Distributed Algorithms for Matching in Hypergraphs
DDistributed Algorithms for Matching in Hypergraphs
Oussama Hanguir [email protected]
Columbia University Clifford Stein ∗ [email protected] Columbia University
Abstract
We study the d -Uniform Hypergraph Matching ( d -UHM) problem: given an n -vertex hy-pergraph G where every hyperedge is of size d , find a maximum cardinality set of disjointhyperedges. For d ≥
3, the problem of finding the maximum matching is
N P -complete, andwas one of Karp’s 21
N P -complete problems. In this paper we are interested in the problemof finding matchings in hypergraphs in the massively parallel computation (MPC) model thatis a common abstraction of MapReduce-style computation. In this model, we present the firstthree parallel algorithms for d -Uniform Hypergraph Matching, and we analyse them in termsof resources such as memory usage, rounds of communication needed, and approximation ratio.The highlights include: • A O (log n )-round d -approximation algorithm that uses O ( nd ) space per machine. • A 3-round, O ( d )-approximation algorithm that uses ˜ O ( √ nm ) space per machine. • A 3-round algorithm that computes a subgraph containing a ( d − d ) -approximation,using ˜ O ( √ nm ) space per machine for linear hypergraphs, and ˜ O ( n √ nm ) in general.For the third algorithm, we introduce the concept of HyperEdge Degree Constrained Subgraph(HEDCS), which can be of independent interest. We show that an HEDCS contains a fractionalmatching with total value at least | M ∗ | / ( d − d ), where | M ∗ | is the size of the maximummatching in the hypergraph. Moreover, we investigate the experimental performance of thesealgorithms both on random input and real instances. Our results support the theoretical boundsand confirm the trade-offs between the quality of approximation and the speed of the algorithms. ∗ Research partially supported by NSF grants CCF-1714818 and CCF-1822809. a r X i v : . [ c s . D S ] S e p Introduction
As massive graphs become more ubiquitous, the need for scalable parallel and distributed algorithmsthat solve graph problems grows as well. In recent years, we have seen progress in many graphproblems (e.g. spanning trees, connectivity, shortest paths [4, 5]) and, most relevant to this work,matchings [22, 27]. A natural generalization of matchings in graphs is to matchings in hypergraphs .Hypergraph Matching is an important problem with many applications such as capital budgeting,crew scheduling, facility location, scheduling airline flights [53], forming a coalition structure inmulti-agent systems [50] and determining the winners in combinatorial auctions [49] (see [56] for apartial survey). Although matching problems in graphs are one of the most well-studied problems inalgorithms and optimization, the NP-hard problem of finding a maximum matching in a hypergraphis not as well understood.In this work, we are interested in the problem of finding matchings in very large hypergraphs,large enough that we cannot solve the problem on one computer. We develop, analyze and experi-mentally evaluate three parallel algorithms for hypergraph matchings in the MPC model. Two ofthe algorithms are generalizations of parallel algorithms for matchings in graphs. The third algo-rithm develops new machinery which we call a hyper-edge degree constrained subgraph (HEDCS),generalizing the notion of an edge-degree constrained subgraph (EDCS). The EDCS has been re-cently used in parallel and dynamic algorithms for graph matching problems [6, 16, 17]. We willshow a range of algorithm tradeoffs between approximation ratio, rounds, memory and computa-tion, evaluated both as worst case bounds, and via computational experiments.More formally, a hypergraph G is a pair G = ( V, E ) where V is the set of vertices and E is theset of hyperedges. A hyperedge e ∈ E is a nonempty subset of the vertices. The cardinality of ahyperedge is the number of vertices it contains. When every hyperedge has the same cardinality d , the hypergraph is said to be d - uniform . A hypergraph is linear if the intersection of any twohyperedges has at most one vertex. A hypergraph matching is a subset of the hyperedges M ⊆ E such that every vertex is covered at most once, i.e. the hyperedges are mutually disjoint. Thisnotion generalizes matchings in graphs. The cardinality of a matching is the number of hyperedgesit contains. A matching is called maximum if it has the largest cardinality of all possible matchings,and maximal if it is not contained in any other matching. In the d -Uniform Hypergraph MatchingProblem (also referred to as Set Packing or d -Set Packing), a d -uniform hypergraph is given andone needs to find the maximum cardinality matching.We adopt the most restrictive MapReduce-like model of modern parallel computation among[4, 11, 29, 40], the Massively Parallel Computation (MPC) model of [11]. This model is widely usedto solve different graph problems such as matching, vertex cover [2, 6, 7, 27, 44], independent set[27, 34], as well as many other algorithmic problems. In this model, we have k machines (processors)each with space s . N is the size of the input and our algorithms will satisfy k · s = ˜ O ( N ), whichmeans that the total space in the system is only a polylogarithmic factor more than the input size.The computation proceeds in rounds. At the beginning of each round, the data (e.g. vertices andedges) is distributed across the machines. In each round, a machine performs local computationon its data (of size s ), and then sends messages to other machines for the next round. Crucially,the total amount of communication sent or received by a machine is bounded by s , its space. Forexample, a machine can send one message of size s , or s messages of size 1. It cannot, however,broadcast a size s message to every machine. Each machine treats the received messages as theinput for the next round. Our model limits the number of machines and the memory per machine tobe substantially sublinear in the size of the input. On the other hand, no restrictions are placed onthe computational power of any individual machine. The main complexity measure is therefore thememory per machine and the number of rounds R required to solve a problem, which we consider2o be the “parallel time” of the algorithm. For the rest of the paper, G ( V, E ) is a d -uniformhypergraph with n vertices and m hyperedges, and when the context is clear, we will simply referto G as a graph and to its hyperedges as edges. M M ( G ) denotes the maximum matching in G , and µ ( G ) = | M M ( G ) | is the size of that matching. We define d G ( v ) to be the degree of a vertex v (thenumber of hyperedges that v belongs to) in G . When the context is clear, we will omit indexingby G and simply denote it d ( v ). We design and implement algorithms for the d -UHM in the MPC model. We will give three differentalgorithms, demonstrating different trade-offs between the model’s parameters. Our algorithms areinspired by methods to find maximum matchings in graphs, but require developing significant newtools to address hypergraphs. We are not aware of previous algorithms for hypergraph matchingin the MPC model. First we generalize the randomized coreset algorithm of [7] which finds an3-rounds O (1)-approximation for matching in graphs. Our algorithm partitions the graph intorandom pieces across the machines, and then simply picks a maximum matching of each machine’ssubgraph. We show that this natural approach results in a O ( d )-approximation. While thealgorithmic generalization is straightforward, the analysis requires several new ideas. Theorem 4.2 (restated).
There exists an
M P C algorithm that with high probability computesa (3 d ( d −
1) + 3 + (cid:15) ) -approximation for the d -UHM problem in 3 M P C rounds on machines ofmemory s = ˜ O ( √ nm ) . Our second result concerns the MPC model with per-machine memory O ( d · n ). We adaptthe sampling technique and post-processing strategy of [44] to construct maximal matchings inhypergraphs, and are able to show that in d -uniform hypergraphs, this technique yields a maximalmatching, and thus a d -approximation to the d -UHM problem in O (log n ) rounds. Theorem 5.1 (restated).
There exists an
M P C algorithm that given a d -uniform hypergraph G ( V, E ) with high probability computes a maximal matching in G in O (log n ) M P C rounds onmachines of memory s = Θ( d · n ) . Our third result generalizes the edge degree constrained subgraphs (EDCS), originally intro-duced by Bernstein and Stein [16] for maintaining large matchings in dynamic graphs, and later usedfor several other problems including matching and vertex cover in the streaming and MPC model[6, 17]. We call these generalized subgraphs hyper-edge degree constrained subgraphs (HEDCS).We show that they exist for specific parameters, and that they contain a good approximation forthe d -UHM problem. We prove that an HEDCS of a hypergraph G , with well chosen parameters,contain a fractional matching with a value at least dd − d +1 µ ( G ), and that the underlying fractionalmatching is special in the sense that for each hyperedge, it either assigns a value of 1 or a value lessthan some chosen (cid:15) . We call such a fractional matching an (cid:15) -restricted fractional matching. ForTheorem 6.15, we compute an HEDCS of the hypergraph in a distributed fashion. This procedurerelies on the robustness properties that we prove for the HEDCSs under sampling. Theorem 6.4 (restated informally).
Let G be a d -uniform hypergraph and < (cid:15) < . Thereexists an HEDCS that contains an (cid:15) -restricted fractional matching M Hf with total value at least µ ( G ) (cid:16) dd − d +1 − (cid:15) (cid:17) . Theorem 6.15 (restated).
There exists an
M P C algorithm that given a d -uniform hypergraph G ( V, E ) , where | V | = n and | E | = m , can construct an HEDCS of G in 2 M P C rounds onmachines of memory s = ˜ O ( n √ nm ) in general and s = ˜ O ( √ nm ) for linear hypergraphs. orollary 6.16 (restated). There exists an
M P C algorithm that with high probability achievesa d ( d − /d ) -approximation to the d -Uniform Hypergraph Matching in 3 rounds. Table 1 summarizes our results.
Approximation ratio Rounds Memory per machine Computation per round3 d ( d −
1) + 3 3 ˜ O ( √ nm ) Exponential d O (log n ) O ( dn ) Polynomial d ( d − /d ) O ( n √ nm ) in general Polynomial˜ O ( √ nm ) for linear hypergraphs Table 1: Our parallel algorithms for the d -uniform hypergraph matching problem. Experimental results.
We implement our algorithms in a simulated MPC model environmentand test them both on random and real-world instances. Our experimental results are consistentwith the theoretical bounds on most instances, and show that there is a trade-off between theextent to which the algorithms use the power of parallelism and the quality of the approximations.This trade-off is illustrated by comparing the number of rounds and the performance of thealgorithms on machines with the same memory size. See Section 7 for more details.
Our techniques.
