MAX CUT in Weighted Random Intersection Graphs and Discrepancy of Sparse Random Set Systems
Sotiris Nikoletseas, Christoforos Raptopoulos, Paul Spirakis
aa r X i v : . [ c s . D M ] S e p MAX CUT in Weighted Random Intersection Graphs andDiscrepancy of Sparse Random Set Systems
Sotiris Nikoletseas , Christoforos Raptopoulos , and Paul Spirakis Computer Engineering & Informatics Department, University of Patras, [email protected], [email protected] Department of Computer Science, University of Liverpool, [email protected] 4, 2020
Abstract
Let V be a set of n vertices, M a set of m labels, and let R be an m × n matrix of independentBernoulli random variables with probability of success p . A random instance G ( V, E, R T R ) ofthe weighted random intersection graph model is constructed by drawing an edge with weight[ R T R ] v,u between any two vertices u, v for which this weight is strictly larger than 0. In thispaper we study the average case analysis of Weighted Max Cut , assuming the input is aweighted random intersection graph, i.e. given G ( V, E, R T R ) we wish to find a partition of V into two sets so that the total weight of the edges having exactly one endpoint in each set ismaximized.In particular, we initially employ a classical approach by proving concentration of the weightof a maximum cut of G ( V, E, R T R ) around its expected value, and then showing that, whenthe number of labels is much smaller than the number of vertices (in particular, m = n α , α < n = m and constant average degree (i.e. p = Θ(1) n ),we show that with high probability, a majority type algorithm outputs a cut with weight thatis larger than the weight of a random cut by a multiplicative constant strictly larger than 1.Then, we formally prove a connection between the computational problem of finding a (weighted)maximum cut in G ( V, E, R T R ) and the problem of finding a 2-coloring that achieves minimumdiscrepancy for a set system Σ with incidence matrix R (i.e. minimum imbalance over all setsin Σ). We exploit this connection by proposing a (weak) bipartization algorithm for the case m = n, p = Θ(1) n that, when it terminates, its output can be used to find a 2-coloring withminimum discrepancy in a set system with incidence matrix R . Finally,we prove that, withhigh probability the latter 2-coloring corresponds to a bipartition with maximum cut-weight in G ( V, E, R T R ). Given an undirected graph G ( V, E ), the
Max Cut problem asks for a partition of the vertices of G into two sets, such that the number of edges with exactly one endpoint in each set of the partitionis maximized. This problem can be naturally generalized for weighted (undirected) graphs. A1eighted graph is denoted by G ( V, E, W ), where V is the set of vertices, E is the set of edgesand W is a weight matrix, which specifies a weight W i,j = w i,j , for each pair of vertices i, j . Inparticular, we assume that W i,j = 0, for each edge { i, j } / ∈ E . Definition 1 ( Weighted Max Cut ) . Given a weighted graph G ( V, E, W ) , find a partition of V into two (disjoint) subsets A, B , so as to maximize the cumulative weight of the edges of G havingone endpoint in A and the other in B . Weighted Max Cut is fundamental in theoretical computer science and is relevant in variousgraph layout and embedding problems [9]. Furthermore, it also has many practical applications,including infrastructure cost and circuit layout optimization in network and VLSI design [18],minimizing the Hamiltonian of a spin glass model in statistical physics [2], and data clustering [17].In the worst case
Max Cut (and also
Weighted Max Cut ) is
APX -hard, meaning that thereis no polynomial-time approximation scheme that finds a solution that is arbitrarily close to theoptimum, unless P = NP [16].The average case analysis of
Max Cut , namely the case where the input graph is chosen atrandom from a probabilistic space of graphs, is also of considerable interest and is further motivatedby the desire to justify and understand why various graph partitioning heuristics work well inpractical applications. In most research works the input graphs are drawn from the Erd˝os-R´enyirandom graphs model G n,m , i.e. random instances are drawn equiprobably from the set of simpleundirected graphs on n vertices and m edges, where m is a linear function of n (see also [6, 12] forthe average case analysis of Max Cut and its generalizations with respect to other random graphmodels). One of the earliest results in this area is that
