furtherdiscuss the technical aspect of our work in Section 1.2. In hypergraph coloring one is given a hypergraph H ( V, E ) and the goal is to find an assignment of one of q colors to each vertex v ∈ V so that no hyperedge is monochromatic. In the more general list-coloring problem, a list of q allowed colors is specified for each vertex. A graph is q -list-colorable if it has a list-coloring no matter how the lists are assigned to each vertex. The list chromatic number , χ ℓ ( H ) , is thesmallest q for which H is q -list colorable. To formally describe our results, we need some notation.A hypergraph is is k -uniform if every hyperedge contains exactly k variables. An i -cycle in a k -uniformhypergraph is a collection of i distinct hyperedges spanned by at most i ( k − vertices. We say that a k -uniform hypergraph has girth at least g if it contains no i -cycles for ≤ i < g . Note that if a k -uniformhypergraph has girth at least then every two of its hyperedges have at most one vertex in common.A k -uniform hypergraph H is κ -degenerate if the induced subhypergraph of all subsets of its vertex sethas a vertex of degree at most κ . The degeneracy of a hypergraph H is the smallest value of κ for which H is κ -degenerate. Note that it is known that κ -degenerate hypergraphs are ( κ + 1) -list colorable and that thedegeneracy of a hypergraph can be computed efficiently by an algorithm that repeatedly removes minimumdegree vertices. Indeed, to list-color a κ -degenerate hypergraph we repeatedly find a vertex with (remaining)degree at most κ , assign to it a color that does not appear in any of its neighbors so far, and remove it from3he hypergraph. Clearly, if the lists assigned to each vertex are of size at least κ + 1 this procedure alwaysterminates successfully. Definition 1.5.
For δ ∈ (0 , , we say that a k -uniform hypergraph H ( V, E ) of average degree d is δ -girth-reducible if its vertex set can be partitioned in two sets, U and V \ U , such that:(a) subhypergraph H [ U ] is (cid:0) d ln d (cid:1) k − -degenerate;(b) subhypergraph H [ V \ U ] has maximum degree at most (1 + δ ) d and is of girth at least ;(c) every vertex in V \ U has at most δ (cid:0) d ln d (cid:1) k − neighbors in U . Note that given a δ -girth-reducible hypergraph we can efficiently find the promised partition ( U, V \ U ) as follows. We start with U := U , where U is the set of vertices that either have degree at least (1 + δ ) d , orthey are contained in a cycle of length at most . Let ∂U denote the vertices in V \ U that violate property (c).While ∂U = ∅ , update U as U := U ∪ ∂U . The correctness of the process lies in the fact that in each stepwe add to the current U a set of vertices that must be in the low-degeneracy part of the hypergraph. Observealso that this process allows us to efficiently check whether a hypergraph is δ -girth-reducible. Theorem 1.6.
For any constants δ ∈ (0 , and k ≥ , there exists d δ,k > such that if H is a δ -girth-reducible, k -uniform hypergraph of average degree d , then χ ℓ ( H ) ≤ (1 + ǫ )( k − (cid:18) d ln d (cid:19) k − , where ǫ = 4 δ = O ( δ ) . Furthermore, if H is a hypergraph on n vertices then there exists a deterministicalgorithm that constructs such a coloring in time polynomial in n . As we have already discussed, girth-reducibility is a pseudo-random property which is admitted byalmost all sparse k -uniform hypregraphs. To establish this fact formally, we need some further notation.The random k -uniform hypergraph H ( k, n, p ) is obtained by choosing each of the (cid:0) nk (cid:1) k -element subsetsof a vertex set V ( | V | = n ) independently with probability p . The chosen subsets are the hyperedges ofthe hypergraph. Note that for k = 2 we have the usual definition of the random graph G ( n, p ) . Wesay that H ( k, n, p ) has a certain property A almost surely or with high probability , if the probability that H ∈ H ( k, n, p ) has A tends to as n → ∞ . Theorem 1.7.
For any constants δ ∈ (0 , , k ≥ , there exists d δ,k > such that for every constant d ≥ d δ,k , almost surely, the random hypergraph H ( k, n, d/ (cid:0) nk − (cid:1) ) is δ -girth-reducible. Remark 1.1.
Note that, for k, d constants, a very standard argument reveals that H ( k, n, d/ (cid:0) nk − (cid:1) ) isessentially equivalent to H ( k, n, kdn ) , namely the uniform distribution over k -uniform hypergraphs with n vertices and exactly kdn hyperedges. Thus, Theorem 1.7 extends to that model as well. Theorem 1.7 follows by simple, although somewhat technical, considerations on properties of sparserandom hypergraphs, which are mainly inspired by the the results of Alon, Krivelevich and Sudakov [8]and Łuczak [31]. Notice that for constant d , the random hypergraph H ( k, n, d/ (cid:0) nk − (cid:1) ) is far from regularas its average degree is roughly d , while its maximum degree is in the order of ln n/ ln ln n . This is themain reason why the results of Kim and Molloy match the algorithmic barrier only for regular graphs. It isworth mentioning though that, as soon as d = Ω(log n ) , the maximum degree of G ( n, d/n ) also becomesapproximately d , which allows recent extensions [2, 15] of Molloy’s result regarding the chromatic numberof (deterministic) locally spare graphs to successfully color G ( n, d/n ) using (1 + o (1)) d/ ln d colors for4alues of d that range from Ω(log n ) to at least ( n ln n ) . We are not aware of analogous results for k -uniform hypergraphs (it appears that the techniques of [2, 15, 33] do not easily extend to k > ).Theorem 1.6 is derived almost immediately by the following extension of the classical result of Kim [26]to hypergraphs. Theorem 1.8.
Let H by any k -uniform hypergraph, k ≥ , of maximum degree ∆ and girth at least . Forall ǫ > , there exist a positive constant ∆ ǫ,k such that if ∆ ≥ ∆ ǫ,k , then χ ℓ ( H ) ≤ (1 + ǫ )( k − (cid:18) ∆ln ∆ (cid:19) k − . (1) Furthermore, if H is a hypergraph on n vertices then there exists a deterministic algorithm that constructssuch a coloring in time polynomial in n . The bound of Theorem 1.8 is within a factor of k − of the algorithmic barrier for coloring randomregular k -uniform hypergraphs [9, 21], while it holds for every hypergraph of girth at least . Further, whenit applies, it improves upon a theorem of Frieze and Mubayi [20] for list-coloring simple, triangle-free, k -uniform hypergraphs, who showed (1) with an unspecified large leading constant (of order at least Ω( k ) ).We remark that the techniques of Frieze and Mubayi are based on the proof of Johansson [22] for coloringtriangle-free graphs of maximum degree ∆ using O (∆ / ln ∆) colors, which is already suboptimal withrespect to the algorithmic barrier for random regular graphs. Therefore, it is highly unlikely that their resultcan be significantly improved to (nearly) match the algorithmic barrier, even for small k . As we show in the end of this section, Theorem 1.6 follows fairly easily from Theorem 1.8, so we focus ondescribing the main approach for proving the latter.The intuition behind the proof of Theorem 1.8 comes from the following observation, which we explainin terms of graph coloring for simplicity. Let G be a triangle-free graph of degree ∆ , and assume thateach of its vertices is assigned an arbitrary list of q colors. Fix a vertex v of G , and consider the randomexperiment in which the neighborhood of v is properly list-colored randomly. Since G contains no triangles,this amounts to assigning to each neighbor of v a color from its list randomly and independently. Assumingthat q ≥ q ∗ := (1 + ǫ )∆ / ln ∆ , the expected number of available colors for v , i.e., the colors from thelist of v that do not appear in any of its neighbors, is at least q (1 − /q ) ∆ = ω (∆ ǫ/ ) . In fact, a simpleconcentration argument reveals that the number of available colors for v in the end of this experiment isat least ∆ ǫ/ with probability that goes to as ∆ grows. To put it differently, as long as q ≥ q ∗ , the vastmajority of valid ways to list-color the neighborhood of v “leaves enough room” to color v without creatingany monochromatic edges.A completely analogous observation regarding the ways to properly color the neighborhood of a vertexcan be made for k -uniform hypergraphs. In order to exploit it we employ the so-called semi-random method ,which is the main tool behind some of the strongest graph coloring results, e.g., [22, 23, 24, 25, 34, 38],including the one of Kim [26]. The idea is to gradually color the hypergraph in iterations until we reach apoint where we can finish the coloring with a simple, e.g., greedy, algorithm. In its most basic form, eachiteration consists of the following simple procedure (using graph vertex coloring as a canonical example):Assign to each vertex a color chosen uniformly at random; then uncolor any vertex that receives the samecolor as one of its neighbors. Using the Lov´asz Local Lemma [17] and concentration inequalities, onetypically shows that, with positive probability, the resulting partial coloring has useful properties that allowfor the continuation of the argument in the next iteration. (In fact, using the Moser-Tardos algorithm [36] However, the “shattering” phenomenon [1] has only been rigorously established for constant d . v , each color c in the list of v , and each integer r ∈ [ k − , weshould keep track of a lower bound on the number of adjacent to v hyperedges that have r uncolored verticesand k − − r vertices colored c . Clearly, these parameters are not independent of each other throughoutthe process, and so the main challenge is to design and analyze a coloring procedure in which all of them,simultaneously, evolve essentially randomly.We conclude this section with a proof of Theorem 1.6, based on Theorem 1.8. Proof of Theorem 1.6.