For Theorem 4.2 and Theorem 6.15, we use the concept of composable coresets,which has been employed in several distributed optimization models such as the streaming andMapReduce models [1, 8, 9, 10, 37, 47]. Roughly speaking, the main idea behind this techniqueis as follows: first partition the data into smaller parts. Then compute a representative solution,referred to as a coreset, from each part. Finally, obtain a solution by solving the optimizationproblem over the union of coresets for all parts. We use a randomized variant of composablecoresets, first introduced in [46], where the above idea is applied on a random clustering of thedata. This randomized variant has been used after for Graph Matching and Vertex Cover [6, 7], aswell as Column Subset Selection [18]. Our algorithm for Theorem 4.2 is similar to previous works,but the analysis requires new techniques for handling hypergraphs. Theorem 5.1 is a relativelystraightforward generalization of the corresponding result for matching in graphs.The majority of our technical innovation is contained in Theorem 6.4 and 6.15. Our generalapproach is to construct, in parallel, an HEDCS that will contain a good approximation of themaximum matching in the original graph and that will fit on one machine. Then we can run anapproximation algorithm on the resultant HEDCS to come up with a good approximation to themaximum matching in this HEDCS and hence in the original graph. In order to make this approachwork, we need to generalize much of the known EDCS machinery [16, 17] to hypergraphs. Thisendeavor is quite involved, as almost all the proofs do not generalize easily and, as the resultsshow, the resulting bounds are weaker than those for graphs. We first show that HEDCSs exist,and that they contain large fractional matchings. We then show that they are useful as a coreset,which amounts to showing that even though there can be many different HEDCSs of some fixedhypergraph G ( V, E ), the degree distributions of every HEDCS (for the same parameters) are almostidentical. In other words, the degree of any vertex v is almost the same in every HEDCS of G . Weshow also that HEDCS are robust under edge sampling, in the sense that edge sampling from aHEDCS yields another HEDCS. These properties allow to use HEDCS in our coresets and parallelalgorithm in the rest of the paper. 4 Related Work
Hypergraph Matching.
The problem of finding a maximum matching in d -uniform hypergraphsis NP-hard for any d ≥ d = 3 [39]. The most natural approach for anapproximation algorithm for the hypergraph matching problem is the greedy algorithm: repeatedlyadd an edge that doesn’t intersect any edges already in the matching. This solution is clearlywithin a factor of d from optimal: from the edges removed in each iteration, the optimal solutioncan contain at most d edges (at most one for each element of the chosen edge). It is also easy toconstruct examples showing that this analysis is tight for the greedy approach. All the best knownapproximation algorithms for the Hypergraph Matching Problem in d -uniform hypergraphs arebased on local search methods [13, 15, 20, 30, 36]. The first such result by Hurkens and Schrijver[36] gave a ( d + (cid:15) )-approximation algorithm. Halldorsson [30] presented a quasi-polynomial( d +23 )-approximation algorithm for the unweighted d -UHM. Sviridenko and Ward [54] establisheda polynomial time ( d +23 )-approximation algorithm that is the first polynomial time improvementover the ( d + (cid:15) ) result from [36]. Cygan [21] and Furer and Yu [26] both provide a ( d +13 + (cid:15) )polynomial time approximation algorithm, which is the best approximation guarantee known sofar. On the other hand, Hazan, Safra and Schwartz [35] proved that it is hard to approximate d -UHM problem within a factor of Ω( d/ log d ). Matching in parallel computation models.
The study of the graph maximum matching prob-lem in parallel computation models can be traced back to PRAM algorithms of 1980s [3, 38, 45].Since then it has been studied in the LOCAL and MPC models, and (1 + (cid:15) )- approximation can beachieved in O (log log n ) rounds using a space of O ( n ) [6, 22, 27]. The question of finding a maximalmatching in a small number of rounds has also been considered in [28, 44]. Recently, Behnezhad et al . [12] presented a O (log log ∆) round algorithm for maximal matching with O ( n ) memoryper machine. While we are not aware of previous work on hypergraph matching in the MPCmodel, finding a maximal hypergraph matching has been considered in the LOCAL model. BothFischer et al . [24] and Harris [33] provide a deterministic distributed algorithm that computes O ( d )-approximation to the d -UHM. Maximum Independent Set.
The d -UHM is strongly related to different variants of the In-dependent Set problem as well as other combinatorial problems. The maximum independent set(MIS) problem on degree bounded graphs can be mapped to d -UHM when the degree bound is d [14, 31, 32, 55]. The d -UHM problem can also be studied under a more general problem of maxi-mum independent set on ( d + 1)-claw-free graphs [13, 20]. (See [19] of connections between d -UHMand other combinatorial optimization problems). O ( d ) -approximation In this section, we generalize the randomized composable coreset algorithm of Assadi and Khanna[7]. They used a maximum matching as a coreset and obtained a O (1)-approximation. We usea hypergraph maximum matching and we obtain a O ( d )-approximation. We first define a k -partitioning and then present our greedy approach. Definition 4.1 (Random k -partitioning) . Let E be an edge-set of a hypergraph G ( V, E ) . We saythat a collection of edges E (1) , . . . , E ( k ) is a random k -partition of E if the sets are constructed byassigning each edge e ∈ E to some E ( i ) chosen uniformly at random. A random k -partition of E naturally results in partitioning the graph G into k subgraphs G (1) , . . . , G ( k ) where G ( i ) := G ( V, E ( i ) )5 or all i ∈ [ k ] . Let G ( V, E ) be any d -uniform hypergraph and G (1) , . . . , G ( k ) be a random k -partitioning of G .We describe a simple greedy process for combining the maximum matchings of G ( i ) , and prove thatthis process results in a O ( d )-approximation of the maximum matching of G . Algorithm 1:
Greedy Construct a random k -partitioning of G across the k machines. Let M (0) := ∅ . for i = 1 to k do Set
M M ( G ( i ) ) to be an arbitrary hypergraph maximum matching of G ( i ) . Let M ( i ) be a maximal matching obtained by adding to M ( i − the edges of M M ( G ( i )that do not violate the matching property. end return M := M ( k ) . Theorem 4.2.
Greedy computes, with high probability, a (cid:0) d ( d −
1) + 3 + o (1) (cid:1) -approximation forthe d -UHM problem in 3 M P C rounds on machines of memory s = ˜ O ( √ nm ) . In the rest of the section, we present the proof of Theorem 4.2. Let c = d ( d − , we show that M ( k ) ≥ c · µ ( G ) w.h.p, where M ( k ) is the output of Greedy . The randomness stems from the factthat the matchings M ( i ) (for i ∈ { , . . . , k } ) constructed by Greedy are random variables dependingon the random k -partitioning. We adapt the general approach in [7] for d -uniform hypergraphs,with d ≥
3. Suppose at the beginning of the i -th step of Greedy , the matching M ( i − is of size o ( µ ( G ) /d ). One can see that in this case, there is a matching of size Ω( µ ( G )) in G that is entirelyincident on vertices of G that are not matched by M ( i − . We can show that in fact Ω( µ ( G ) / ( d k ))edges of this matching are appearing in G ( i ) , even when we condition on the assignment of the edgesin the first ( i −
1) graphs. Next we argue that the existence of these edges forces any maximummatching of G ( i ) to match Ω( µ ( G ) / ( d k )) edges in G ( i ) between the vertices that are not matchedby M ( i − . These edges can always be added to the matching M ( i − to form M ( i ) . Therefore,while the maximal matching in Greedy is of size o ( µ ( G )), we can increase its size by Ω( µ ( G ) / ( d k ))edges in each of the first k/ µ ( G ) /d ) at the end. Thefollowing Lemma 4.3 formalizes this argument. Lemma 4.3.
For any i ∈ [ k/ , if M ( i − ≤ c · µ ( G ) , then, w.p. − O (1 /n ) , M ( i ) ≥ M ( i − + 1 − d ( d − c − o (1) k · µ ( G ) . Before proving the lemma, we define some notation. Let M ∗ be an arbitrary maximum matchingof G . For any i ∈ [ k ], we define M ∗
W.p. − O (1 /n ) , for any i ∈ [ k ] : M ∗
Fix an i ∈ [ k ]; each edge in M ∗ is assigned to G (1) , . . . , G ( i − , w.p. ( i − /k ,hence in expectation, size of M ∗
3] and the set of edges for E (1) , . . . E ( i − . This also fixesthe matching M ( i − while the set of edges in E ( i ) , . . . , E ( k ) together with the matching M ( i ) are still random variables. We further condition on the event that after fixing the edges in E (1) , . . . , E ( i − , | M ∗
There exists a matching in G ( V, E new ) of size (cid:16) k − i +1 − o ( i ) k − d ( d − c (cid:17) · µ ( G ) thatavoids the vertices of V new ( M old ) .Proof of Claim 4.3. By the assumption that | M ∗
1) + 3), we ensure that − d ( d − c = c and in either case, the matching computed by Greedy is of size at least µ ( G ) / (3 d ( d −
1) + 3) − o ( µ ( G )), and this proves that Greedy is a O ( d )-approximation. All is left is to prove that Greedy can be implemented in 3 rounds with a memory of˜ O ( √ nm ) per machine. Let k = (cid:112) mn be the number of machines, each with a memory of ˜ O ( √ nm ).We claim that Greedy can run in three rounds. In the first round, each machine randomly partitionsthe edges assigned to it across the k machines. This results in a random k -partitioning of the graphacross the machines. In the second round, each machine sends a maximum matching of its inputto a designated central machine M ; as there are k = (cid:112) mn machines and each machine is sending˜ O ( n ) size coreset, the input received by M is of size ˜ O ( √ nm ) and hence can be stored entirely onthat machine. Finally, M computes the answer by combining the matchings.We conclude this section by stating that computing a maximum matching on every machineis only required for the analysis, i.e., to show that there exists a large matching in the union ofcoresets. In practice, we can use an approximation algorithm to obtain a large matching from thecoresets. In our experiments, we use a maximal matching instead of maximum. O (log n ) -rounds d -approximation algorithm In this section, we show how to compute a maximal matching in O (log n ) MPC rounds, by gener-alizing the algorithm in [44] to d -uniform hypergraphs. The algorithm provided by Lattanzi el al. [44] computes a maximal matching in graphs in O (log n ) if s = Θ( n ), and in at most (cid:98) c/(cid:15) (cid:99) iterationswhen s = Θ( n (cid:15) ), where 0 < (cid:15) < c is a fixed constant. Harvey et al. [34] show a similar result.The algorithm first samples O ( s ) edges and finds a maximal matching M on the resultingsubgraph (we will specify a bound on the memory s later). Given this matching, we can safelyremove edges that are in conflict (i.e. those incident on nodes in M ) from the original graph G . Ifthe resulting filtered graph H is small enough to fit onto a single machine, the algorithm augments M with a matching found on H . Otherwise, we augment M with the matching found by recursingon H . Note that since the size of the graph reduces from round to round, the effective samplingprobability increases, resulting in a larger sample of the remaining graph. Theorem 5.1.