Max Cut undergoes a phase transitionon G n,γn at γ = [7], in that the difference between the number of edges of the graph and theMax-Cut size is O (1), for γ < , while it is Ω( n ), when γ > . For large values of γ , it wasproved in [3] that the maximum cut size of G n,γn normalized by the number of vertices n reachesan absolute limit in probability as n → ∞ , but it was not until recently that the latter limit wasestablished and expressed analytically in [8], using the interpolation method; in particular, it wasshown to be asymptotically equal to ( γ + P ∗ q γ ) n , where P ∗ ≈ . G n,γn . An efficient approximation scheme in this case was designed in [7], and it wasproved that, with high probability, this scheme constructs a cut with at least (cid:0) γ + 0 . √ γ (cid:1) n =(1 + 0 . √ γ ) γ n edges, noting that γ n is the size of a random cut (in which each vertex is placedindependently and equiprobably in one of the two sets of the partition). Whether there exists anefficient approximation scheme that can close the gap between the approximation guarantee of [7]and the limit of [8] remains an open problem.In this paper, we study the average case analysis of Weighted Max Cut when input graphsare drawn from the generalization of another well-established model of random graphs, namelythe weighted random intersection graphs model (the unweighted version of the model was initiallydefined in [14]). In this model, edges are formed through the intersection of label sets assignedto each vertex and weights correspond to the size of these intersections. The definition belowhighlights a connection between weighted random intersection graphs and random set systems.
Definition 2 (Weighted random intersection graph) . Consider a universe M = { , , . . . , m } oflabels and a set of n vertices V . We define the m × n representation matrix R whose entries are ndependent Bernoulli random variables with probability of success p . For ℓ ∈ M and v ∈ V , wesay that vertex v has chosen label ℓ iff R ℓ,v = 1 . Furthermore, we draw an edge between two verticesiff they have selected at least one label in common. The weighted graph G = ( V, E, R T R ) is then arandom instance of the weighted random intersection graphs model G n,m,p . Random intersection graphs are relevant to and capture quite nicely social networking; ver-tices are the individual actors and labels correspond to specific types of interdependency. Otherapplications include oblivious resource sharing in a (general) distributed setting, efficient and se-cure communication in sensor networks [20], interactions of mobile agents traversing the web etc.(see e.g. the survey papers [5, 15] for further motivation and recent research related to randomintersection graphs). One of the most celebrated results in this area is equivalence (measured interms of total variation distance) of random intersection graphs and Erd˝os-R´enyi random graphswhen the number of labels satisfies m = n α , α > α ≥
3. Similarities of the two probabilistic spaces also exist even for smaller values of α >
Weighted MaxCut under the weighted random intersection graphs model, for the range m = n α , α ≤ Max Cut has not been considered in theliterature so far when the input is a drawn from the random intersection graphs model, and thusthe asymptotic behaviour of the maximum cut remains unknown especially for the range of valueswhere random intersection graphs and Erd˝os-R´enyi random graphs differ the most. Furthermore,studying a model where we can implicitly control its intersection number (indeed m is an obviousupper bound on the number of cliques that can cover all edges of the graph) may help understandalgorithmic bottlenecks for finding maximum cuts in Erd˝os-R´enyi random graphs.Second, we note that the representation matrix R of a weighted random intersection graphnaturally defines a random set system Σ consisting of m sets Σ = { L , . . . , L m } , where L ℓ is theset of vertices that have chosen label ℓ ; we say that R is the incidence matrix of Σ. We study Weighted Max Cut through the prism of a connection