Let ǫ = 4 δ . Given lists of colors of size (1+ ǫ )( k − (cid:0) d ln d (cid:1) k − for each vertex of H ,we first color the vertices of U using the greedy algorithm which exploits the low degeneracy of H [ U ] . Noweach vertex in V − U has at most δ (cid:0) d ln d (cid:1) k − forbidden colors in its list as it has at most that many neighborsin U . We delete these colors from the list. Observe that if we manage to properly color the induced subgraph H [ V \ U ] using colors from the updated lists, then we are done since every hyperedge with vertices both in U and V \ U will be automatically “satisfied”, i.e., it cannot be monochromatic. Notice now that the updatedlist of each vertex still contains at least (1 + 3 δ )( k − (cid:0) d ln d (cid:1) k − colors, for sufficiently large d . Since theinduced subgraph H [ V \ U ] is of girth at least and of maximum degree at most (1 + δ ) d , it is efficiently (1 + δ )( k − (cid:16) (1+ δ ) d ln((1+ δ ) d ) (cid:17) k − -list-colorable for sufficiently large d per Theorem 1.8. This concludes theproof since (1 + δ )(1 + δ ) k − < (1 + 3 δ ) . The paper is organized as follows. In Section 2 we present the necessary background. In Section 3 wepresent the algorithm and state the key lemmas for the proof of Theorem 1.8, while in Section 4 we give thefull details. Finally, in Section 5 we prove Theorem 1.7.
In this section we give some background on the technical tools that we will use in our proofs.
We will find useful the so-called lopsided version of the Lov´asz Local Lemma [17, 18].6 heorem 2.1.
Consider a set B = { B , B , . . . , B m } of (bad) events. For each B ∈ B , let D ( B ) ⊆ B\{ B } be such that Pr[ B | T C ∈ S C ] ≤ Pr[ B ] for every S ⊆ B\ ( D ( B ) ∪{ B } ) . If there is a function x : B → (0 , satisfying Pr[ B ] ≤ x ( B ) Y C ∈ D ( B ) (1 − x ( C )) for all B ∈ B , (2) then the probability that none of the events in B occurs is at least Q B ∈B (1 − x ( B )) > . In particular, we will need the following two corollaries of Theorem 2.1. For their proofs, the reader isreferred to Chapter 19 in [35].
Corollary 2.2.
Consider a set B = { B , . . . , B m } of (bad) events. For each B ∈ B , let D ( B ) ⊆ B \ { B } be such that Pr[ B | T C ∈ S C ] ≤ Pr[ B ] for every S ⊆ B \ ( D ( B ) ∪ { B } ) . If for every B ∈ B :(a) Pr[ B ] ≤ ;(b) P C ∈ D ( B ) Pr[ C ] ≤ ,then the probability that none of the events in B occurs is strictly positive. Corollary 2.3.
Consider a set B = { B , B , . . . , B m } of (bad) events such that for each B ∈ B :(a) Pr[ B ] ≤ p < ;(b) B is mutually independent of a set of all but at most ∆ of the other events.If p ∆ ≤ then with positive probability, none of the events in B occur. We will also need the following version of Talagrand’s inequality [37] whose proof can be found in [35].
Theorem 2.4.
Let X be a non-negative random variable, not identically , which is determined by n inde-pendent trials T , . . . , T n , and satisfying the following for some c, r > :1. changing the outcome of any trial can affect X by at most c , and2. for any s , if X ≥ s then there is a set of at most ws trials whose outcomes certify that X ≥ s ,then for any ≤ t ≤ E [ X ] , Pr[ | X − E [ X ] | > t + 60 c p w E [ X ]] ≤ − t c w E [ X ] . In this section we describe the algorithm of Theorem 1.8. As we already explained, our approach is basedon the semi-random method. For an excellent exposition both of the method and Kim’s result the reader isreferred to [35].We assume without loss of generality that ǫ < . Also, it will be convenient to define the parameter δ := (1 + ǫ )( k − − , so that the list of each vertex initially has at least (1 + δ )( ∆ln ∆ ) k − colors.We analyze each iteration of our procedure using a probability distribution over the set of (possiblyimproper) colorings of the uncolored vertices of H where, additionally, each vertex is either activated or de-activated. We call a pair of coloring and activation bits assignments for the uncolored vertices of hypergraph H a state . 7et V i denote the set of uncolored vertices in the beginning of the the i -th iteration. (Initially, all verticesare uncolored.) For each v ∈ V i we denote by L v = L v ( i ) the list of colors of v . Further, we say that acolor c ∈ L v is available for v in a state σ if assigning c to v does not cause any hyperedge whose initiallyuncolored vertices are all activated in σ to be monochromatic.For each vertex v , color c ∈ L v and iteration i , we define a few quantities of interest that our algorithmwill attempt to control. Let ℓ i ( v ) be the size of L v . Further, for each r ∈ [ k ] , let D i,r ( v, c ) denote the set ofhyperedges h that contain v and (i) exactly r vertices { u , . . . , u r } ⊆ h \ { v } are uncolored and c ∈ L u j forevery j ∈ [ r ] ; (ii) the rest k − − r vertices other than v are colored c . We define t i,r ( v, c ) = | D i,r ( v, c ) | .As it is common in the applications of the semi-random method, we will not attempt to keep track of thevalues of ℓ i ( v ) and t i,r ( v, c ) , r ∈ [ k − , for every vertex v and color c but, rather, we will focus on theirextreme values. In particular, we will define appropriate L i , T i,r such that we can show that, for each i , thefollowing property holds at the beginning of iteration i : Property P(i):
For each vertex v ∈ V i , color c ∈ L v and r ∈ [ k − , ℓ i ( v ) ≥ L i ,t i,r ( v, c ) ≤ T i,r . As a matter of fact, it would be helpful for our analysis (though not necessary) if the inequalities definedin P ( i ) were actually tight. Given that P ( i ) holds, we can always enforce this stronger property in astraightforward way as follows. First, for each vertex v such that ℓ i ( v ) > L i we choose arbitrarily ℓ i ( v ) − L i colors from its list and remove them. Then, for each vertex v and color c ∈ L i such that t i,r ( v, c ) < T i,r we add to the hypergraph T i,r − t i,r ( v, c ) new hyperedges of size r + 1 that contain v and r new “dummy”vertices. (As it will be evident from the proof, we can always assume that L i , T i,r are integers, since ouranalysis is robust to replacing L i , T i,r with ⌊ L i ⌋ and T i,r with ⌈ T i,r ⌉ .) We assign each dummy vertex a listof L i colors: L i − of them are new and do not appear in the list of any other vertex, and the last one is c . Remark 3.1.
Dummy vertices are only useful for the purposes of our analysis and can be removed at theend of the iteration. Indeed, one could use the technique of “equalizing coin flips” instead. For more detailssee e.g., [35].
Overall, without loss of generality, at each iteration i our goal will be to guarantee that P ( i + 1) holdsassuming Q ( i ) . Property Q(i):
For each vertex v ∈ V i , color c ∈ L v and r ∈ [ k − , ℓ i ( v ) = L i ,t i,r ( v, c ) = T i,r . An iteration.
For the i -th iteration we will apply the Local Lemma with respect to the probability distribu-tion induced by assigning to each vertex v ∈ V i a color chosen uniformly at random from L v and activating v with probability α = K ln ∆ , where K = (100 k k ) − .The partial coloring of the hypergraph, set V i +1 , and the lists of colors for each uncolored vertex in thebeginning of iteration i + 1 are induced as follows. Let σ be the output state of the i -th iteration. The listof each vertex v ∈ V i +1 ⊆ V i , L v ( i + 1) , is induced from L v ( i ) by removing every non-available color c ∈ L v ( i ) for v in σ . We obtain the partial coloring φ for the hypergraph and set V i +1 for the beginning ofiteration i + 1 by removing the color from every vertex v ∈ V i which is either deactivated or is assigned anon-available for it color in σ . 8 ontrolling the parameters of interest. Next we describe the recursive definitions for L i and T i,r which,as we already explained, will determine the behavior of the parameters ℓ i ( v ) and t i,r ( v, c ) , respectively.Initially, L = (1 + δ ) (cid:0) ∆ln ∆ (cid:1) k − , T ,k − = ∆ and T ,r = 0 for every r ∈ [ k − . Letting Keep i = k − Y r =1 (cid:18) − (cid:18) αL i (cid:19) r (cid:19) T i,r , (3)we define L i +1 = L i · Keep i − L / i , (4) T i +1 ,r = k − X j = r T i,j · (cid:18) jr (cid:19) (Keep i (1 − α Keep i )) r (cid:18) α Keep i L i (cid:19) j − r ! +3 k r α − r +1 L ri k − X ℓ =1 T i,ℓ L ℓi (ln ∆) ℓ + k − X j = r (cid:18) jr (cid:19) α j − r T i,j L j − ri / . (5)To get some intuition for the recursive definitions (4), (5), observe that Keep i is the probability thata color c ∈ L v ( i ) is present in L v ( i + 1) as well. Note further that this implies that the expected valueof ℓ i +1 ( v, c ) is L i · Keep i , a fact which motivates (4). Calculations of similar flavor for E [ t i +1 ,r ( v, c )] motivate (5). The key lemmas.
We are almost ready to state the main lemmas that will guarantee that our procedureeventually reaches a partial list-coloring of H with favorable properties that will allow us to extend it to afull list-coloring. Before doing so, we need to settle a subtle issue that has to do with the fact that t i +1 ,r ( v, c ) is not sufficiently concentrated around its expectation. To see this, notice for example that t i +1 , ( v, c ) dropsto zero if v is assigned c . (Similarly, for r ∈ { , . . . , k − } , if v is assigned c then t i +1 ,r ( v, c ) can beaffected by a large amount.) To deal with this problem we will focus instead on variable t ′ i +1 ,r ( v, c ) , i.e.,the number of hyperedges h that contain v and (i) exactly k − r − vertices of h \ { v } are colored c in theend of iteration i ; (ii) the rest r vertices of h \ { v } did not retain their color during iteration i and, further, c would be available for them if we ignored the color assigned to v . Observe that if c is not assigned to v then t i +1 ,r ( v, c ) = t ′ i +1 ,r ( v, c ) and t ′ i +1 ,r ( v, c ) ≥ t i +1 ,r ( v, c ) otherwise.The first lemma that we prove estimates the expected value of the parameters at the end of the i -thiteration. Its proof can be found in Section 4. Lemma 3.1.