Given a d -uniform hypergraph G ( V, E ) , Iterated-Sampling omputes a maximalmatching in G with high probability in O (log n ) M P C rounds on machines of memory s = Θ( d · n ) . Next we present the proof of Theorem 5.1. We first show that after sampling edges, the size ofthe graph G [ I ] induced by unmatched vertices decreases exponentially with high probability, andtherefore the algorithm terminates in O (log n ) rounds. Next we argue that M is indeed a maximalmatching by showing that if after terminating, there is an edge that’s still unmatched, this willyield a contradiction. This is formalized in the following lemmas. Lemma 5.2.
Let E (cid:48) ⊂ E be a set of edges chosen independently with probability p . Then withprobability at least − e − n , for all I ⊂ V either | E [ I ] | < n/p or E [ I ] ∩ E (cid:48) (cid:54) = ∅ .Proof of Lemma 5.2. Fix one such subgraph, G [ I ] = ( I, E [ I ]) with | E [ I ] | ≥ n/p . The probabilitythat none of the edges in E [ I ] were chosen to be in E (cid:48) is (1 − p ) | E [ I ] | ≤ (1 − p ) n/p ≤ e − n . Since8 lgorithm 2: Iterated-Sampling Set M := ∅ and S = E . Sample every edge e ∈ S uniformly with probability p = s |S| d to form E (cid:48) . If | E (cid:48) | > s the algorithm fails. Otherwise give the graph G ( V, E (cid:48) ) as input to a singlemachine and compute a maximal matching M (cid:48) on it. Set M = M ∪ M (cid:48) . Let I be the unmatched vertices in G and G [ I ] the induced subgraph with edges E [ I ]. If E [ I ] > s , set S := E [ I ] and return to step 2. Compute a maximal matching M (cid:48)(cid:48) on G [ I ] and output M = M ∪ M (cid:48)(cid:48) .there are at most 2 n total possible induced subgraphs G [ I ] (because each vertex is either matchedor unmatched), the probability that there exists one that does not have an edge in E (cid:48) is at most2 n e − n ≤ e − n . Lemma 5.3. If s ≥ d · n then Iterated-Sampling runs for at most O (log n ) iterations with highprobability.Proof of Lemma 5.3. Fix an iteration i of Iterated-Sampling and let p be the sampling probabilityfor this iteration. Let E i be the set of edges at the beginning of this iteration, and denote by I bethe set of unmatched vertices after this iteration. From Lemma 5.2, if | E [ I ] | ≥ n/p then an edgeof E [ I ] will be sampled with high probability. Note that no edge in E [ I ] is incident on any edge in M (cid:48) . Thus, if an edge from E [ I ] is sampled then Iterated-Sampling would have chosen this edge tobe in the matching. This contradicts the fact that no vertex in I is matched. Hence, | E [ I ] | ≤ n/p with high probability.Now consider the first iteration of the algorithm, let G ( V , E ) be the induced graph on theunmatched nodes after the first step of the algorithm. The above argument implies that | E | ≤ d · n | E | s ≤ d · n | E | s ≤ | E | . Similarly | E | ≤ d · n | E | s ≤ (10 d · n ) | E | s ≤ | E | . After i iterations : | E i | ≤ | E | i , and the algorithm will terminate after O (log n ) iterations. Lemma 5.4.
Iterated-Sampling finds a maximal matching of G with high probability.Proof of Lemma 5.4. First consider the case that the algorithm does not fail. Suppose there is anedge e = { v , . . . , v d } ∈ E such that none of the v i ’s are matched in the final matching M thatthe algorithm output. In the last iteration of the algorithm, since e ∈ E and its endpoints are notmatched, then e ∈ E [ I ]. Since this is the last run of the algorithm, a maximal matching M (cid:48)(cid:48) of G [ I ] is computed on one machine. Since M (cid:48)(cid:48) is maximal, at least one of the v i ’s must be matchedin it. All of the edges of M (cid:48)(cid:48) get added to M in the last step. This yields a contradiction.Next, consider the case that the algorithm fails. This occurs due to the set of edges E (cid:48) having sizelarger than the memory in some iteration of the algorithm. Note that E [ | E (cid:48) | ] = | S |· p = s/ d ≤ s/ | E (cid:48) | ≥ s with probability smaller than2 s ≥ − d · n (since s ≥ d · n ). By Lemma 5.3 the algorithm completes in at most O (log n ) rounds,thus the total failure probability is bounded by O (log n · − · n ) using the union bound.We are now ready to show that Iterated-Sampling can be implemented in O (log n ) MPC rounds. Proof of Theorem 5.1.
We show that
Iterated-Sampling can be implemented in the
M P C modelwith machines of memory Θ( dn ) and O (log n ) MPC rounds. Combining this with Lemma 5.4 onthe correctness of Iterated-Sampling , we immediately obtain Theorem 5.1 for the case of s = Θ( dn ).Every iteration of Iterated-Sampling can be implemented in O (1) MPC rounds. Suppose the edges9re initially distributed over all machines. We can sample every edge with the probability p (ineach machine) and then send the sampled edges E (cid:48) to the first machine. With high probability wewill have | E (cid:48) | = O ( n ), so we can fit the sampled edges in the first machine. Therefore, the samplingcan be done in one M P C round. Computing the subgraph of G induced by I can be done in 2rounds; one round to send the list of unmatched vertices I from the first machine to all the othermachines, and the other round to compute the subgraph G [ I ], and send it to a single machine if itfits, or start the sampling again. O ( d ) -approximation using HEDCSs In graphs, edge degree constrained subgraphs (EDCS) [16, 17] have been used as a local conditionfor identifying large matchings, and leading to good dynamic and parallel algorithms [6, 16, 17].These papers showed that an EDCS exists, that it contains a large matching, that it is robust tosampling and composition, and that it can be used as a coreset.In this section, we present a generalization of EDCS for hypergraphs. We prove that an HEDCSexists, that it contains a large matching, that it is robust to sampling and composition, and thatit can be used as a coreset. The proofs and algorithms, however, require significant developmentsbeyond the graph case. We first present definitions of a fractional matching and HEDCS.
Definition 6.1 (Fractional matching) . In a hypergraph G ( V, E ) , a fractional matching is a mappingfrom y : E (cid:55)→ [0 , such that for every edge e we have ≤ y e ≤ and for every vertex v : (cid:80) e ∈ v y e ≤ . The value of such fractional matching is equal to (cid:80) e ∈ E y e . For (cid:15) > , a fractionalmatching is an (cid:15) -restricted fractional matching if for every edge e : y e = 1 or y e ∈ [0 , (cid:15) ] . Definition 6.2.
For any hypergraph G ( V, E ) and integers β ≥ β − ≥ a hyperedge degree constraintsubgraph HEDCS ( H, β, β − ) is a subgraph H := ( V, E H ) of G satisfying: • (P1): For any hyperedge e ∈ E H : (cid:80) v ∈ e d H ( v ) ≤ β. • (P2): For any hyperedge e (cid:54)∈ E H : (cid:80) v ∈ e d H ( v ) ≥ β − . We show via a constructive proof that a hypergraph contains an HEDCS when the parametersof this HEDCS satisfy the inequality β − β − ≥ d − Lemma 6.3.
Any hypergraph G contains an HEDCS ( G, β, β − ) for any parameters β − β − ≥ d − .Proof. Consider the following simple procedure for creating an HEDCS H of a given hypergraph G : start by initializing H to be equal to G . And while H is not an HEDCS( G, β, β − ), find an edge e which violates one of the properties of HEDCS and fix it. Fixing the edge e implies removing itfrom H if it was violating Property (P1) and adding it to H if it was violating Property (P2) . Theoutput of the above procedure is clearly an HEDCS of graph G . However, a-priori it is not clearthat this procedure ever terminates as fixing one edge e can result in many edges violating theHEDCS properties, potentially undoing the previous changes. In the following, we use a potentialfunction argument to show that this procedure always terminates after a finite number of steps,hence implying that a HEDCS ( G, β, β − ) always exists.We define the following potential function Φ:Φ := ( 2 d β − d − d ) · (cid:88) v ∈ V d H ( v ) − (cid:88) e ∈ H (cid:88) u ∈ e d H ( u )10e argue that in any step of the procedure above, the value of Φ increases by at least 1.Since the maximum value of Φ is at most O ( d n · β ), this immediately implies that this procedureterminates in O ( d n · β ) iterations.Define Φ = ( d β − d − d ) · (cid:80) v ∈ V d H ( v ) and Φ = (cid:80) e ∈ H (cid:80) u ∈ e d H ( u ). Let e be the edge we choose tofix at this step, H b be the subgraph before fixing the edge e , and H a be the resulting subgraph.Suppose first that the edge e was violating Property (P1) of HEDCS. As the only change is in thedegrees of vertices v ∈ e , Φ decreases by (2 β − ( d − (cid:80) v ∈ e d H b ( v ) ≥ β + 1originally (as e was violating Property (P1) of HEDCS), and hence after removing e , Φ increasesby β + 1. Additionally, for each edge e u incident upon u ∈ e in H a , after removing the edge e , (cid:80) v ∈ e u d H a ( v ) decreases by one. As there are at least (cid:80) u ∈ e d H a ( u ) = (cid:80) u ∈ e d H b ( u ) − d ≥ β − ( d −
1) choicesfor e u , this means that in total, Φ increases by at least 2 β + 1 − ( d − e .Now suppose that the edge e was violating Property (P2) of HEDCS instead. In this case,degree of vertices u ∈ e all increase by one, hence Φ increases by 2 β − ( d − e was violating Property (P2) we have (cid:80) v ∈ e d H b ( v ) ≤ β − −
1, so the addition of edge e decreases Φ by at most (cid:80) v ∈ e d H a ( v ) = (cid:80) v ∈ e d H b ( v ) + d ≤ β − − d . Moreover, for each edge e u incident upon u ∈ e , after adding the edge e , (cid:80) v ∈ e u d H a ( v ) increases by one and since there areat most (cid:80) v ∈ e d H b ( v ) ≤ β − − e u , Φ decreases in total by at most 2 β − − d . Thetotal variation in Φ is therefore equal to 2 β − ( d − − (2 β − − d ) = 3 + 2( β − β − − d ) . So if β − β − ≥ d −
1, we have that Φ increases by at least 1 after fixing edge e .The main result in this section shows that an HEDCS of a graph contains a large (cid:15) -restrictedfractional matching that approximates the maximum matching by a factor less than d . Theorem 6.4.