with the discrepancy of such random setsystems. In particular, given a set system Σ with incidence matrix R , its discrepancy is definedas disc(Σ) = min x ∈{± } n max L ∈ Σ (cid:12)(cid:12)P v ∈ L x v (cid:12)(cid:12) = k Rx k ∞ , i.e. it is the minimum imbalance of allsets in Σ over all 2-colorings x . Recent work on the discrepancy of random set systems defined asabove [13] has shown that, when the number of labels (sets) m is roughly o ( √ n ) and p = Ω(log n ) m ,the discrepancy of Σ is at most 1 with high probability. The proof of the main result in [13] is basedon Fourier analysis of an m -dimensional random variable, and improves upon earlier results [10].The design of an efficient algorithm that can find a 2-coloring having discrepancy O (1) in thisrange still remains an open problem. Approximation algorithms for a similar model for randomset systems were designed and analyzed in [1]; however, the algorithmic ideas there do not applyin our case. In this paper, we introduce the model of weighted random intersection graphs and we study theaverage case analysis of
Weighted Max Cut through the prism of discrepancy of random setsystems. We formalize the connection between these two combinatorial problems for the case ofarbitrary weighted intersection graphs in Theorem 1. We prove that, given a weighted intersectiongraph G = ( V, E, R T R ) with representation matrix R , and a set system with incidence matrix3 , such that disc(Σ) ≤
1, a 2-coloring has maximum cut weight in G if and only if it achievesminimum discrepancy in Σ. In particular, Theorem 1 applies in the range of values consideredin [13] (i.e. n = Ω( n log m ) and p = Ω(log n ) m ), and thus any algorithm that finds a maximum cut in G ( V, E, R T R ) with large enough probability can also be used to find a 2-coloring with minimumdiscrepancy in a set system Σ with incidence matrix R , with the same probability of success.We then consider weighted random intersection graphs in the case m = n α , α ≤
1, and weprove that the maximum cut weight of a random instance G ( V, E, R T R ) of G n,m,p concentratesaround its expected value (see Theorem 2). In particular, with high probability over the choices of R , Max-Cut ( G ) ∼ E R [ Max-Cut ( G )], where E R denotes expectation with respect to R . The proofuses standard probabilistic tools and the Efron-Stein inequality for upper bounding the varianceof the maximum cut. As a consequence of our concentration result, we prove in Theorem 3 that,in the case α <
1, a random 2-coloring (i.e. biparition) x ( rand ) in which each vertex chooses itscolor independently and equiprobably, has cut weight asymptotically equal to Max-Cut ( G ), withhigh probability over the choices of x ( rand ) and R .The latter result on random cuts allows us to focus on the case m = n (i.e. α = 1), and p = cn ,for some constant c (see also the discussion at the end of Section 3.1), where the assumptionsof Theorem 3 do not hold. It is worth noting that, in this range of values, the expected weightof a fixed edge in a weighted random intersection graph is equal to mp = Θ(1 /n ), and thus wehope that our work here will serve as an intermediate step towards understanding and overcomingalgorithmic bottlenecks for finding maximum cuts in sparse Erd˝os-R´enyi random graphs. In par-ticular, we analyze a Majority Cut Algorithm 1 that extends the algorithmic idea of [7] to weightedintersection graphs as follows: vertices are colored sequentially (each color +1 or − t -th vertex is colored opposite to the signof P i ∈ [ t − [ R T R ] i,t x i , namely the total available weight of its incident edges, taking into accountcolors of adjacent vertices. Our average case analysis of the Majority Cut Algorithm shows that,when m = n and p = cn , for c >