Let S i = P k − ℓ =1 T i,ℓ L ℓi (ln ∆) ℓ and Y i,r = P k − j = r T i,j L ji . If Q ( i ) holds and for all < j < i, r ∈ [ k − , L j ≥ (ln ∆) k − , T i,r ≥ (ln ∆) k − , then, for every vertex v ∈ V i +1 and color c ∈ L v :(a) E [ ℓ i +1 ( v )] = ℓ i ( v ) · Keep i ;(b) E [ t ′ i +1 ,r ( v, c )] ≤ k − X j = r T i,j ( v, c ) · (cid:18) jr (cid:19) (Keep i (1 − α Keep i )) r (cid:18) α Keep i L i (cid:19) j − r ! + 3 k r α − r +1 L ri S i + O ( Y i ) . The next step is to prove strong concentration around the mean for our random variables per the follow-ing lemma. Its proof can be found in Section 4. 9 emma 3.2. If Q ( i ) holds and L i , T i,r ≥ (ln ∆) k − , r ∈ [ k − , then for every vertex v ∈ V i +1 andcolor c ∈ L v ,(a) Pr h | ℓ i +1 ( v ) − E [ ℓ i +1 ( v )] | < L / i i < ∆ − ln ∆ ;(b) Pr " t ′ i +1 ,r ( v, c ) − E [ t ′ i +1 ,r ( v, c )] > (cid:18)P k − j = r (cid:0) jr (cid:1) α j − r T i,j L j − ri (cid:19) / < ∆ − ln ∆ . Armed with Lemmas 3.1, 3.2, a straightforward application of the symmetric Local Lemma, i.e., Corol-lary 2.3, reveals the following.
Lemma 3.3.
With positive probability, P ( i ) holds for every i such that for all < j < i : L j , T j,r ≥ (ln ∆) k − and T j,k − ≥ k L k − j . The proof of Lemma 3.3 can be found in Section 4.In analyzing the recursive equations (4), (5), it would be helpful if we could ignore the “error terms”.The next lemma shows that this is indeed possible. Its proof can be found in Section 4.
Lemma 3.4.
Define L ′ = (1 + δ ) (cid:0) ∆ln ∆ (cid:1) k − , T ′ ,k − = ∆ , T ′ ,r = 0 for r ∈ [ k − , and recursively define L ′ i +1 = L ′ i · Keep i ,T ′ i +1 ,r = k − X j = r T ′ i,j · (cid:18) jr (cid:19) (Keep i · (1 − α Keep i )) r (cid:18) α Keep i L ′ i (cid:19) j − r ! +3 k r α − r +1 L ri k − X ℓ =1 T i,ℓ L ℓi (ln ∆) ℓ . If for all < j < i , L j ≥ (ln ∆) k − , T j,r ≥ (ln ∆) k − for every r ∈ [ k − , and T j,k − ≥ L k − j k ,then(a) | L i − L ′ i | ≤ ( L ′ i ) ;(b) | T i,r − T ′ i,r | ≤ ( T ′ i,r ) r r +1 . Remark 3.2.
Note that
Keep i in Lemma 3.4 is still defined in terms of L i , T i,r and not L ′ i , T ′ i,r . Note alsothat in the definition of T ′ i +1 ,r , the second summand is a function of T i,ℓ , L i , ℓ ∈ [ r − , and not T ′ i,ℓ , L ′ i . Using Lemma 3.4 we are able to prove the following in Section 4.
Lemma 3.5.
There exists i ∗ = O (ln ∆ ln ln ∆) such that(a) For all < i ≤ i ∗ , T i,r > (ln ∆) k − , L i ≥ ∆ k − , and T i,k − ≥ k L k − i ;(b) T i ∗ +1 ,r ≤ k L ri ∗ +1 , for every r ∈ [ k − and L i ∗ +1 ≥ ∆ ǫ/ k − ǫ/ . Lemmas 3.3, 3.5 and 3.6 imply Theorem 1.8.
Lemma 3.6.
Let σ be the state promised by Lemma 3.5. Given σ , we can find a full list-coloring of H inpolynomial time in the number of vertices of H . roof of Theorem 1.8. We carry out i ∗ iterations of our procedure. If P ( i ) fails to hold for any iteration i ,then we halt. By Lemmas 3.3 and 3.5, P ( i ) (and, therefore, Q ( i ) ) holds with positive probability for eachiteration and so it is possible to perform i ∗ iterations. Further, the fact that our LLL application is withinthe scope of the so-called variable setting [36] implies that the deterministic version of the Moser-Tardosalgorithm [36, 11] applies and, thus, we can perform i ∗ iterations in polynomial time.After i ∗ iterations we can apply the algorithm of Lemma 3.6 and complete the list-coloring of the inputhypergraph. Let U σ denote the set of uncolored vertices in σ , and U σ ( h ) the subset of U σ that belongs to a hyperedge h .Our goal is to color the vertices in U σ to get a full list-coloring.Towards that end, let L v = L v ( σ ) denote the list of colors for v at σ , and D r ( v, c ) := D i ∗ +1 ,r ( v, c ) the set of hyperedges (of size t i ∗ +1 ,r ( v, c ) ) with r uncolored vertices in σ whose vertices “compete” for c with v , and recall the conclusion of Lemma 3.5. Let µ be the probability distribution induced by giving eachvertex v ∈ U σ a color from L v uniformly at random. For every hyperedge h and color c ∈ T u ∈ h L u wedefine A h,c to be the event that all vertices of h are colored c . Let A be the family of these (bad) events, andobserve that for every A h,c ∈ A : µ ( A h,c ) ≤ Q v ∈U σ ( h ) | L v ( σ ) | < for large enough ∆ , since L i ∗ +1 = L i ∗ +1 (∆) ∆ →∞ −−−−→ ∞ .Moreover, let I ( A h,c ) denote the set of all bad events A h ′ ,c ′ , where h ′ = h , such that either U σ ( h ) ∩U σ ( h ′ ) = ∅ , or c ′ is not in the list of colors of the (necessarily unique) uncolored vertex that h and h ′ share.Notice that conditioning on any the non-occurrence of any set S ⊆ I ( A h,c ) does not increase the probabilityof A h,c .Let D ( A h,c ) := A \ I ( A h,c ) . Lemma 3.6 follows from Corollary 2.2 (and can be made constructiveusing the deterministic version of the Moser-Tardos algorithm [36, 11]) as, for every A h,c ∈ A : X A ∈ D ( A h,c ) µ ( A ) ≤ X v ∈U σ ( h ) X c ′ ∈ L v k − X r =1 X h ′ ∈ D r ( v,c ′ ) µ (cid:0) A h ′ ,c ′ (cid:1) = X v ∈U σ ( h ) X c ′ ∈ L v k − X i =1 X h ′ ∈ D r ( v,c ′ ) Q u ∈U σ ( h ′ ) | L u |≤ max v ∈U σ ( h ) k | L v | X c ′ ∈ L v k − X r =1 | D r ( v, c ′ ) | L ri ∗ +1 (6) ≤ k k max v ∈U σ ( h ) L ri ∗ +1 · | L v || L v | · L ri ∗ +1 (7) ≤ < , (8)for large enough ∆ , concluding the proof. Note that in (6) we used the facts that every hyperedge has at most k vertices and L i ∗ +1 ≥ ∆ ǫ/ k − ǫ/ , and in (7) we used the fact that | D r ( v, c ′ ) | ≤ T ri ∗ +1 ≤ k L ri ∗ +1 .11 Hypergraph list-coloring proofs
In this section we prove Lemmas 3.1, 3.2, 3.3, 3.4, 3.5.We start by showing a couple of important lemmas that will be helpful for these proofs. It will beconvenient to define R i,r = T i,r L ri , R ′ i,r = T ′ i,r ( L ′ i ) r for every r ∈ [ k − . Lemma 4.1.
If for all < j < i, r ∈ [ k − , L j , T j,r ≥ (ln ∆) k − , then R i,r ≤ k k − − r ) ln ∆ . Proof.