Let G be a d -uniform hypergraph and ≤ (cid:15) < . Let H := HEDCS ( G, β, β (1 − λ )) with λ = (cid:15) and β ≥ d d − · λ − . Then H contains an (cid:15) -restricted fractional matching M Hf with totalvalue at least µ ( G )( dd − d +1 − (cid:15)d − ) . In order to prove Theorem 6.4, we will need the following two lemmas. The first lemma weprove is an algebraic result that will help us bound the contribution of vertices in the (cid:15) -restrictedfractional matching. The second lemma identifies additional structure on the HEDCS, that wewill use to construct the fractional matching of the theorem. For proofs of both lemmas, seeAppendix A.2
Lemma 6.5.
Let φ ( x ) = min { , ( d − xd ( β − x ) } . If a , . . . a d ≥ and a + . . . + a d ≥ β (1 − λ ) for some λ ≥ , then d (cid:80) i =1 φ ( a i ) ≥ − · λ . Lemma 6.6.
Given any
HEDCS ( G, β, β (1 − λ )) H , we can find two disjoint sets of vertices X and Y that satisfy the following properties:1. | X | + | Y | = d · µ ( G ) .2. There is a perfect matching in Y using edges in H . . Letting σ = | Y | d + (cid:80) x ∈ X φ ( d H ( x )) ,we have that σ ≥ µ ( G )(1 − λ ) .4. All edges in H with vertices in X have at least one other vertex in Y , and have vertices onlyin X and Y . We are now ready to prove Theorem 6.4.
Proof of theorem 6.4.
Suppose we have two sets X and Y satisfying the properties of Lemma 6.6.We construct an (cid:15) -restricted fractional matching M Hf using the edges in H such that val ( M Hf ) ≥ ( dd − d + 1 − (cid:15)d − µ ( G ) , where val ( M Hf ) is the value of the fractional matching M Hf . Now, by Property 2 of Lemma 6.6, | Y | contains a perfect matching M HY using edges in H . Let Y − be a subset of Y obtained by randomlysampling exactly 1 /d edges of M HY and adding their endpoints to Y − Let Y ∗ = Y \ Y − and observethat | Y − | = | Y | /d and | Y ∗ | = d − d | Y | .Let H ∗ be the subgraph of H induced by X ∪ Y ∗ (each edge in H ∗ has vertices in only X and Y ∗ ). We define a fractional matching M H ∗ f on the edges of H ∗ in which all edges have value at most (cid:15) . We will then let our final fractional matching M Hf be the fractional matching M H ∗ f joined withthe perfect matching in H of Y − (so M Hf assigns value 1 to the edges in this perfect matching). M Hf is, by definition, an (cid:15) -restricted fractional matching.We now give the details for the construction of M H ∗ f . Let V ∗ = X ∪ Y ∗ be the vertices of H ∗ ,and let E ∗ be its edges. For any vertex v ∈ V ∗ , define d ∗ H ( v ) to be the degree of v in H ∗ . Recallthat by Property 4 of Lemma 6.6, if x ∈ X then all the edges of H incident to x go to Y (but somemight go to Y − ). Thus, for x ∈ X , we have E [ d ∗ H ( x )] ≥ d H ( x )( d − d .We now define M H ∗ f as follows. For every x ∈ X , we arbitrarily order the edges of H incident to x , and then we assign/add a value of min (cid:110) (cid:15) | X ∩ e | , | X ∩ e | β − d H ( x ) (cid:111) to these edges one by one, stoppingwhen either val ( x ) (the sum of values assigned to vertices incident to x ) reaches 1 or there are nomore edges in H incident to x , whichever comes first. In the case that val ( x ) reaches 1 the lastedge might have added value less than min (cid:110) (cid:15) | X ∩ e | , | X ∩ e | β − d H ( x ) (cid:111) , where e is the last edge to beconsidered.We now verify that M H ∗ f is a valid fractional matching in that all vertices have value at most1. This is clearly true of vertices x ∈ X by construction. For a vertex y ∈ Y ∗ , it suffices toshow that each edge incident to y receives a value of at most 1 /d H ( y ) ≤ /d ∗ H ( y ). To see this,first note that the only edges to which M H ∗ f assigns non-zero values have at least two endpointsin X × Y ∗ . Any such edge e receives value at most min (cid:8) (cid:15), (cid:80) x ∈ X ∩ e | X ∩ e | β − d H ( x ) (cid:9) , but since e isin M H ∗ f and so in H , we have by Property (P1) of an HEDCS that d H ( y ) ≤ β − d H ( x ), and so (cid:80) x ∈ X ∩ e | X ∩ e | β − d H ( x ) ≤ | X ∩ e | | X ∩ e | d H ( y ) ≤ d H ( y ) .By construction, for any x ∈ X , we have that the value val ( x ) of x in M H ∗ f satisfies : val ( x ) = min (cid:40) , (cid:88) e (cid:51) x min (cid:26) (cid:15) | X ∩ e | , | X ∩ e | β − d H ( x ) (cid:27)(cid:41) ≥ min (cid:26) , d ∗ H ( x ) · min (cid:26) (cid:15)d − , d − β − d H ( x ) (cid:27)(cid:27) . M H ∗ f : as val ( M H ∗ f ) ≥ (cid:80) x ∈ X val ( x ) . For convenience, we use val (cid:48) ( x ) = min (cid:110) , d ∗ H ( x ) · min (cid:110) (cid:15), β − d H ( x ) (cid:111)(cid:111) such that val ( x ) ≥ val (cid:48) ( x ) d − val ( M H ∗ f ) ≥ d − (cid:88) x ∈ X val (cid:48) ( x ) , (2)Next we present a lemma, proved in Appendix A.2 that bounds val (cid:48) ( x ) for each vertex. Lemma 6.7.
For any x ∈ X , E [ val (cid:48) ( x )] ≥ (1 − λ ) φ ( d H ( x )) . This last lemma, combined with (2), allows us to lower bound the value of M H ∗ f : val ( M H ∗ f ) ≥ d − (cid:88) x ∈ X val (cid:48) ( x ) ≥ − λd − (cid:88) x ∈ X φ ( d H ( x )) . Note that we have constructed M Hf by taking the fractional value in M H ∗ f and adding theperfect matching on edges from Y − . The latter matching has size | Y − | d = | Y | d , and the value of M Hf is bounded by: val ( M Hf ) ≥ d − − λ ) (cid:88) x ∈ X φ ( d H ( x )) + | Y | d = 1 d − (cid:0) (1 − λ ) (cid:88) x ∈ X φ ( d H ( x )) + | Y | d (cid:1) − | Y | d ( d − ≥ d − − λ )(1 − λ ) µ ( G ) − | Y | d ( d − ≥ d − − λ ) µ ( G ) − | Y | d ( d − . To complete the proof, recall that Y contains a perfect matching in H of | Y | /d edges, so if | Y | d ≥ dd ( d − µ ( G ) then there already exists a matching in H of size at least dd ( d − µ ( G ), andthe theorem is true. We can thus assume that | Y | /d < (cid:0) dd ( d − (cid:1) µ ( G ), in which case the previousequation yields that: val ( M Hf ) ≥ d − − λ ) µ ( G ) − | Y | d ( d − ≥ ( 1 − λd − µ ( G ) − µ ( G )( d − d ( d −
1) + 1)= (cid:0) dd ( d −
1) + 1 − λd − (cid:1) µ ( G ) . In both cases we get that val ( M Hf ) ≥ ( dd − d + 1 − λd − µ ( G ) . .1 Sampling and constructing an HEDCS in the MPC model Our results in this section are general and applicable to every computation model. We provestructural properties about the HEDCSs that will help us construct them in the MPC model. Weshow that the degree distributions of every HEDCS (for the same parameters β and λ ) are almostidentical. In other words, the degree of any vertex v is almost the same in every HEDCS of thesame hypergraph G . We show also that HEDCS are robust under edge sampling, i.e. that edgesampling from and HEDCS yields another HEDCS, and that the degree distributions of any twoHEDCS for two different edge sampled subgraphs of G is almost the same no matter how the twoHEDCS are selected. In the following lemma, we argue that any two HEDCS of a graph G (forthe same parameters β , β − ) are “somehow identical” in that their degree distributions are close toeach other. In the rest of this section, we fix the parameters β , β − and two subgraphs A and B that are both HEDCS( G, β, β − ). Lemma 6.8. (Degree Distribution Lemma). Fix a d -uniform hypergraph G ( V, E ) and parameters β , β − = (1 − λ ) · β (for λ small enough). For any two subgraphs A and B that are HEDCS ( G, β, β − ) ,and any vertex v ∈ V , then | d A ( v ) − d B ( v ) | = O ( √ n ) λ / β .Proof. Suppose that we have d A ( v ) = kλβ for some k and that d B ( v ) = 0. We will show that ifthe k = Ω( √ nλ / ), then this will lead to a contradiction. Let e be one of the kλβ edges that areincident to v in A . e (cid:54)∈ B so (cid:80) u (cid:54) = v d B ( u ) ≥ (1 − λ ) β . From these (1 − λ ) β edges, at most (1 − kλ ) β can be in A in order to respect (P1) , so at least we will have ( k − λβ edges in B \ A , thus wehave now covered kλβ + ( k − λβ edges in both A and B . Let’s keep focusing on the edge e , andespecially on one of its ( k − λβ incident edges in B \ A . Let e be such an edge. e ∈ B \ A ,therefore (cid:80) v (cid:48) ∈ e d A ( v (cid:48) ) ≥ (1 − λ ) β . The edges incident to e in A that we have covered so far areat most (1 − kλ ) β , therefore we still need at least ( k − λ new edges in A to respect (P1) . Outof these ( k − λ edges, at most λβ can be in B (because e has already (1 − λ ) β covered edgesincident to it in B ). Therefore at least ( k − λβ are in A \ B . Thus, we have so far covered atleast kλβ + ( k − λβ + ( k − λβ . One can see that we can keep doing this until we cover at least k ( k +1)2 λβ edges in both A and B . The number of edges in each of A and B cannot exceed n · β (each vertex has degree ≤ β ), therefore we will get a contradiction if k ( k +1)2 λβ > nβ , which holdsif k > √ nλ / . Therefore k ≤ √ nλ / , and d A ( v ) = kλβ ≤ √ nλ / β. The next corollary shows that if the hypergraph is linear (every two hyperedges intersect in atmost on vertex), then the degree distribution is closer. The proof is in Appendix A.3.