1, with high probability over the choices of R , the expected weightof the constructed cut is at least 1 + β times larger than the expected weight of a random cut,for some constant β = β ( c ) ≥ q πc − o (1). The fact that the lower bound on beta is inverselyproportional to c / was to be expected, because, as p increases, the approximation of the maximumcut that we get from the weight of a random cut improves (see also the discussion at the end ofSection 3.1).In Section 5 we propose a framework for finding maximum cuts in weighted random intersectiongraphs for m = n and p = cn , for constant c , by exploiting the connection between Weighted MaxCut and the problem of discrepancy minimization in random set systems. In particular, we designa Weak Bipartization Algorithm 2, that takes as input an intersection graph with representationmatrix R and outputs a subgraph that is “almost” bipartite. In fact, the input intersection graphis treated as a multigraph composed by overlapping cliques formed by the label sets L ℓ = { v : R ℓ,v = 1 } , ℓ ∈ M . The algorithm attempts to destroy all odd cycles of the input (except from oddcycles that are formed by labels with only two vertices) by replacing each clique induced by somelabel set L ℓ by a random maximal matching. In Theorem 5 we prove that, with high probabilityover the choices of R , if the Weak Bipartization Algorithm terminates, then its output can be usedto construct a 2-coloring that has minimum discrepancy in a set system with incidence matrix R ,which also gives a maximum cut in G ( V, E, R T R ). It is worth noting that this does not follow fromTheorem 1, because a random set system with incidence matrix R has discrepancy larger than1 with (at least) constant probability when m = n and p = cn . Our proof relies on a structural4roperty of closed 0-strong vertex-label sequences (loosely defined as closed walks of edges formedby distinct labels) in the weighted random intersection graph G ( V, E, R T R ) (Lemma 1). We believethat this part of our work may also be of interest regarding the design of efficient algorithms forfinding minimum disrepancy colorings in random set systems. We denote weighted undirected graphs by G ( V, E, W ); in particular, V = V ( G ) (resp. E = E ( G ))is the set of vertices (resp. set of edges) and W = W ( G ) is the weight matrix, i.e. W i,j = w i,j isthe weight of (undirected) edge { i, j } ∈ E . We allow W to have non-zero diagonal entries, as thesedo not affect cut weights. We also denote the number of vertices by n , and we use the notation[ n ] = { , , . . . , n } . We also use this notation to define parts of matrices, for example W [ n ] , denotesthe first column of the weight matrix.A bipartition of the sets of vertices is a partition of V into two sets A, B such that A ∩ B = ∅ and A ∪ B = V . Bipartitions correspond to 2-colorings, which we denote by vectors x such that x i = +1 if i ∈ A and x i = − i ∈ B .Given a weighted graph G ( V, E, W ), we denote by Cut ( G, x ) the weight of a cut defined by abipartition x , namely Cut ( G, x ) = P { i,j }∈ E : i ∈ A,j ∈ B w i,j = P { i,j }∈ E w i,j ( x i − x j ) . The maximumcut of G is Max-Cut ( G ) = max x ∈{− , +1 } n Cut ( G, x ).For a weighted random intersection graph G ( V, E, R T R ) with representation matrix R , wedenote by S v the set of labels chosen by vertex v ∈ V , i.e. S v = { ℓ : R ℓ,v = 1 } . Furthermore, wedenote by L ℓ the set of vertices having chosen label ℓ , i.e. L ℓ = { v : R ℓ,v = 1 } . Using this notation,the weight of an edge { v, u } ∈ E is | S v ∪ S u | ; notice also that this is equal to 0 when { v, u } / ∈ E .We also note here that we may also think of a weighted random intersection graph as a simpleweighted graph where, for any pair of vertices v, u , there are | S v ∩ S u | simple edges between them.A set system Σ defined on a set V is a family of sets Σ = { L , L , . . . , L m } , where L ℓ ⊆ V, ℓ ∈ [ m ]. The incidence matrix of Σ is an m × n matrix R = R (Σ), where for any ℓ ∈ [ m ] , v ∈ [ n ], R ℓ,v = 1 if v ∈ S ℓ and 0 otherwise. The discrerpancy of Σ with respect to a 2-coloring x ofthe vertices in V is disc(Σ , x ) = max ℓ ∈ [ m ] (cid:12)(cid:12)P v ∈ V R ℓ,v x v (cid:12)(cid:12) = k Rx k ∞ . The discrepancy of Σ isdisc(Σ) = min x ∈{− , +1 } n disc(Σ , x ).It is well-known that the cut size of a bipartition of the set of vertices of a graph G ( V, E ) intosets A and B is given by P { i,j }∈ E ( x i − x j ) , where x i = +1 if i ∈ A and x i = − i ∈ B .This can be naturally generalized for multigraphs and also for weighted graphs. In particular, the Max-Cut size of a weighted graph G ( V, E, W ) is given by Max-Cut ( G ) = max x ∈{− , +1 } n X { i,j }∈ E W i,j ( x i − x j ) . (1)In particular, we get the following Corollary (refer to Section 6.1 for the proof): Corollary 1.