We proceed by induction. The case i = 1 is straightforward to verify since R ,r = 0 for every r ∈ [ k − , while R ,k − = ln ∆(1+ δ ) k − . Therefore, we inductively assume the claim for i , and consider thecase i + 1 . Note that the inductive hypothesis implies that Keep i = Ω(1) since − x ≥ e − x − for every x ≥ and, thus, Keep i ≥ exp − k − X r =1 T i,r ( α − L i ) r − ! ≥ exp − Kk k − − δ k ! , (9)for sufficiently large ∆ . R i +1 ,r = k − X j = r T i,j L ri +1 · (Keep i (1 − α Keep i )) r (cid:18) jr (cid:19) (cid:18) α Keep i L i (cid:19) j − r ! + 1 L ri +1 k − X j = r (cid:18) jr (cid:19) α j − r T i,r L j − ri / + 3 k r α − r +1 (cid:18) L i L i +1 (cid:19) r k − X ℓ =1 T i,ℓ L ℓi (ln ∆) ℓ = k − X j = r T i,j L ri (cid:16) Keep i − L − / i (cid:17) r · (Keep i (1 − α Keep i )) r (cid:18) jr (cid:19) (cid:18) α Keep i L i (cid:19) j − r + O (cid:16) (ln ∆) − k − r + (cid:17) = k − X j = r R i,j · (1 − α Keep i ) r (cid:18) − L − / i Keep i (cid:19) r (cid:18) jr (cid:19) ( α Keep i ) j − r + O (cid:16) (ln ∆) − k − r + (cid:17) (10) ≤ k − X j = r (cid:18) R i,j · (cid:18) − α Keep i (cid:19) r (cid:18) jr (cid:19) ( α Keep i ) j − r (cid:19) + O (cid:16) (ln ∆) − k − r + (cid:17) (11) ≤ (cid:18) − α Keep i (cid:19) r R i,r + k − X j = r +1 (cid:18) jr (cid:19) R i,j α j − r + O (cid:16) (ln ∆) − k − r + (cid:17) ≤ (cid:18) − α Keep i (cid:19) k k − − r ) ln ∆ + k − X j = r +1 (cid:18) jr (cid:19) k k − − j ) K j − r (ln ∆) j − r − + O (cid:16) (ln ∆) − k − r + (cid:17) (cid:18) − α Keep i (cid:19) (cid:16) k k − − r ) ln ∆ + k k − − ( r +1)) ( r + 1) K (cid:17) + O (cid:18) (cid:19) ≤ k k − − r ) ln ∆ − K Keep i k k − − r ) − k k − − ( r +1)) ( r + 1) ! + O (cid:18) (cid:19) ≤ k k − − r ) ln ∆ , (12)for sufficiently large ∆ , concluding the proof. Note that in (11) we used the facts that Keep i = Ω(1) , L i , T i,r ≥ (ln ∆) k − . In deriving (12) we used the fact that K = (100 k k ) − and, therefore, the secondterm is a negative constant, the inductive hypothesis, and that L i ≥ (ln ∆) k − and Keep i = Ω(1) .A straightforward corollary of Lemma 4.1 is the following. Corollary 4.2. If L i , T i,r ≥ (ln ∆) k − and R i,k − ≥ k , then C := exp − Kk k − − δ k ! ≤ Keep i ≤ − K k − k (ln ∆) k − . Proof.
The lower bound follows directly from (9). The upper bound follows from our assumption that R i,k − ≥ k which implies that Keep i ≤ e − P k − r =1 α r R i,r ≤ e − α k − R i,k − ≤ e − Kk − k k − < − K k − k (ln ∆) k − , for sufficiently large ∆ . Lemma 4.3. If L j , T j,r ≥ (ln ∆) k − for all < j ≤ i , then for every r ∈ [ k − : R ′ i,r ≤ (1 − αC ) r ( i − ln ∆ · (1 + δk ) k − − r (1 + δ − δk ) k − C k − − r k − Y p = r ( p + 1) . Proof of Lemma 4.3.
We proceed by induction. The base cases are easy to verify since R ′ ,r = 0 for every r ∈ [ k − and R ′ ,k − = ln ∆(1+ δ ) k − .We first focus on the case r = k − . We assume that the claim is true for i − and consider i . Notethat the inductive hypothesis, the fact that L j ≥ (ln ∆) k − and Keep j = Ω(1) , imply that k r α − r +1 L ri − L ri k − X ℓ =1 T i − ,ℓ ( L i − ) ℓ (ln ∆) ℓ ≤ k − , (13)13or sufficiently large ∆ , for every r ∈ [ k − and j < i . Therefore, R ′ i,k − ≤ R ′ i − ,k − (1 − α Keep i − ) k − + 1(ln ∆) k − ≤ R ′ i − ,k − (1 − α Keep i − ) k − (1 − α Keep i − ) k − + (1 − α Keep i − ) k − (ln ∆) k − + 1(ln ∆) k − ≤ R ′ i − ,k − (1 − αC ) k − + (1 − αC ) k − (ln ∆) k − + 1(ln ∆) k − ≤ . . . ≤ (1 − αC ) ( i − k − R ,k − + 1(ln ∆) k − i − X ℓ =0 (1 − αC ) ( k − ℓ ≤ (1 − αC ) ( i − k − ln ∆(1 + δ ) k − + 1(ln ∆) k − ≤ (1 − αC ) ( i − k − ln ∆(1 + δ − δk ) k − , for sufficiently large ∆ , concluding the proof for the case r = k − .We now focus on r ∈ [ k − . As the first step, we observe that R ′ ,r ≤ k − X j = r T ′ ,j ( L ′ ) r · (Keep (1 − α Keep )) r (cid:18) jr (cid:19) (cid:18) α Keep L ′ (cid:19) j − r ! + 1(ln ∆) k − = R ′ ,k − · (Keep (1 − α Keep )) r (cid:18) k − r (cid:19) ( α Keep ) k − − r + 1(ln ∆) k − ≤ (ln ∆) − ( k − − r ) (1 + δ ) k − K k − − r (cid:18) k − r (cid:19) + 1(ln ∆) k − (14) ≤ (1 + δk ) k − − r (ln ∆) − ( k − − r ) (1 + δ − δk ) k − K k − − r k − Y p = r ( p + 1) , concluding the proof of the base cases.Assume that the claim holds for all pairs ( r ′ , i ′ ) , where r ′ ∈ { r, . . . , k − } and i ′ ≤ i − . It suffices toprove that it also holds for the pair ( r, i ) , where i > and r ∈ [ k − . To see this, observe that R ′ i,r ≤ k − X j = r T ′ i − ,j ( L ′ i ) r · (cid:0) Keep i − (cid:0) − α Keep i − (cid:1)(cid:1) r (cid:18) jr (cid:19) (cid:18) α Keep i − L ′ i − (cid:19) j − r ! + 1(ln ∆) k − = k − X j = r (cid:18) T ′ i − ,j ( L ′ i − ) j · Keep j − ri − (cid:0) − α Keep i − (cid:1) r (cid:18) jr (cid:19) α j − r (cid:19) + 1(ln ∆) k − = (cid:0) − α Keep i − (cid:1) r k − X j = r (cid:18) R ′ i − ,j (cid:18) jr (cid:19) ( α Keep i − ) j − r (cid:19) + 1(ln ∆) k − (1 − αC ) r k − X j = r (cid:18) R ′ i − ,j (cid:18) jr (cid:19) α j − r (cid:19) + 1(ln ∆) k − ≤ (1 − αC ) r R ′ i − ,r + k − X j = r +1 (cid:18) jr (cid:19) K j − r (ln ∆) − ( j − r ) (1 − αC ) j ( i − r (1 + δk ) k − − j (1 + δ − δk ) k − C k − − j k − Y p = j ( p + 1)+ 1(ln ∆) k − ≤ (1 − αC ) r R ′ i − ,r + k − X j = r +1 (cid:18) jr (cid:19) K j − r (ln ∆) − ( j − r ) (1 − αC ) j ( i − r (1 + δk ) k − − j (1 + δ − δk ) k − C k − − j k − Y p = j ( p + 1)+ k − X j = r +1 (cid:18) jr (cid:19) K j − r (ln ∆) − ( j − r ) (1 − αC ) j ( i − r (1 + δk ) k − − j (1 + δ − δk ) k − C k − − j k − Y p = j ( p + 1)+ 1 + (1 − αC ) r (ln ∆) k − (15) ≤ (1 − αC ) r R ′ i − ,r + k − X j = r +1 (cid:18) jr (cid:19) K j − r (ln ∆) − ( j − r ) (1 − αC ) j ( i − r (1 + δk ) k − − j (1 + δ − δk ) k − C k − − j k − Y p = j ( p + 1)+ k − X j = r +1 (cid:18) jr (cid:19) K j − r (ln ∆) − ( j − r ) (1 − αC ) j ( i − r (1 + δk ) k − − j (1 + δ − δk ) k − C k − − j k − Y p = j ( p + 1)+ k − X j = r +1 (cid:18) jr (cid:19) K j − r (ln ∆) − ( j − r ) (1 − αC ) j ( i − r (1 + δk ) k − − j (1 + δ − δk ) k − C k − − j k − Y p = j ( p + 1)+ 1 + (1 − αC ) r + (1 − αC ) r (ln ∆) k − (16) ≤ . . . ≤ (1 − αC ) ( i − r R ′ ,r + k − X j = r +1 (cid:18) jr (cid:19) K j − r (ln ∆) − ( j − r ) k − Y p = j ( p + 1) (1 + δ/k ) k − − j (1 + δ − δk ) k − C k − − j i − X ℓ =1 (1 − αC ) j ( i − ℓ − ℓr + P i − ℓ =0 (1 − αC ) r ( ℓ − (ln ∆) k − (17) ≤ (1 − αC ) ( i − r (ln ∆) − ( k − − r ) (1 + δ ) k − K k − − r (cid:18) k − r (cid:19) + O (cid:18) k − (cid:19) + k − X j = r +1 (cid:18) jr (cid:19) K j − r (ln ∆) − ( j − r ) k − Y p = j ( p + 1) (1 + δ/k ) k − − j (1 + δ − δk ) k − i − X ℓ =1 (1 − αC ) j ( i − ℓ − ℓr (18)15 K k − − r (ln ∆) − ( k − − r ) (1 + δ − δk ) k − (1 − αC ) ( i − r (cid:18) k − r (cid:19) ++ k − X j = r +1 (cid:18) jr (cid:19) K j − r (ln ∆) − ( j − r ) k − Y p = j ( p + 1) (1 + δ/k ) k − − j (1 + δ − δk ) k − C k − − j i − X ℓ =1 (1 − αC ) ( i − r +( i − ℓ − j − r ) ≤ (1 − αC ) ( i − r (1 + δ − δk ) k − k − X j = r +1 (cid:18) jr (cid:19) K j − r (ln ∆) − ( j − r ) k − Y p = j ( p + 1) (1 + δ/k ) k − − j C k − − j X ℓ ≥ (1 − αC ) ℓ ( j − r ) = (1 − αC ) ( i − r (1 + δ − δk ) k − k − X j = r +1 (cid:18) jr (cid:19) K j − r (ln ∆) − ( j − r ) Q k − p = j ( p + 1) C k − − j (1 + δ/k ) k − − j − (1 − αC ) j − r ≤ (1 − αC ) r ( i − ln ∆ · (1 + δk ) k − − r (1 + δ − δk ) k − C k − − r k − Y p = r ( p + 1) , (19)for sufficiently large ∆ , concluding the proof. Note that in order to get (15) we upper bound R ′ i − ,r inthe same way we upper bounded R ′ i,r . We keep using the same idea to bound R ′ i − ,r , R ′ i − ,r , . . . until weget (17), and we obtain (18) by using (14).We are now ready to prove Lemmas 3.1, 3.2, 3.3, 3.4 and 3.5. Proof of part (a) . For every color c ∈ L v ( i ) , Pr[ c ∈ L v ( i + 1)] = k − Y r =1 Y h ∈ D i,r ( v,c ) − Y u ∈ ( h \{ v } ) ∩ V i αℓ i ( u ) = k − Y r =1 (cid:18) − (cid:18) αL i (cid:19) r (cid:19) T i,r = Keep i , (20)where for the second equality we used our assumption that Q ( i ) holds. Therefore, the proof of the first partof the lemma follows from linearity of expectation. Proof of part (b) . Recall the definition of t ′ i +1 ,r ( v, c ) and note that only hyperedges in S k − j = r D i,j ( v, c ) canbe potentially counted by t ′ i +1 ,r ( v, c ) . In particular, unless every uncolored vertex of an edge h ∈ D i,j ( v, c ) , j ≥ r , is assigned the same color with v in iteration i , then if h is counted by t ′ i +1 ,r ( c ) , it is also counted by t i +1 ,r ( c ) . Therefore, E [ t ′ i +1 ,r ( v, c )] ≤ E [ t i +1 ,r ( v, c )] + O k − X j = r T i,j L ji , (21)and so we will focus on bounding E [ t i +1 ,r ( v, c )] .Fix h ∈ D i,j ( v, c ) , where j ≥ r . Our goal will be to show that Pr[ h ∈ D i +1 ,r ( v, c )] ≤ (cid:18) jj − r (cid:19) (Keep i (1 − α Keep i )) r (cid:18) α Keep i L i (cid:19) j − r + 4 r (cid:18) jr (cid:19) Keep j − i α j − r +1 S i L j − ri + O L ji ! , (22)16ince combining (22) with (21) implies the lemma. To see this, observe that T i,j · r (cid:18) jr (cid:19) Keep j − i α j − r +1 S i L j − ri ≤ αr (cid:18) jr (cid:19) · α j T i,j L ji Keep j − i · ( α − L i ) r S i ≤ α (cid:0) jr (cid:1) e( j −
1) ( α − L i ) r S i , (23)and, therefore, E [ t i +1 ,r ( v, c )] ≤ k − X j = r T i,j max h ∈ D i,j ( v,c ) Pr[ h ∈ D i +1 ,r ( v, c )] ≤ k − X j = r T i,j · (cid:18) jr (cid:19) (Keep i (1 − α Keep i )) r (cid:18) α Keep i L i (cid:19) j − r ! +3 k r α ( α − L i ) r S i + O k − X j = r T i,j L ji . Note that in (23) we first used that − x ≤ e − x for every x ≥ (to bound Keep i ), and then that max x x e − ℓx ≤ ℓ e for every ℓ .Towards proving (22), for any vertex u ∈ h \ { v } , consider the events E u, = “ u does not retain its color and c ∈ L u ( i + 1) ” ,E u, = “ u is assigned c and retains its color” . Let also B c be the event that v and j − other uncolored vertices of h receive color c . Since we haveassumed that our hypergraph is of girth at least , for any neighbor u of v and j ∈ { , } , event E u,j ismutually independent of all events E u ′ ,ℓ , ℓ ∈ { , } , u = u ′ , conditional on B c not occurring. Thus, if Pr[ E u,ℓ | B c ] ≤ p ℓ , ℓ ∈ { , } , for every neighboring vertex u of v , we obtain Pr[ h ∈ D i +1 ,r ( v, c )] ≤ (cid:18) jr (cid:19) p r p j − r + 2 k L ji , (24)since Pr[ B c ] ≤ k L − ji .Assume now that we condition on an arbitrary assignment of colors to the uncolored vertices of h sothat B c does not hold. Then, for any neighboring vertex u of v , and sufficiently large ∆ , Pr[ E u, | B c ] ≤ α Keep i L i + 2( L i ln ∆) j =: q + δ , (25)since the probability (conditional on B c ) that u is activated is α , it is assigned c with probability less than L − i ; and it retains c with probability at most Y r ∈ [ k − \{ j } (cid:18) − α r L ri (cid:19) T i,r · − α j L ji ! T i,j − ≤ Keep i + 2( L i ln ∆) j , for sufficiently large ∆ . 17e now claim that Pr[ E u, | B c ] ≤ Keep i (1 − α Keep i ) + (cid:0) αS i + ( L i ln ∆) − j (3 + 4 αS i ) (cid:1) =: q + δ . (26)To see this, at first observe that if u is not activated, then u will not be assigned a color, and the probabilitythat c ∈ L u ( i + 1) conditional on B c is at most Keep i + 2( L i ln ∆) − j . If u is activated and is assigned c ,then the probability that c ∈ L u ( i + 1) and u does not retain c is zero.Finally, suppose that u is activated and is assigned γ = c . For each γ ∈ L u ( i ) \ { c } we compute theprobability that γ / ∈ L u ( i + 1) conditional on that u was activated and assigned γ , B c did not occur and c ∈ L u ( i + 1) .For any ℓ ∈ [ k − and any g ∈ D i,ℓ ( u, γ ) , we consider the probability that every vertex in ( g \ { u } ) ∩ V i is activated and assigned γ , conditional on that u was activated and assigned γ , c ∈ L u ( i + 1) and B c . Sincethe color activations and color assignments are independent over different vertices, this equals the probabilitythat every vertex in ( g \ { u } ) ∩ V i is activated and assigned γ , conditional on the event A g that not everyvertex in ( g \ { u } ) ∩ V i is activated and assigned c . The latter probability equals to Pr[ (every vertex in g ∩ V i \ { u } is activated and assigned γ ) ∧ A g ]Pr[ A g ]= Pr[ (every vertex in g ∩ V i \ { u } is activated and assigned γ ) ]Pr[ A g ] ≤ α ℓ L − ℓi − α ℓ L − ℓi ≤ (cid:18) αL i (cid:19) ℓ + 1 L ℓi (ln ∆) ℓ , for sufficiently large ∆ , since K < . Therefore, the probability of γ / ∈ L u ( i + 1) given that u was activatedand assigned γ , c ∈ L u ( i + 1) and B c is at most − k − Y ℓ =1 Y g ∈ D i,ℓ ( u,γ ) (1 − Pr[ ∀ w ∈ ( g \ { u } ) ∩ V i : w is activated and assigned γ | A g ]) T i,ℓ ( u,γ ) ≤ − k − Y ℓ =1 − (cid:18) αL i (cid:19) ℓ − L ℓi (ln ∆) ℓ ! T i,ℓ ( u,γ ) ≤ − Keep i + 2 k − X ℓ =1 T i,ℓ L ℓi (ln ∆) ℓ , for sufficiently large ∆ .Overall, we have shown that Pr[ E u, | B c ] is at most (1 − α )Keep i + 2( L i ln ∆) − j + α L i − L i (Keep i + 2( L i ln ∆) − j ) − Keep i + 2 k − X ℓ =1 T i,ℓ L ℓi (ln ∆) ℓ ! , ≤ Keep i (1 − α Keep i ) + (cid:0) αS i + ( L i ln ∆) − j (3 + 4 αS i ) (cid:1) , where recall that S i = P k − ℓ =1 T i,ℓ L ℓi (ln ∆) ℓ . 18ombining (24), (25) and (26) we obtain Pr[ h ∈ D i +1 ,r ( v, c )] ≤ (cid:18) jr (cid:19) q r q j − r (cid:0) δ q − (cid:1) r (cid:0) δ q − (cid:1) j − r + 2 k L ji ≤ (cid:18) jr (cid:19) q r q j − r (cid:0) rδ q − (cid:1) (cid:0) j − r ) δ q − (cid:1) + 2 k L ji ≤ (cid:18) jr (cid:19) q r q j − r + 2 r (cid:18) jr (cid:19) q r − q j − r δ + O L ji ! ≤ (cid:18) jr (cid:19) q r q j − r + 4 r (cid:18) jr (cid:19) Keep j − i α j − r +1 S i L j − ri + O L ji ! , concluding the proof. Note that in the second inequality above we used that Keep i is bounded below by aconstant according to the bound in Corollary 4.2 (which only requires the assumptions of Lemma 4.1). Let
Bin( n, p ) denote the binomial random variable that counts the number of successes in n Bernoulli trials,where each trial succeeds with probability p . We will find the following lemma useful (see, e.g., Exercise2.12 in [35]) : Lemma 4.4.