Corollary 6.9.
For d -uniform linear hypergraphs, the degree distribution is tighter, and | d A ( v ) − d B ( v ) | = O (log n ) λβ. Next we prove two lemmas regarding the structure of different HEDCSs across sampled sub-graphs. The first lemma shows that edge sampling an HEDCS results in another HEDCS for thesampled subgraph. The second lemma shows that the degree distributions of any two HEDCS fortwo different edge sampled subgraphs of G is almost the same. Lemma 6.10.
Let H be a HEDCS( G, β H , β − H ) for parameters β H := (1 − λα ) · βp , β − H := β H − ( d − and β ≥ d ( αd ) · λ − · log n such that p < − α . Suppose G p := G Ep ( V, E p ) is an edge sampledsubgraph of G and H p := H ∩ G p ; then, with high probability: . For any vertex v ∈ V : | d H p ( v ) − p · d H ( v ) | ≤ λαd β H p is a HEDCS of G p with parameters ( β, (1 − λ ) · β ) .Proof. Let α (cid:48) := αd . For any vertex v ∈ V , E [ d H p ( v )] = p · d H ( v ) and d H ( v ) ≤ β H by Property (P1) of HEDCS H . Moreover, since each edge incident upon v in H is sampled in H p independently,by the Chernoff bound: P (cid:16) | d H p ( v ) − p · d H ( v ) | ≥ λα (cid:48) β (cid:17) ≤ · exp( − λ β · α (cid:48) ) ≤ n d . In the following, we condition on the event that: | d H p ( v ) − p · d H ( v ) | ≤ λα (cid:48) β . This event happens with probability at least 1 − n d − by above equation and a union bound on | V | = n vertices. This finalizes the proof of the first part of the claim. We are now ready to provethat H p is indeed am HEDCS( G p , β, (1 − λ ) · β ) conditioned on this event. Consider any edge e ∈ H p . Since H p ⊂ H , e ∈ H as well. Hence, we have, (cid:88) v ∈ e d H p ( v ) ≤ p · β H + dλα (cid:48) β = (1 − λα + dλα (cid:48) ) β = β , because αα (cid:48) = d , where the inequality is by Property (P1) of HEDCS H and the equality is by thechoice of β H . As a result, H p satisfies Property (P1) of HEDCS for parameter β . Now consider anedge e ∈ G p \ H p . Since H p = G p ∩ H , e (cid:54)∈ H as well. Hence, (cid:88) v ∈ e d H p ( v ) ≥ p · β − H − dλα (cid:48) β = (1 − λα − dλα (cid:48) ) β − p · ( d − − λα ) β − p · ( d − > (1 − λ ) · β . Lemma 6.11. (HEDCS in Edge Sampled Subgraph). Fix any hypergraph G ( V, E ) and p ∈ (0 , .Let G and G be two edge sampled subgraphs of G with probability p (chosen not necessarilyindependently). Let H and H be arbitrary HEDCSs of G and G with parameters ( β, (1 − λ ) · β ) .Suppose β ≥ d ( αd ) · λ − · log n , then, with probability − n d − , simultaneously for all v ∈ V : | d H ( v ) − d H ( v ) | = O (cid:0) n / (cid:1) λ / β .Proof. Let H be an HEDCS(G, β H , β − H ) for the parameters β H and β − H as defined in the previouslemma. The existence of H follows since β H − ( d − ≥ β − H . Define ˆ H := H ∩ G and ˆ H := H ∩ G .By Lemma 6.10, ˆ H (resp. ˆ H ) is an HEDCS of G (resp. G ) with parameters ( β, (1 − λ ) β ) withprobability 1 − n d − . In the following, we condition on this event. By Lemma 6.8 (DegreeDistribution Lemma), since both H (resp. H ) and ˆ H (resp. ˆ H ) are HEDCSs for G (resp. G ),the degree of vertices in both of them should be “close” to each other. Moreover, since by Lemma6.10 the degree of each vertex in ˆ H and ˆ H is close to p times its degree in H , we can argue thatthe vertex degrees in H and H are close. Formally, for any v ∈ V , we have15 d H ( v ) − d H ( v ) | ≤ | d H ( v ) − d ˆ H ( v ) | + | d ˆ H ( v ) − d ˆ H ( v ) | + | d ˆ H ( v ) − d H ( v ) |≤ O (cid:0) n / (cid:1) λ / β + | d ˆ H ( v ) − pd H ( v ) | + | d ˆ H ( v ) − pd H ( v ) |≤ O (cid:0) n / (cid:1) λ / β + O (1) · λ · β . Corollary 6.12. If G is linear, then | d H ( v ) − d H ( v ) | = O (cid:0) log n (cid:1) λβ . We are now ready to present a parallel algorithm that will use the HEDCS subgraph. We firstcompute an HEDCS in parallel via edge sampling. Let G (1) , . . . , G ( k ) be a random k -partition of agraph G . We show that if we compute an arbitrary HEDCS of each graph G ( i ) (with no coordinationacross different graphs) and combine them together, we obtain a HEDCS for the original graph G .We then store this HEDCS in one machine and compute a maximal matching on it. We presentour algorithm for all range of memory s = n Ω(1) . Lemma 6.13 and Corollary 6.14 serve as a proofto Theorem 6.15.
Algorithm 3:
HEDCS-Matching ( G, s ): a parallel algorithm to compute a O ( d )-approximation matching on a d -uniform hypergraph G with m edges on machines of memory O ( s ) Define k := ms log n , λ := n log n and β := 500 · d · n · log n . G (1) , . . . , G ( k ) := random k -partition of G . for i = 1 to k , in parallel do Compute C ( i ) = HEDCS ( G ( i ) , β, (1 − λ ) · β ) on machine i . end Define the multi-graph C ( V, E C ) with E C := ∪ ki =1 C ( i ) . Compute and output a maximal matching on C . Lemma 6.13.
Suppose k ≤ √ m . Then with high probability1. The subgraph C is an HEDCS ( G, β C , β − C ) for parameters: λ C = O (cid:0) n / (cid:1) λ / , β C = (1 + d · λ C ) · k · β and β − C = (1 − λ − d · λ C ) · k · β .2. The total number of edges in each subgraph G ( i ) of G is ˜ O ( s ) .3. If s = ˜ O ( n √ nm ) , then the graph C can fit in the memory of one machine.Proof of Lemma 6.13.
1. Recall that each graph G ( i ) is an edge sampled subgraph of G with sampling probability p = k . By Lemma 6.11 for graphs G ( i ) and G ( j ) (for i (cid:54) = j ∈ [ k ]) and their HEDCSs C ( i ) and C ( j ) , with probability 1 − n d − , for all vertices v ∈ V : | d C ( i ) ( v ) − d C ( j ) ( v ) | ≤ O (cid:0) n / (cid:1) λ / β . By taking a union bound on all (cid:0) k (cid:1) ≤ n d pairs of subgraphs G ( i ) and G ( j ) for i (cid:54) = j ∈ [ k ], theabove property holds for all i, j ∈ [ k ], with probability at least 1 − n d − . In the following, wecondition on this event. 16e now prove that C is indeed a HEDCS ( G, β C , β − C ). First, consider an edge e ∈ C and let j ∈ [ k ] be such that e ∈ C ( j ) as well. We have (cid:88) v ∈ e d C ( v ) = (cid:88) v ∈ e k (cid:88) i =1 d C ( i ) ( v ) ≤ k · (cid:88) v ∈ e d C ( j ) ( v ) + d · k · λ C · β ≤ k · β + d · k · λ C · β = β C . Hence, C satisfies Property (P1) of HEDCS for parameter β C . Now consider an edge e ∈ G \ C and let j ∈ [ k ] be such that e ∈ G ( j ) \ C ( j ) (recall that each edge in G is sent to exactly onegraph G ( j ) in the random k -partition). We have, (cid:88) v ∈ e d C ( v ) = (cid:88) v ∈ e k (cid:88) i =1 d C ( i ) ( v ) ≥ k · (cid:88) v ∈ e d C ( j ) − d · kλ C β ≥ k · (1 − λ ) · β − d · kλ C β .
2. Let E ( i ) be the edges of G ( i ) . By the independent sampling of edges in an edge sampledsubgraph, we have that E (cid:2) | E ( i ) | (cid:3) = mk = ˜ O ( s ). By Chernoff bound, with probability 1 − k · n ,the size of E ( i ) is ˜ O ( s ) . We can then take a union bound on all k machines in G ( i ) and havethat with probability 1 − /n , each graph G ( i ) is of size ˜ O ( s ).3. The number of edges in C is bounded by n · β c = O ( n · k · β ) = ˜ O ( n ms ) = ˜ O ( s ). Corollary 6.14. If G is linear, then by choosing λ :=
12 log n and β := 500 · d · log n in HEDCS-Matching we have:1. With high probability, the subgraph C is a HEDCS ( G, β C , β − C ) for parameters: λ C = O (log n ) λ , β C = (1 + d · λ C ) · k · β and β − C = (1 − λ − d · λ C ) · k · β .2. If s = ˜ O ( √ nm ) then C can fit on the memory of one machine.Proof of Corollary 6.14.