Let G ( V, E, R T R ) be a weighted intersection graph with representation matrix R .Then, for any x ∈ {− , +1 } n , Cut ( G, x ) = 14 X i,j ∈ [ n ] (cid:2) R T R (cid:3) i,j − k Rx k (2)5 nd so Max-Cut ( G ) = 14 X i,j ∈ [ n ] (cid:2) R T R (cid:3) i,j − min x ∈{− , +1 } n k Rx k , (3) where k · k denotes the 2-norm. In particular, the expectation of the size of a random cut, whereeach entry of x is independently and equiprobably either +1 or -1 is equal to E x [ Cut ( G, x )] = P i = j,i,j ∈ [ n ] (cid:2) R T R (cid:3) i,j , where E x denotes expectation with respect to x . Since P i,j ∈ [ n ] (cid:2) R T R (cid:3) i,j is fixed for any given representation matrix R , the above Corollaryimplies that, to find a bipartition of the vertex set V that corresponds to a maximum cut, we needto find an n -dimensional vector in arg min x ∈{− , +1 } n k Rx k . We thus get the following: Theorem 1.
Let G ( V, E, R T R ) be a weighted intersection graph with representation matrix R and Σ a set system with incidence matrix R . If disc (Σ) ≤ , then x ∗ ∈ arg min x ∈{− , +1 } n k Rx k ifand only if x ∗ ∈ arg min x ∈{− , +1 } n disc (Σ , x ) . In particular, if the minimum discrepancy of Σ is atmost 1, a bipartition corresponds to a maximum cut iff it achieves minimum discrepancy.Proof. Since disc(Σ , x ∗ ) ≤
1, then each component of Rx ∗ is either 0 or 1, for any x ∗ ∈ {− , +1 } n .In particular, for any ℓ ∈ [ m ], [ Rx ∗ ] ℓ is 0 if the number of ones in the ℓ -th row is even and it isequal to 1 otherwise. This is the best one can hope for, since sets with an odd number of elementscannot have discrepancy less than 1. Therefore, k Rx ∗ k is also the minimum possible. In particular,this implies that, in the case disc(Σ , x ∗ ) ≤
1, any 2-coloring that achieves minimum discrepancygives a bipartition that corresponds to a maximum cut and vice versa.Notice that above result is not necessarily true when disc(Σ) >
1, since the minimum of k Rx k could be achieved by 2-colorings with larger discrepancy than the optimal. We are interested in asymptotic analysis, and so we will take n to be large. As it is common in therandom intersection graphs literature, we assume m = n α for some positive constant α . We willfurther assume that α ≤
1, since in this range the distributions of random intersection graphs andErd˝os-R´enyi random graphs models, differ the most. In particular, even though existing proofs thatthe total variation distance of the two distributions tends to 0 work only for the case α ≥ α is between 1 and 3.Concerning the success probability p , we note that, when p = o (cid:16)q nm (cid:17) , direct application ofthe results of [4] suggest that G ( V, E, R T R ) is chordal with high probability, but in fact the sameproofs reveal that a stronger property holds, namely that there is no closed vertex-label sequence(refer to the definition in Section 2.2) having distinct labels. Therefore, in this case, finding abipartition with maximum cut weight is straightforward: indeed, one way to construct a maximumcut is to run our Weak Bipartization Algorithm 2 from Section 5, and then to apply Theorem5 (noting that Weak Bipartization termination