For any c, k, n we have Pr h Bin (cid:16) n, cn (cid:17) ≥ k i ≤ c k k ! . Proof of Part (a) . We will use Theorem 2.4 to show that that the number of colors, ℓ v , which are removedfrom L v during iteration i is highly concentrated.Note that changing the assignment to any neighboring vertex of v can change ℓ v by at most , andchanging the assignment to any other vertex cannot affect ℓ v at all.If ℓ v ≥ s , there are at most s groups of at most k − neighbors of v , so that each vertex in eachgroup received the same color, and each group corresponds to a different color from L v . Thus, the colorassignments and activation choices of these vertices certify that ℓ v ≥ s .Since, according to Corollary 4.2, Keep i = Ω(1) , applying Theorem 2.4 with t = L . i , w = 2 k , c = 1 ,we obtain Pr h | ℓ v − E [ ℓ v ] | > L / i i < ∆ − ln ∆ , for sufficiently large ∆ .Finally, E [ ℓ i +1 ( v )] = ℓ i ( v ) − E [ ℓ v ] implies that Pr h | ℓ i +1 ( v ) − E [ ℓ i +1 ( v )] | > L / i i = Pr h | ℓ v − E [ ℓ v ] | > L / i i < ∆ − ln ∆ . Proof of Part (b) . Recall the definition of D i,r ( v, c ) and let Z i,r ( v, c ) = S k − j = r D i,j ( v, c ) . Let X i +1 ,r ( v, c ) denote the number of hyperedges in Z i,r ( v, c ) which (i) contain exactly r uncolored vertices other than v ;and (ii) the rest of their vertices are assigned c in the end of the i -th iteration. Let also Y i +1 ,r ( v, c ) be thenumber of these hyperedges which they contain an uncolored vertex u = v such that (i) c / ∈ L u ( i + 1) ; and(ii) c would still not be in L u ( i + 1) even if we ignored the color of v .19t is straightforward to verify that t ′ i +1 ,r ( v, c ) = X i +1 ,r ( v, c ) − Y i +1 ,r ( v, c ) . Therefore, by the linearityof expectation, it suffices to show that X i +1 ,r ( v, c ) and Y i +1 ,r ( v, c ) are both sufficiently concentrated. Thisis because Pr t ′ i +1 ,r ( v, c ) − E [ t ′ i +1 ( v, c )] > k − X j = r (cid:18) jr (cid:19) α j − r T i,j L j − ri / , = Pr X i +1 ,r ( v, c ) − E [ X i +1 ( v, c )] − ( Y i +1 ,r ( v, c ) − E [ Y i +1 ,r ( v, c )]) > k − X j = r (cid:18) jr (cid:19) α j − r T i,j L j − ri / , and, therefore, it is sufficient to prove that Pr X i +1 ,r ( v, c ) − E [ X i +1 ,r ( v, c )] > k − X j = r (cid:18) jr (cid:19) α j − r T i,j L j − ri / ≤
12 ∆ − ln ∆ , (27) Pr Y i +1 ,r ( v, c ) − E [ Y i +1 ,r ( v, c )] < − k − X j = r (cid:18) jr (cid:19) α j − r T i,j L j − ri / ≤
12 ∆ − ln ∆ . (28)We first focus on X i +1 ,r ( v, c ) . Let X ′ i +1 ,r ( v, c ) denote the number of hyperedges in Z i,r ( v, c ) which (i)contain exactly r uncolored vertices other than v ; and (ii) the rest of their vertices were activated and assigned c (but did not retain their color necessarily). Further, let W i +1 ,r ( v, c ) denote the random variable that countsall the hyperedges counted by X ′ i +1 ,r ( v, c ) , except for those whose r uncolored vertices (other than v ) wereuncolored because they were activated and received the same color as v . Finally, let W i +1 ,r ( v, c ) be thenumber of hyperedges which (i) contain exactly r vertices that are activated and received the same colorwith v ; and (ii) the rest of their k − − r vertices were activated and assigned c .Observe that X i +1 ,r ( v, c ) ≤ W i +1 ,r ( v, c ) + W i +1 ,r ( v, c ) . The idea here is that we cannot directly applyTalagrand’s inequality to X i +1 ,r ( v, c ) and so we consider W i +1 ,r ( v, c ) , W i +1 ,r ( v, c ) instead.First, we consider W i +1 ,r ( v, c ) . Since our hypergraph is of girth at least , changing a choice for somevertex of a hyperedge h ∈ Z i,r ( v, c ) can only affect whether or not the vertices of h remain uncolored,and thus affect W i +1 ,r ( v, c ) by at most . Furthermore, changing a choice for a vertex outside the onesthat correspond to the hyperedges in Z i,r ( v, c ) can affect at most one vertex of at most one hyperedge in Z i,r ( v, c ) and, therefore, can affect W i +1 ,r ( v, c ) by at most .We claim now that if W i +1 ,r ( v, c ) ≥ s , then there exist at most k s random choices that certify thisevent. To see this, notice that if a hyperedge h is counted by W i +1 ,r ( v, c ) , then for every u ∈ h \ { v } thatremained uncolored, it must be that either u was deactivated, or u is contained in a hyperedge h ′ = h suchthat all the vertices in ( h ′ \ { u } ) ∩ V i were activated and received the same color as u . Moreover, the eventthat a variable u ∈ h \ { v } was activated and received c can be verified by the outcome of two randomchoices. So, overall, we can certify that h was counted by W i +1 ,r ( v, c ) by using the outcome of at most k random choices.Finally, observe that E [ W i +1 ,r ( v, c )] ≤ P k − j = r (cid:0) jr (cid:1) α j − r T i,j L j − ri and, thus, applying Theorem 2.4 with c = 1 ,20 = 2 k and t = (cid:18)P k − j = r (cid:0) jr (cid:1) T i,r L j − ri (cid:19) . / we obtain Pr (cid:12)(cid:12) W i +1 ,r ( v, c ) − E [ W i +1 ,r ( v, c )] (cid:12)(cid:12) > k − X j = r (cid:18) jr (cid:19) α j − r T i,j L j − ri / ≤
14 ∆ − ln ∆ , (29)for sufficiently large ∆ .As far as W i +1 ,r ( v, c ) is concerned, note that it can be bounded above by P k − j = r Bin( T i,j , (cid:0) jr (cid:1) α j L ji ) andobserve that, according to Lemma 4.1, each of these binomial random variables has constant expectation.Since T i,j ≥ (ln ∆) k − for every j ∈ [ k − , Lemma 4.4 implies that Pr (cid:12)(cid:12) W i +1 ,r ( v, c ) − E [ W i +1 ,r ( v, c )] (cid:12)(cid:12) > k − X j = r (cid:18) jr (cid:19) α j − r T i,j L j − ri / ≤
14 ∆ − ln ∆ . (30)Combining (29) and (30) implies (27).We follow the same approach for Y i +1 ,r ( v, c ) . Let Y ′ i +1 ,r ( v, c ) be the number of hyperedges counted by X ′ i +1 ,r ( v, c ) and which also contain an uncolored vertex u = v such that (i) c / ∈ L u ( i + 1) ; and (ii) c wouldstill not be in L u ( i + 1) even if we ignored the color of v . Further, let Y ′′ i +1 ,r ( v, c ) be the random variablethat counts all the hyperedges counted by Y ′ i +1 ,r ( v, c ) , except for those whose r uncolored vertices wereuncolored because they were activated and received the same color as v , and observe that Y i +1 ,r ( v, c ) ≤ Y ′ i +1 ,r ( v, c ) ≤ Y ′′ i +1 ,r ( v, c ) + W i +1 ,r ( v, c ) .Moreover, if Y ′′ i +1 ,r ( v, c ) ≥ s , then there exist at most (2 k + 2 k ) s random choices that certify thisevent. To see this, observe that for each hyperedge h counted by Y i +1 ,r ( v, c ) , we need the output of at most k choices to certify that it is counted by X ′ i +1 ,r ( v, c ) , and the output of at most k extra random choicesto certify that there is a vertex u ∈ h \ { v } for which c / ∈ L u ( i + 1) , and c would still not be in L u ( i + 1) even if we ignored the color of v .Finally, E [ Y ′′ i +1 ,r ( v, c )] ≤ P k − j = r (cid:0) jr (cid:1) α j − r T i,j L j − ri and, therefore, an almost identical argument to the casefor X i +1 ,r ( v, c ) implies (28). We will use induction on i . Property P (1) clearly holds, so we assume that property P ( i ) holds and weprove that with property P ( i + 1) holds with positive probability. Recall our discussion in the previoussection in which we argued that we can assume without loss of generality that property Q ( i ) holds.For every v and c ∈ L v let A v be the event that ℓ i +1 ( v ) < L i +1 and B rv,c to be the event that t i +1 ,r ( v, c ) > T i +1 ,r . Clearly, if we can avoid these bad events with positive probability, then P ( i + 1) holds.Recall that, according to Lemma 4.1, R i,r = O (ln ∆) for every r ∈ [ k − . Recall further thatfor any vertex v and color c such that at the beginning of iteration i + 1 , v is uncolored and c ∈ L v ,we have t i +1 ,r ( v, c ) = t ′ i +1 ,r ( v, c ) for every r ∈ [ k − . Therefore by Lemma 3.1, if B rv,c holds then t ′ i +1 ,r ( v, c ) − E [ t ′ i +1 ,r ( v, c )] > P k − j = r (cid:16)(cid:0) jr (cid:1) α j − r T i,j L i,j (cid:17) / .By Lemma 3.2, the probability of any of our bad events is at most ∆ − ln ∆ . Furthermore, each badevent f v ∈ { A v , B rv,c } event is determined by the colors assigned to vertices of distance at most from v .Therefore, f v is mutually independent of all but at most ( k ∆) (1 + δ ) (cid:0) ∆ln ∆ (cid:1) k − < ∆ other bad events.For ∆ sufficiently large, ∆ − ln ∆ ∆ < and so the proof is concluded by applying Corollary 2.3.21 .4 Proof of Lemma 3.4 Since L i < L ′ i , for the first part of the lemma it suffices to prove that L ′ i ≤ L i + ( L ′ i ) / . Towards that end,at first we observe that for sufficiently large ∆ , Corollary 4.2 and the fact that K = k k imply: Keep / i − Keep i ≥ (cid:18) − K k − k (ln ∆) k − (cid:19) / − (cid:18) − K k − k (ln ∆) k − (cid:19) ≥ (cid:18) − · K k − k (ln ∆) k − (cid:19) − (cid:18) − K k − k (ln ∆) k − (cid:19) = K k − k (ln ∆) k − . (31)Note that in deriving the first inequality we used the fact that the function x / − x is decreasing on theinterval [ C, since K is sufficiently small. For the second one, we used the Taylor Series for (1 − y ) / around y = 0 .We now proceed by using induction. The base case is trivial, so assume that the statement is true for i ,and consider i + 1 . Since, by our assumption, L i ≥ (ln ∆) k − we obtain L ′ i +1 = Keep i L ′ i ≤ Keep i (cid:16) L i + ( L ′ i ) / (cid:17) (32) = L i +1 + L / i + Keep i ( L ′ i ) / ≤ L i +1 + L / i + Keep / i ( L ′ i ) / − K k − k (ln ∆) k − ( L ′ i ) / (33) ≤ L i +1 + ( L ′ i +1 ) / + L / i − K k − k (ln ∆) k − ( L ′ i ) / < L i +1 + ( L ′ i +1 ) / , (34)for sufficiently large ∆ . Note that in deriving (32) we used the inductive hypothesis; for (33) we used (31);and for (34) the fact that L i ≥ (ln ∆) k − and the inductive hypothesis.The proof of the second part of the lemma is similar. That is, we observe that it suffices to show that T ′ i,r ≥ T i,r − ( T ′ i,r ) r r +1 and proceed by using induction. Again, the base case is trivial, so we assume thestatement is true for i , and consider i + 1 . We obtain the following. T ′ i +1 ,r = k − X j = r T ′ i,j · (Keep i (1 − α Keep i )) r (cid:18) jr (cid:19) (cid:18) α Keep i L ′ i (cid:19) j − r ! + 3 k r α − r +1 L ri k − X ℓ =1 T i,ℓ L ℓi (ln ∆) ℓ ≥ k − X j = r (cid:16) T i,j − ( T ′ i,j ) r r +1 (cid:17) (Keep i (1 − α Keep i )) r (cid:18) jr (cid:19) (cid:18) α Keep i L ′ i (cid:19) j − r ! + 3 k r α − r +1 L ri k − X ℓ =1 T i,ℓ L ℓi (ln ∆) ℓ (35)22 T i +1 ,r − k − X j = r (cid:18) jr (cid:19) α j − r T i,j L j − ri / − k − X j = r T i,j (Keep i (1 − α Keep i )) r ( α Keep i ) j − r L j − ri − L ′ i ) j − r ! − k − X j = r ( T ′ i,j ) r r +1 (Keep i (1 − α Keep i )) r (cid:18) jr (cid:19) (cid:18) α Keep i L ′ i (cid:19) j − r ≥ T i +1 ,r − k − X j = r (cid:18) jr (cid:19) α j − r T i,j L j − ri / − k − X j = r +1 (Keep i (1 − α Keep i )) r ( α Keep i ) j − r T i,j L j − ri O (cid:18) L − i (cid:19) − (Keep i (1 − α Keep i )) r/ ( T ′ i +1 ,r ) r r +1 (36) ≥ T i +1 ,r − (cid:18) − K k − k (ln ∆) k − (cid:19) ( T ′ i +1 ,r ) r r +1 − O (cid:18) L r − i + L r/ i + T / i,r (cid:19) (37) ≥ T i +1 ,r − ( T ′ i +1 ,r ) r r +1 , (38)for sufficiently large ∆ , concluding the proof of the lemma. Note that in deriving (36) we used the first partof Lemma 3.4, and that k − X j = r ( T ′ i,j ) r r +1 (Keep i (1 − α Keep i )) r (cid:18) jr (cid:19) (cid:18) α Keep i L ′ i (cid:19) j − r = (Keep i (1 − α Keep i )) r r +1 k − X j = r ( T ′ i,j ) r r +1 (Keep i (1 − α Keep i )) r r +1 (cid:18) jr (cid:19) (cid:18) α Keep i L ′ i (cid:19) j − r ≤ (Keep i (1 − α Keep i )) r r +1 k − X j = r ( T ′ i,j ) r r +1 (Keep i (1 − α Keep i )) r r +1 (cid:18) jr (cid:19) (cid:18) α Keep i L ′ i (cid:19) j − r ! r r +1 ≤ (Keep i (1 − α Keep i )) r r +1 k − X j = r ( T ′ i,j ) (Keep i (1 − α Keep i )) r (cid:18) jr (cid:19) (cid:18) α Keep i L ′ i (cid:19) j − r r r +1 = (Keep i (1 − α Keep i )) r r +1 ( T ′ i,r +1 ) r r +1 , for sufficiently large ∆ . In deriving (37) we used Lemma 4.1 and the fact that, using Corollary 4.2, weobtain (Keep i (1 − α Keep i )) r r +1 ≤ (cid:18) − K k − k (ln ∆) k − (cid:19) r r +1 ≤ (cid:18) − K k − k (ln ∆) k − (cid:19) = (cid:18) − K k − k (ln ∆) k − (cid:19) . Finally, to derive (38) at first we observe that R i,ℓ = Ω (cid:16) k − − r (cid:17) , by definition (recall e.g., (10)) andour assumption that R i,k − = Ω(1) , and, thus, L i = O (cid:18) T r i,r (ln ∆) k − − rr (cid:19) . Then, we use our assumption23hat T i,ℓ ≥ (ln ∆) k − for every ℓ ∈ [ k − and the inductive hypothesis to conclude that ( T ′ i +1 ,r ) r r +1 k (ln ∆) k − = ω (cid:18) L r − i + L r/ i + T / i,r (cid:19) . We proceed by induction. Let η := ǫ/ k − ǫ/ . We will assume that L j ≥ ∆ η , T j,r ≥ (ln ∆) k − forall ≤ j ≤ i < i ∗ , and prove that L i +1 ≥ ∆ η , T i +1 ,r ≥ (ln ∆) k − . Towards that end, it will be usefulto focus on the family of ratios R i,r , r ∈ [ k − ]. Note that, according to Lemma 3.4, this family is well-approximated by the family R ′ i,r , r ∈ [ k − . In particular, recalling Lemma 4.3 and applying Lemma 3.4we obtain: R i,r ≤ R ′ i,r · T ′ i,r ) − r +1 (cid:0) − ( L ′ i ) − / (cid:1) r ≤ (1 − αC ) r ( i − ln ∆ · Q k − p = r ( p + 1)(1 + δ − . δk ) k − C k − − r , (39)for sufficiently large ∆ , since L i , T i,r ≥ (ln ∆) k − .Using (39) and the fact that − x > e − x − for x ≥ we can get an improved lower bound for Keep i as follows. Keep i ≥ exp − − δk k ) k − X r =1 α r R i,r ! ≥ exp − (cid:0) δ − . δk (cid:1) k − k − X r =1 (1 − αC ) r ( i − K r Q k − p = r ( p + 1)(ln ∆) r − C k − − r ! , (40)for sufficiently large ∆ .Using (40) we get i − Y j =1 Keep j ≥ exp − (cid:0) δ − . δk (cid:1) k − k − X r =1 K r Q k − p = r ( p + 1)(ln ∆) r − C k − − r i − X j =1 (1 − αC ) r ( j − ≥ exp − (cid:0) δ − . δk (cid:1) k − k − X r =1 K r Q k − p = r ( p + 1)(ln ∆) r − C k − − r · − (1 − αC ) r !! ≥ exp − C − ( k − (1 + δk )( k − (cid:0) δ − . δk (cid:1) k − ! ≥ exp (cid:18) − ln ∆(1 + ǫ )( k − (cid:19) (41)for sufficiently large ∆ .Using (41) we can now bound L ′ i as follows. L ′ i = L ′ i − Y j =1 Keep j ≥ (1 + δ ) (cid:18) ∆ln ∆ (cid:19) k − ∆ − ǫ k − ≥ ∆ η , (42)24or sufficiently large ∆ . Thus, L ′ i never gets too small for the purposes of our analysis. Lemma 3.4 impliesthat neither does L i .The proof is concluded by observing that (39) implies that R i,r , r ∈ [ k − , becomes smaller than k for i = O (ln ∆ ln ln ∆) . In this section we present the proof of Theorem 1.7. To do so, we build on ideas of Alon, Krivelevichand Sudakov [8] and show that the random hypergraph H ( k, n, d/ (cid:0) nk − (cid:1) ) almost surely admits a few usefulfeatures.The first lemma we prove states that all subgraphs of H ( k, n, d/ (cid:0) nk − (cid:1) ) with not too many vertices aresparse and, therefore, of small degeneracy. Lemma 5.1.