1. Similarly to Lemma 6.13 and by using corollary 6.12, we know that for graphs G ( i ) and G ( j ) (for i (cid:54) = j ∈ [ k ]) and their HEDCSs C ( i ) and C ( j ) , with high probability , for all vertices v ∈ V : | d C ( i ) ( v ) − d C ( j ) ( v ) | ≤ O (cid:0) log n (cid:1) λβ. By taking a union bound on all (cid:0) k (cid:1) ≤ n d pairs of subgraphs G ( i ) and G ( j ) for i (cid:54) = j ∈ [ k ], theabove property holds for all i, j ∈ [ k ], with probability at least 1 − n d − . In the following,we condition on this event. Showing that C is indeed a HEDCS ( G, β C , β − C ) follows by thesame analysis from the proof of Lemma 6.13.17. The number of edges in C is bounded by n · β c = O ( n · k · β ) = O ( n · k ) = ˜ O ( nms ) = ˜ O ( s ).The previous lemmas allow us to formulate the following theorem. Theorem 6.15.
HEDCS-Matching constructs a HEDCS of G in 3 M P C rounds on machines ofmemory s = ˜ O ( n √ nm ) in general and s = ˜ O ( √ nm ) for linear hypergraphs. Corollary 6.16.
HEDCS-Matching achieves a d ( d − /d ) -approximation to the d -UniformHypergraph Matching in 3 rounds with high probability.Proof of Corollary 6.16. We show that with high probability, C verifies the assumptions of theorem6.4. From Lemma 6.13, we get that with high probability, the subgraph C is a HEDCS ( G, β C , β − C )for parameters: λ C = O (cid:0) n / (cid:1) λ / , β C = (1 + d · λ C ) · k · β and β − C = (1 − λ − d · λ C ) · k · β. Wecan see that β c ≥ d d − · λ − c . Therefore by Theorem 6.4, C contains a ( d − d )-approximate (cid:15) -restricted matching. Since the integrality gap of the d -UHM is at most d − d (see [19] fordetails), then C contains a ( d − d ) -approximate matching. Taking a maximal matching in C multiplies the approximation factor by at most d . Therefore, any maximal matching in C is a d ( d − d ) -approximation. To understand the relative performance of the proposed algorithms, we conduct a wide variety ofexperiments on both random and real-life data [43, 51, 57]. We implement the three algorithms
Greedy , Iterated-Sampling and
HEDCS-Matching using Python, and more specifically relying on themodule pygraph and its class pygraph.hypergraph to construct and perform operations on hyper-graphs. We simulate the MPC model by computing a k -partitioning and splitting it into k differentinputs. Parallel computations on different parts of the k -partitioning are handled through the useof the multiprocessing library in Python. We compute the optimal matching through an IntegerProgram for small instances of random uniform hypergraphs ( d ≤
10) as well as geometric hy-pergraphs. The experiments were conducted on a 2.6 GHz Intel Core i7 processor and 16 GBRAM workstation. The datasets differ in their number of vertices, hyperedges, vertex degree andhyperedge cardinality. In the following tables, n and m denote the number of vertices and numberof hyperedges respectively, and d is the size of hyperedges. For Table 3, the graphs might havedifferent number of edges and ¯ m denotes the average number of edges. k is the number of machinesused to distribute the hypergraph initially. We limit the number of edges that a machine can storeto mk . In the columns Gr, IS and HEDCS, we store the average ratio between the size of thematching computed by the algorithms Greedy , Iterated-Sampling and
HEDCS-Matching respectively,and the size of a benchmark. This ratio is computed by the percentage
ALGBENCHMARK , where
ALG is the output of the algorithm, and
BEN CHM ARK denotes the size of the benchmark. Thebenchmarks include the size of optimal solution when it is possible to compute, or the size of amaximal matching computed via a sequential algorithm. I denotes the number of instances ofrandom graphs that we generated for fixed n , m and d . β and β − are the parameters used toconstruct the HEDCS subgraphs in HEDCS-Matching . These subgraphs are constructed using theprocedure in the proof of Lemma 6.3. 18 .1 Experiments with random hypergraphs
We perform experiments on two classes of d -uniform random hypergraphs. The first containsrandom uniform hypergraphs, and the second contains random geometric hypergraphs. Random Uniform Hypergraphs.
For a fixed n , m and d , each potential hyperedge is sampledindependently and uniformly at random from the set of vertices. In Table 2, we use the size ofa perfect matching nd as a benchmark, because a perfect matching in random graphs exists withprobability 1 − o (1) under some conditions on m, n and d . If d ( n, m ) = m · dn is the expected degreeof a random uniform hypergraph, Frieze and Janson [25] showed that d ( n,m ) n / → ∞ is a sufficientcondition for the hypergraph to have a perfect matching with high probability. Kim [42] furtherweakened this condition to d ( n,m ) n / (5+2 / ( d − → ∞ . We empirically verify, by solving the IP formulation,that for d = 3 , d = 10, our random graphs contain a perfect matching.In Table 2, the benchmark is the size of a perfect matching, while in Table 5, it is the size of agreedy maximal matching. In terms of solution quality (Tables 2 and 5) HEDCS-Matching performsconsistently better than
Greedy , and
Iterated-Sampling performs significantly better than the othertwo. None of the three algorithms are capable of finding perfect matchings for a significant numberof the runs. When compared to the size of a maximal matching,
Iterated-Sampling still performsbetter, followed by
HEDCS-Matching . However, the ratio is smaller when compared to a maximalmatching, which is explained by the deterioration of the quality of greedy maximal matching as n and d grow. Dufosse et al . [23] confirm that the approximation quality of a greedy maximalmatching on random graphs that contain a perfect matching degrades as a function of n and d .The performance of the algorithms decreases as d grows, which is theoretically expected sincetheir approximations ratio are both proportional to d . The number of rounds for Iterated-Sampling grows slowly with n , which is consistent with O (log n ) bound. Recall that the number of roundsfor the other two algorithms is constant and equal to 3. Geometric Random Hypergraphs.
The second class we experimented on is random geometrichypergraphs. The vertices of a random geometric hypergraph (RGH) are randomly sampled fromthe uniform distribution of the space [0 , . A set of d different vertices v , . . . , v d ∈ V forms ahyperedge if, and only if, the distance between any v i and v j is less than a previously specifiedparameter r ∈ (0 , r and n fully characterize a RGH. We fix d = 3 and generatedifferent geometric hypergraphs by varying n and r . We compare the output of the algorithmsto the optimal solution that we compute through the IP formulation. Table 3 shows that theperformance of our three algorithms is almost similar with Iterated-Sampling outperforming
Greedy and
HEDCS-Matching as the size of the graphs grows. We also observe that random geometrichypergraphs do not contain perfect matchings, mainly because of the existence of some outliervertices that do not belong to any edge. The number of rounds of
Iterated-Sampling still grows with n , confirming the theoretical bound and the results on random uniform hypergraphs.19 m d k β β − Rounds IS15 200 3 5 500 77.6% 86.6% 82.8% 5 3 3.830 400 5 78.9% 88.1% 80.3% 7 4 4.56100 3200 10 81.7% 93.4% 83.1% 5 2 5.08300 4000 10 78.8% 88.7% 80.3% 8 6 7.0550 800 5 6 500 66.0% 76.2% 67.0% 16 11 4.89100 2,800 10 68.0% 79.6% 69.8% 16 11 4.74300 4,000 10 62.2% 75.1% 65.5% 10 5 6.62500 8,000 16 63.3% 76.4% 65.6% 10 5 7.62500 15,000 10 16 500 44.9% 58.3% 53.9% 20 10 6.691,000 50,000 20 47.3% 61.3% 50.5% 20 10 8.252,500 100,000 20 45.6% 59.9% 48.2% 20 10 8.115,000 200,000 20 45.0% 59.7% 47.8% 20 10 7.891,000 50,000 25 25 100 27.5% 34.9% 30.8% 75 50 8.102,500 100,000 25 26.9% 34.0% 27.0% 75 50 8.265,000 250,000 30 26.7% 33.8% 28.8% 75 50 8.2310,000 500,000 30 26.6% 34.1% 28.2% 75 50 8.465,000 250,000 50 30 100 22.4% 30.9% 27.9% 100 50 10.2210,000 500,000 30 22.2% 31.0% 26.5% 100 50 10.1515,000 750,000 30 20.9% 30.8% 26.4% 100 50 10.2625,000 1,000,000 30 20.9% 30.8% 26.4% 100 50 10.29
Table 2: Comparison on random instances with perfect matching benchmark, of size nd . n r ¯ m d k β β − Rounds IS100 0.2 930.1 ±
323 3 5 100 88.3% 89.0% 89.6% 3 5 4.1100 0.25 1329.5 ±
445 10 88.0% 89.0% 89.5% 3 5 5.2250 0.15 13222 ± ± ± Table 3: Comparison on random geometric hypergraphs with optimal matching benchmark.
PubMed and Cora Datasets.
We employ two citation network datasets, namely the Coraand Pubmed datasets [51, 57]. These datasets are represented with a graph, with vertices beingpublications, and edges being a citation links from one article to another. We construct thehypergraphs in two steps 1) each article is a vertex; 2) each article is taken as a centroid andforms a hyperedge to connect those articles which have citation links to it (either citing it orbeing cited). The Cora hypergraph has an average edge size of 3 . ± .
1, while the average in thePubmed hypergraph is 4 . ± .
7. The number of edges in both is significantly smaller than thenumber of vertices, therefore we allow each machine to store only mk + mk edges. We randomlysplit the edges on each machine, and because the subgraphs are small, we are able to computethe optimal matchings on each machine, as well as on the whole hypergraphs. We perform tenruns of each algorithm with different random k -partitioning and take the maximum cardinalityobtained. Table 4 shows that none of the algorithms is able to retrieve the optimal matching. Thisbehaviour can be explained by the loss of information that using parallel machines implies. Wesee that Iterated-Sampling , like in previous experiments, outperforms the other algorithms due tohighly sequential design.
HEDCS-Matching particularly performs worse than the other algorithms,mainly because it fails to construct sufficiently large HEDCSs.20 ocial Network Communities.