For every constant k ≥ , there exists d k > such that for any constant d ≥ d k , the randomhypergraph H ( k, n, d/ (cid:0) nk − (cid:1) ) has the following property almost surely: Every s ≤ nd − k − vertices of H span fewer than (cid:16) d (ln d ) (cid:17) k − s hyperedges. Therefore, any subhypergraph of H induced by a subset V ⊂ V of size | V | ≤ nd − k − , is k (cid:16) d (ln d ) (cid:17) k − -degenerate.Proof. Letting r = (cid:16) d (ln d ) (cid:17) k − , we see that the probability that there exists a subset V ⊂ V which violatesthe statement of the lemma is at most nd − k − X i = r k − (cid:18) ni (cid:19)(cid:18)(cid:0) ik (cid:1) ri (cid:19) d (cid:0) nk − (cid:1) ! ri ≤ nd − k − X i = r k − " e ni (cid:18) e i k − r (cid:19) r d (cid:0) nk − (cid:1) ! r i (43) ≤ nd − k − X i = r k − e k − ( k − (cid:18) dr (cid:19) k − e i k − dr (cid:0) nk − (cid:1) ! r − k − i = o (1) , for sufficiently large d . Note that in the lefthand side of (43) we used the fact that any subset of vertices ofsize s < r k − cannot violate the assertion of the lemma, since it can span at most s k < rs hyperedges. Inderiving the final inequality we used that for any pair of integers α, β , we have that (cid:0) αβ (cid:1) ≥ (cid:16) αβ (cid:17) β .Next we show that, as far as the number of vertices of H ( k, n, d/ (cid:0) nk − (cid:1) ) that have a constant degree c isconcerned, the degree of each vertex of H is essentially a Poisson random variable with mean d . Lemma 5.2.
For constants c ≥ , k ≥ and d , let X c denote the number of vertices of degree c in H ( k, n, d/ (cid:0) nk − (cid:1) ) . Then, for c = O (1) , with high probability, X c ≤ d c e − d c ! n (cid:18) O (cid:18) log n √ n (cid:19)(cid:19) . Proof.
The lemma follows from standard ideas for estimation of the degree distribution of random graphs(see for example the proof of Theorem 3.3 in [19] for the case k = 2 ). In particular, assume that the vertices25f H ( k, n, d/ (cid:0) nk − (cid:1) ) are labeled , , . . . , n . Then, E [ X c ] = n Pr[deg(1) = c ]= n (cid:18)(cid:0) n − k − (cid:1) c (cid:19) d (cid:0) nk − (cid:1) ! c − d (cid:0) nk − (cid:1) ! ( n − k − ) − c ≤ n (cid:16)(cid:0) n − k − (cid:1)(cid:17) c c ! O c (cid:0) n − k − (cid:1) !! d (cid:0) nk − (cid:1) ! c exp − (cid:18)(cid:18) n − k − (cid:19) − c (cid:19) d (cid:0) nk − (cid:1) ! ≤ n d c e − d c ! (cid:18) O (cid:18) n k − (cid:19)(cid:19) . To show concentration of X c around its expectation, we will use Chebyshev’s inequality. In order to doso, we need to estimate Pr[deg(1) = deg(2) = c ] . For ℓ ∈ { , . . . , c } , let E ℓ , denote the event that thereexist exactly ℓ hyperedges that contain both vertices and . Then, letting p = d ( nk − ) , we see that Pr[deg(1) = deg(2) = c ] ≤ c X ℓ =0 Pr h E ℓ , i (cid:18)(cid:0) n − k − (cid:1) c − ℓ (cid:19) p c (1 − p )( n − k − ) − c ! = c X ℓ =0 (cid:18)(cid:0) n − k − (cid:1) ℓ (cid:19) p ℓ (1 − p )( n − k − ) − ℓ (cid:18)(cid:0) n − k − (cid:1) c − ℓ (cid:19) p c (1 − p )( n − k − ) − c ! = Pr[deg(1) = c ] · Pr[deg(2) = c ] (cid:18) O (cid:18) n k − (cid:19)(cid:19) . Therefore,
Var[ X c ] = n X i =1 n X j =1 (Pr[deg( i ) = c, deg( j ) = c ] − Pr[deg(1) = c ] Pr[deg(2) = c ]) ≤ X i = j =1 O (cid:18) n k − (cid:19) + E [ X c ] = An , for some constant A = A ( c, d ) .Finally, applying the Chebyshev’s inequality, we obtain that, for any t > , Pr (cid:2) | X c − E [ X c ] | ≥ t √ n (cid:3) ≤ At , and, thus, the proof is concluded by choosing t = log n .Lemma 5.2 implies the following useful corollary. Corollary 5.3.
For any constants δ ∈ (0 , , k ≥ , d > , let X = X ( δ, k, d ) denote the random variableequal to the number of vertices in H ( k, n, d/ (cid:0) nk − (cid:1) ) whose degree is in [(1 + δ ) d, k − k − d ] . Thereexists a constant d δ > such that if d ≥ d δ then, almost surely, X ≤ nd . Proof.
Let X r denote the number of vertices of degree r in H ( k, n, d/ (cid:0) nk − (cid:1) ) . Since k, d are constants,using Lemma 5.2 and Stirling’s approximation we see that, almost surely, k − k − d X r =(1+ δ ) d X r ≤ n (cid:18) O (cid:18) log n √ n (cid:19)(cid:19) k − k − d X r =(1+ δ ) d d r e − d r ! ≤ n (1 + δ ) k − k − d X r =(1+ δ ) d d r e − d √ πr (cid:0) r e (cid:1) r ≤ nd , for sufficiently large d and n . 26sing Lemma 5.1 and Corollary 5.2 we show that, almost surely, only a small fraction of vertices of H ( k, n, d/ (cid:0) nk − (cid:1) ) have degree that significantly exceeds its average degree. Lemma 5.4.
For every constants k ≥ and δ ∈ (0 , , there exists d k,δ > such that for any constant d ≥ d k,δ , all but at most nd vertices of the random hypergraph H ( k, n, d/ (cid:0) nk − (cid:1) ) have degree at most (1 + δ ) d , almost surely.Proof. Corollary 5.3 implies that the number of vertices with degree in the interval [(1 + δ ) d, k − k − d ] is at most nd , for sufficiently large d .Suppose now there are more than nd vertices with degree at least k − k − d . Denote by S a setcontaining exactly nd such vertices. According to Lemma 5.1, almost surely, the induced subhypergraph H [ S ] has at most e ( H [ S ]) ≤ (cid:18) d (ln d ) (cid:19) k − | S | = nd − k − (ln d ) k − hyperedges. Therefore, the number of hyperedges between the sets of vertices S and V \ S is at least k − k − d | S | − ke ( H [ S ]) ≥ . k − k − nd =: N. However, the probability that H ( k, n, d/ (cid:0) nk − (cid:1) ) contains such a subhypergraph is at most (cid:18) n nd (cid:19)(cid:18) n k d N (cid:19) d (cid:0) nk − (cid:1) ! N ≤ (cid:0) e d (cid:1) nd n k e d N · d (cid:0) nk − (cid:1) ! N = o (1) , for sufficiently large d . Note that in deriving the final equality we used that for any pair of integers α, β , wehave that (cid:0) αβ (cid:1) ≥ (cid:16) αβ (cid:17) β . Therefore, almost surely there are at most nd vertices in G with degree greater than k − k − d , concluding the proof.Finally, we show that the neighborhood of a typical vertex of H ( k, n, d/ (cid:0) nk − (cid:1) ) is locally tree-like. Lemma 5.5.
For every constants k ≥ , δ ∈ (0 , , almost surely, the random hypergraph H ( k, n, d/ (cid:0) nk − (cid:1) ) has a subset U ⊆ V ( H ) of size at most n − δ such that the induced hypergraph H [ V \ U ] is of girth at least .Proof. Let Y , Y , Y , denote the number of -, - and -cycles in H ( n, k, d/ (cid:0) nk − (cid:1) ) , respectively. A straight-forward calculation reveals that for i ∈ { , , } : E [ Y i ] ≤ i ( k − X s =1 (cid:18) ns (cid:19)(cid:18)(cid:0) sk − (cid:1) i (cid:19) d (cid:0) nk − (cid:1) ! i ≤ i ( k − n i ( k − (cid:18) ( i ( k − k − e ( k − k − in k − (cid:19) i = O (1) . By Markov’s inequality this implies that Y + Y + Y ≤ n −√ δ almost surely. Denote by U the union of all -, - and - cycles in H . Then the induced subhypergraph H [ V \ U ] has girth at least and, almost surely, | U | ≤ n − δ .We are now ready to prove Theorem 1.7. 27 roof of Theorem 1.7. Our goal will be to find a subset U ⊂ V of size | U | ≤ nd − k − such that the inducedsubgraph H [ V \ U ] is of girth at least and maximum degree at most (1 + δ ) d and, further, every vertex v in V \ U has at most k (cid:16) d (ln d ) (cid:17) k − = o (cid:16) d (ln d ) (cid:17) k − neighbors in U . Note that in this case, accordingto Lemma 5.1, H [ U ] is k (cid:16) d (ln d ) (cid:17) k − -degenerate, concluding the proof assuming d is sufficiently large. Asimilar idea has been used in [7, 8, 31].Towards that end, let U be the set of vertices of degree more than (1 + δ ) d , and U the set of verticesthat are contained in a -, - or a -cycle. Notice that U , U , can be found in polynomial time and, accordingto Lemmas 5.4 and 5.5, the size of U := | U ∪ U | is at most nd for sufficiently large n and d .We now start with U := U and as long as there exists a vertex v ∈ V \ U having at least k (cid:16) d (ln d ) (cid:17) k − neighbors in U we do the following. Let S v = { u , u , . . . , u N } be the neighbors of v in U . We choose anarbitrary hyperedge h that contains v and u and update U and S v by defining U := U ∪ h and S v := S v \ h .We keep repeating this operation until S v is empty.This process terminates with | U | < nd − k − because, otherwise, we would get a subset U ⊂ V of size | U | = nd − k − spanning more than k (cid:18) nd k − − | U | (cid:19) × k (cid:18) d (ln d ) (cid:19) k − × k > nd k − × (cid:18) d (ln d ) (cid:19) k − hyperedges, for sufficiently large d . According to Lemma 5.1 however, H does not contain any such setalmost surely. The author is grateful to Dimitris Achlioptas and Irit Dinur for detailed comments and feedback.
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