We include two larger real-world datasets, orkut-groups andLiveJournal, from the Koblenz Network Collection [43]. We use two hypergraphs that were con-structed from these datasets by Shun [52]. Vertices represent individual users, and hyperedgesrepresent communities in the network. Because membership in these communities does not requirethe same commitment as collaborating on academic research, these hypergraphs have different char-acteristics from co-citation hypergraphs, in terms of size, vertex degree and hyperedge cardinality.We use the size of a maximal matching as a benchmark. Table 4 shows that
Iterated-Sampling still provides the best approximation.
HEDCS-Sampling performs worse than
Greedy on Live-journal, mainly because the ratio mn is not big enough to construct an HEDCS with a large matching. Name n m k
Gr IS HEDCS Rounds ISCora 2,708 1,579 2 75.0% 83.2% 63.9% 6PubMed 19,717 7,963 3 72.0% 86.6% 62.4% 9Orkut 2,32 × ×
10 55.6% 66.1% 58.1% 11Livejournal 3,20 × × Table 4: Comparison on co-citation and social network hypergraphs.
In the majority of our experiments,
Iterated-Sampling provides the best approximation to the d -UHM problem, which is consistent with its theoretical superiority. On random graphs, HEDCS-Matching performs consistently better than
Greedy , even-though
Greedy has a better theoreticalapproximation ratio. We suspect it is because the O ( d )-approximation bound on HEDCS-Matching is loose. We conjecture that rounding an (cid:15) − restricted matching can be done efficiently, which wouldimprove the approximation ratio. The performance of the three algorithms decreases as d grows.The results on the number of rounds of Iterated-Sampling also are consistent with the theoreticalbound. However, due to its sequential design, and by centralizing the computation on one singlemachine while using the other machines simply to coordinate,
Iterated-Sampling not only takes morerounds than the other two algorithms, but is also slower when we account for the absolute runtimeas well as the runtime per round. We compared the runtimes of our three algorithms on a set ofrandom uniform hypergraphs. Figure 1 and 2 show that the absolute and per round run-times of
Iterated-Sampling grow considerably faster with the size of the hypergraphs. We can also see that
HEDCS-Sampling is slower than
Greedy , since the former performs heavier computation on eachmachine. This confirms the trade-off between the extent to which the algorithms use the power ofparallelism and the quality of the approximations.
We have presented the first algorithms for the d -UHM problem in the MPC model. Our theoreticaland experimental results highlight the trade-off between the approximation ratio, the necessarymemory per machine and the number of rounds it takes to run the algorithm. We have alsointroduced the notion of HEDCS subgraphs, and have shown that an HEDCS contains a goodapproximation for the maximum matching and that they can be constructed in few rounds in theMPC model. We believe better approximation algorithms should be possible, especially if we cangive better rounding algorithms for a (cid:15) -restricted fractional hypergraph matching. For future work,it would be interesting to explore whether we can achieve better-than- d approximation in the MPCmodel in a polylogarithmic number of rounds. Exploring algorithms relying on vertex sampling21nstead of edge sampling might be a good candidate. In addition, our analysis in this paper isspecific to unweighted hypergraphs, and we would like to extend this to weighted hypergraphs.Figure 1: Runtime of the three algorithms when d = 3 and m = 20 · d · n Figure 2: Total runtime per round (maximum over all machines in every round) of the threealgorithms when d = 3 and m = 20 · d · n eferences [1] S. Abbar, S. Amer-Yahia, P. Indyk, S. Mahabadi, and K. R. Varadarajan. Diverse nearneighbor problem. In Proceedings of the twenty-ninth annual symposium on Computationalgeometry , pages 207–214. ACM, 2013.[2] K. J. Ahn and S. Guha. Access to data and number of iterations: Dual primal algorithmsfor maximum matching under resource constraints.
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Omitted proofs
A.1 Proof of Lemma 6.7
Lemma 6.7.
For any x ∈ X , E [ val (cid:48) ( x )] ≥ (1 − λ ) φ ( d H ( x )) .Proof. We distinguish three cases: • d H ( x ) ≤ βd : in this case β − d H ( x ) ≤ d ( d − β < (cid:15) and d H ∗ ( x ) ≤ d H ( x ) ≤ β − d H ( x ). This impliesthat val (cid:48) ( x ) = d H ∗ ( x ) β − d H ( x ) , so that : E [ val (cid:48) ( x )] ≥ d − d · d H ( x ) β − d H ( x ) = φ ( d H ( x )) . Now consider the case in which d H ( x ) > βd . Then E [ d H ∗ ( x )] ≥ ( d − d H ( x ) d > ( d − d · β ≥ · λ − >> (cid:15) − , because β ≥ d d − · λ − , and by a Chernoff Bound : P (cid:104) d H ∗ ( x ) < (1 − λ d − d H ( x ) d (cid:105) ≤ exp ( − E [ d H ∗ ( x )]( λ
12 ) ≤ exp ( − λ − ) ≤ λ/ . (3) • Let us now consider the case d H ( x ) > βd and min (cid:110) (cid:15), β − d H ( x ) (cid:111) = (cid:15) . With probability at least(1 − λ ), we have that: d H ∗ ( x ) ≥ ( β − (cid:15) )(1 − λ d − d >> (cid:15) − . Thus with probability at least (1 − λ ), we have that d H ∗ ( x ) (cid:15) > E [ val (cid:48) ( x )] ≥ (1 − λ ≥ (1 − λ φ ( x ) . • The only case that we need to check is d H ( x ) ≥ βd and min (cid:110) (cid:15), β − d H ( x ) (cid:111) = β − d H ( x ) , so that val (cid:48) ( x ) = min (cid:110) , d H ∗ ( x ) β − d H ( x ) (cid:111) . Again we have that with probability at least (1 − λ ) : d H ∗ ( x ) β − d H ( x ) ≥ d − d d H ( x ) β − d H ( x ) (1 − λ ≥ (1 − λ φ ( d H ( x )) . In other words, with probability at least (1 − λ ), we have val ( x ) ≥ (1 − λ ) φ ( d H ( x )), so that E [ val ( x )] ≥ (1 − λ ) φ ( d H ( x )) > (1 − λ ) φ ( d H ( x )). We just showed that in all cases E [ val (cid:48) ( x )] ≥ (1 − λ ) φ ( d H ( x )) 27 .2 Proof of Lemma 6.5 and Lemma 6.6 Lemma 6.5.
Let φ ( x ) = min { , ( d − xd ( β − x ) } . If a , . . . a d ≥ and a + . . . + a d ≥ β (1 − λ ) for some λ ≥ , then d (cid:80) i =1 φ ( a i ) ≥ − · λ .Proof. We will provide a proof for d = 3 that is easy to generalize. We first show that if a + b + c ≥ β ,then φ ( a ) + φ ( b ) + φ ( c ) ≥
1. The claim is true if φ ( a ) ≥ φ ( b ) ≥ φ ( c ) ≥
1. Suppose that φ ( a ) < φ ( b ) < φ ( c ) <
1. Then : φ ( a ) + φ ( b ) + φ ( c ) = d − d (cid:16) aβ − a + bβ − b + cβ − c (cid:17) ≥ d − d (cid:16) ab + c + ba + c + ca + b (cid:17) . By Nesbitt’s Inequality we know that ab + c + ba + c + ca + b ≥ dd − , and therefore φ ( a ) + φ ( b ) + φ ( c ) ≥ d > d (cid:88) i =1 a i (cid:80) j (cid:54) = i a j ≥ dd − . So if d (cid:80) i =1 a i ≥ β , then d (cid:80) i =1 φ ( a i ) ≥
1. Now, let φ (cid:48) ( x ) = ddx φ ( x ). To complete the proof, it is sufficientto show that we always have φ (cid:48) ( x ) ≤ β . To prove this inequality, note that if x ≥ d d − β then φ ( x ) = 1 and thus φ (cid:48) ( x ) = 0 . Now, if x ≤ d d − β then: φ (cid:48) ( x ) = d − d ddx xβ − x = d − d β ( β − x ) , which is increasing in x and maximized at x = d d − β , in which case φ (cid:48) ( x ) = (2 d − d ( d −
1) 1 β ≤ β . In theend we get: d (cid:88) i =1 φ ( a i ) ≥ − · λ . Lemma 6.6.
Given any
HEDCS ( G, β, β (1 − λ )) H , we can find two disjoint sets of vertices X and Y that satisfy the following properties:1. | X | + | Y | = d · µ ( G ) .2. There is a perfect matching in Y using edges in H .3. Letting σ = | Y | d + (cid:80) x ∈ X φ ( d H ( x )) ,we have that σ ≥ µ ( G )(1 − λ ) .4. All edges in H with vertices in X have at least one other vertex in Y , and have vertices onlyin X and Y . roof. Let M G be some maximum integral matching in G . Some of the edges in M G are in H , while others are in G \ H . Let X contain all vertices incident to edges in M G ∩ ( G \ H ),and let Y contain all vertices incident to edges in M G ∩ H . We now show that X and Y satisfy the first three properties of the lemma. Property 1 is satisfied because X ∪ Y con-sists of all matched vertices in M G . Property 2 is satisfied by definition of Y . To see thatProperty 3 is satisfied, remark that the vertices of Y each contribute exactly d . Now, X consists of | X | /d disjoint edge in G \ H , and by Property P2 of a HEDCS, for each such edge e : (cid:80) x ∈ e d H ( x ) ≥ β (1 − λ ) and by Lemma 6.5, we have (cid:80) x ∈ e φ ( d H ( x )) ≥ (1 − λ ) and each oneof these vertices contributes in average at least − λd to σ , just as desired. Loosely speaking, φ ( d H ( x )) will end up corresponding to the profit gained by vertex x in the fractional matching M Hf .Consider Y and X from above. These sets might not satisfy the Property 4 (that all edgesin H with an endpoint in X have at least one other endpoint in Y ). Can we transform theseinto sets X and Y , such that the first three properties still hold and there are no hyperedgeswith endpoints in X and V \ ( X ∪ Y ); at this stage, however, there will be possibly edgesin H with different endpoints in X . To construct X , Y , we start with X = X and Y = Y ,and present a transformation that terminates with X = X and Y = Y . Recall that X has aperfect matching using edges in G \ H . The set X will maintain this property throughout thetransformation, and each vertex x ∈ X has always a unique mate e (cid:48) . The construction does thefollowing : as long as there exists an edge e in H containing x and only endpoints in X and V \ ( X ∪ Y ), let e (cid:48) be the mate of x , we then remove the endpoints of e (cid:48) from X and add theendpoints of e to Y . Property 1 is maintained because we have removed d vertices from X andadded d to Y . Property 2 is maintained because the vertices we added to Y were connected byan edge in H . Property 3 is maintained because X clearly still has a perfect matching in G \ H ,and for the vertices { x , x , . . . , x d } = e (cid:48) , the average contribution is still at least − λd , as above.We continue this process while there is an edge with endpoints in X and V \ ( X ∪ Y ). Theprocess terminates because each time we are removing d vertices from X and adding d verticesto Y . We end up with two sets X and Y such that the first three properties of the lemma aresatisfied and there are no edges with endpoints in X and V \ ( X ∪ Y ). This means that forany edges in H incident to X , this edge is either incident to Y as well or incident to only points in X .We now set X = X and Y = Y and show how to transform X and Y into two setsthat satisfy all four properties of the lemma. Recall that X still contains a perfect matchingusing edges in G \ H ; denote this matching M GX . Our final set, however, will not guaranteesuch a perfect matching. Let M HX be a maximal matching in X using edges in H (with edgesnot incident to Y , because they already satisfy Property 4). Consider the edge set E ∗ X = M GX ∪ M HX .We now perform the following simple transformation, we remove the endpoints of the edgesin M HX from X and add them directly to Y . Property 1 is preserved because we are deleting andadding the same number of vertices from X and to Y respectively. Property 2 is preserved be-cause the endpoints we add to Y are matched in M HX and thus in H . We will see later for Property 3.Let’s check if Property 4 is preserved. To see this, note that we took M XH to be maximalamong edges of H that contain only endpoints in X , and moved all the matched vertices in M HX to Y . Thus all vertices that remain in X are free in M XH , and so there are no edges between onlyendpoints in X after the transformation. There are also no edges in H containing endpoints from X and V \ ( X ∪ Y ), because originally they don’t exist for X = X d = 4. In blue the edges of M GX and in yellow the edges of M HX Let’s check if the Property 3 is preserved. As before, this involves showing that after thetransformation, the average contribution of a vertex in X ∪ Y to σ is at least − λd . (Because everyvertex in X is incident to an edge in E ∗ X , each vertex is accounted for in the transformation.)Now, all vertices that were in Y remain in Y , so their average contribution remains at 1 /d . Wethus need to show that the average contribution to σ among vertices in X remains at least − λd after the transformation.Let n = | M GX | and k = | M HX | . Since M GX is a perfect matching on X , we always have k ≤ n .If n = k , then all vertices of X are transferred to Y and clearly their average contribution to σ is d ≥ − λd . Now consider when k ≤ n −
1. Let the edges of M HX be { e , . . . e k } and these of M GX be { e (cid:48) , . . . e (cid:48) n } . Let X (cid:48) be the set of vertices that remain in X . Because the edges of M GX are not H ,the by two properties of HEDCS : (cid:88) ≤ i ≤ n, x ∈ e (cid:48) i d H ( x ) ≥ nβ (1 − λ ) , and (cid:88) ≤ i ≤ k, x ∈ e i d H ( x ) ≤ kβ . The sum of the degrees of vertices in X (cid:48) can be written as the following difference: (cid:88) x ∈ X (cid:48) d H ( x ) = (cid:88) ≤ i ≤ n, x ∈ e (cid:48) i d H ( x ) − (cid:88) ≤ i ≤ k, x ∈ e i d H ( x ) (4) ≥ nβ (1 − λ ) − kβ = ( n − k ) · β − nβλ . Now we prove that (5) implies that the contribution of vertices from X (cid:48) on average at least − λd . Claim A.1. (cid:80) x ∈ X (cid:48) φ ( d H ( x )) ≥ ( n − k ) − n · λ .
30y claim A.1, the average contribution among the vertices that are considered is( n − k ) − n · λ + knd = 1 − λd , which proves Property 3 and completes the proof of the lemma. Proof of claim A.1.
Let’s denote m := n − k . Recall that the number of vertices in X (cid:48) is equal to md and φ ( x ) = d − d xβ − x . Here we will prove it for λ = 0. This means that we will prove that (cid:88) x ∈ X (cid:48) d H ( x ) ≥ m · β ⇒ (cid:88) x ∈ X (cid:48) φ ( d H ( x )) ≥ m . If m = n − k = 1, then the result clearly holds by Lemma 6.5. Let’s suppose k < n − m ≥
2. By Lemma 6.5, we know that: md − md (cid:88) x ∈ X (cid:48) d H ( x ) m · β − d H ( x ) ≥ . (5)We also know that : mβ − aβ − a = m + ( m − aβ − a , and (6) aβ − a = m · amβ − a + ( m − a ( mβ − a )( β − a ) . (7)Combining (5) and (7), we get: (cid:88) x ∈ X (cid:48) d H ( x ) β − d H ( x ) ≥ m dmd − (cid:88) x ∈ X (cid:48) ( m − d H ( x ) ( mβ − d H ( x ))( β − d H ( x )) . which leads to: (cid:88) x ∈ X (cid:48) φ ( d H ( x )) ≥ d − d (cid:88) x ∈ X (cid:48) d H ( x ) β − d H ( x ) ≥ m · m ( d − md − d − d (cid:88) x ∈ X (cid:48) ( m − d H ( x ) ( mβ − d H ( x ))( β − d H ( x )) ≥ m + d − d (cid:88) x ∈ X (cid:48) ( m − d H ( x ) ( mβ − d H ( x ))( β − d H ( x )) − m − md − . (8)By convexity of the function x (cid:55)→ x ( mβ − x )( β − x ) : (cid:88) x ∈ X (cid:48) d H ( x ) ( mβ − d H ( x ))( β − d H ( x )) ≥ md ( (cid:80) d H ( x )) md ) ( mβ − (cid:80) d H ( x )) md ( β − (cid:80) d H ( x )) md ) ≥ mβ ( md − d − . (9)Where the last inequality is due to (cid:80) x ∈ X (cid:48) d H ( x ) ≥ m · β Therefore, when β ≥ d the right hand side of (8) becomes: m + d − d (cid:88) x ∈ X (cid:48) d H ( x ) ( mβ − d H ( x ))( β − d H ( x )) − md − ≥ m + 1 md − (cid:18) mβd − (cid:19) ≥ m. .3 Proof of Corollary 6.9 Corollary 6.9.
For d -uniform linear hypergraphs, the degree distribution is tighter, and | d A ( v ) − d B ( v ) | = O (log n ) λβ. Proof.
For linear hypergraphs, assume that d A ( v ) = kλβ with k = Ω(log n ) and d B ( v ) = 0. Thedifference in the analysis is that, for every edge e belonging to the kλβ edges that are incident to v in A , we can find a new set of at least ( k − ( d + 1)) λβ edges in B \ A . In fact, for such an edge e , every one of the (1 − λ ) β edges that verify (cid:80) u (cid:54) = v d B ( u ) ≥ (1 − λ ) β intersect e in exactly on vertexthat is not v . The same goes for the subset of at least ( k − λβ that are in B \ A, these edgesalready intersect e , and can at most have one intersection in between them. At most d ( d −
1) ofthese edges can be considered simultaneously for different e , . . . , e d from the kλβ edges incidentto v . Therefore, for every edge e , we can find at least a set of ( k − λβd ( d − ≥ ( k − ( d + 1)) λβ new edges that in B \ A . This means that at this point we have already covered kλβ ( k − ( d + 1)) λβ edges in both A and B . One can see that we can continue covering new edges just like in theprevious lemma, such that at iteration l , the number of covered edges is at least k ( k − ( d + 1))( k − d + 1)) . . . ( k − l ( d + 1))( λβ ) l , for l ≤ k − d +1 . It is easy to see that for l = k − d +1 , we will have k ( k − ( d + 1))( k − d + 1)) . . . ( k − l ( d + 1))( λβ ) l > nβ if k = Ω(log n ), which will be a contradiction. B Figures and Tables n m d k β λ
15 200 3 5 500 79.1% 87.5% 82.3% 5 330 400 5 82.6% 91.3% 84.4% 7 4100 3200 10 83.9% 96.2% 88.2% 5 2300 4000 10 81.1% 92.0% 86.3% 4 250 800 5 6 500 76.0% 89.1% 78.5% 16 11100 2,800 10 77.9% 92.1% 81.9% 16 11300 4,000 10 77.8% 93.9% 87.1% 10 5500 8,000 16 79.2% 94.3% 85.9% 10 5500 15,000 10 16 500 71.8% 90.6% 79.2% 20 101,000 50,000 20 73.8% 92.3% 81.5% 20 102,500 100,000 20 72.2% 91.5% 80.7% 20 105,000 200,000 20 72.5% 90.7% 79.8% 20 101,000 5,0000 25 20 100 68.2% 87.5% 75.6% 75 502,500 100,000 25 69.0% 87.9% 74.3% 75 505,000 250,000 30 67.8% 87.3% 75.1% 75 5010,000 500,000 30 67.2% 86.9% 73.7% 75 505,000 250,000 50 30 100 67.4% 86.6% 74.0% 100 5010,000 500,000 30 68.1% 87.1% 73.4% 100 5015,000 750,000 30 66.9% 86.2% 73.2% 100 5025,000 1,000,000 30 67.3% 86.0% 72.8% 100 50Table 5: Comparison on random uniform instances with maximal matching benchmark.32
Chernoff Bound
Let X , . . . , X n be independent random variables taking value in [0 ,
1] and X := n (cid:80) i =1 X i . Then, forany δ ∈ (0 , P r (cid:16) | X − E [ X ] | ≥ δ E [ X ] (cid:17) ≥ · exp (cid:16) − δ E [ X ]3 (cid:17) . D Nesbitt’s Inequality
Nesbitt’s inequality states that for positive real numbers a , b and c , ab + c + ba + c + ca + b ≥ , with equality when all the variables are equal. And generally, if a , . . . , a n are positive realnumbers and s = n (cid:80) i =1 , then: n (cid:88) i =1 a i s − a i ≥ nn − , with equality when all the a ii