condition trivially holds, since the set C odd ( G ( b ) )defined in Section 5 is empty). Furthermore, even though we consider weighted graphs, we willalso assume that mp = O (1), since otherwise G ( V, E, R T R ) will be almost complete with high6robability (notice that the probability that there is an edge between two vertices is 1 − (1 − p ) m ,which tends to 1 for mp = ω (1)). Therefore, we will assume that C q nm ≤ p ≤ C √ m , forarbitrary positive constants C , C ; in particular, C can be as small as possible, and C can be aslarge as possible, provided C √ m ≤ We define a closed vertex-label sequence as σ := v , ℓ , v , ℓ , · · · , v k , ℓ k , v k +1 = v , where k = | σ | isthe number of labels in σ , v i ∈ V , ℓ i ∈ M , { v i , v i +1 } ⊆ L ℓ i , for all i ∈ [ k ]. We will also say thatlabel ℓ is strong if | L ℓ | ≥
3, otherwise it is weak . For a given closed vertex-label sequence σ , andany integer λ ∈ [ | σ | ], we will say that σ is λ -strong if | L ℓ i | ≥
3, for λ indices i ∈ [ | σ | ].The proof of the following Lemma uses the first moment method (see Section 6.2 for the details). Lemma 1.
Let G ( V, E, R T R ) be a random instance of the G n,m,p model, with m = n , and p = cn ,for some constant c > . With high probability over the choices of R , 0-strong closed vertex-labelsequences in G do not have labels in common. We claim that the above property also holds if we replace 0-strong with λ -strong, for anyconstant λ , but this stronger version is not necessary for our analysis. In this section we prove that the size of the maximum cut in a weighted random intersection graphconcentrates around its expected value. We note however, that the following Theorem does notprovide an explicit formula for the expected value of the maximum cut.
Theorem 2.
Let G ( V, E, R T R ) be a random instance of the G n,m,p model with m = n a , α ≤ ,and C q nm ≤ p ≤ C √ m , for arbitrary positive constants C , C , and let R be its representationmatrix. Then Max-Cut ( G ) ∼ E R [ Max-Cut ( G )] , where E R denotes expectation with respect to R ,i.e. Max-Cut ( G ) concentrates around its expected value.Proof. Let G = G ( V, E, R T R ) be a weighted random intersection graph, andlet D denote the (random) diagonal matrix containing all diagonal elements of R T R . In particular, equation (3) of Corollary 1 can be written as Max-Cut ( G ) = (cid:16)P i = j,i,j ∈ [ n ] (cid:2) R T R (cid:3) i,j − min x ∈{− , +1 } n x T (cid:0) R T R − D (cid:1) x (cid:17) . Furthermore, for any given R ,notice that, if we select each element of x independently and equiprobably from {− , +1 } , then E x [ x T (cid:0) R T R − D (cid:1) x ] = 0, where E x denotes expectation with respect to x . By the probabilisticmethod, we thus have min x ∈{− , +1 } n x T (cid:0) R T R − D (cid:1) x ≤
0, implying the following bound:14 X i = j,i,j ∈ [ n ] (cid:2) R T R (cid:3) i,j ≤ Max-Cut ( G n,m,p ) ≤ X i = j,i,j ∈ [ n ] (cid:2) R T R (cid:3) i,j , (4)where the second inequality follows trivially by observing that P i = j,i,j ∈ [ n ] (cid:2) R T R (cid:3) i,j equals thesum of the weights of all edges.By linearity of expectation, we have E R hP i = j,i,j ∈ [ n ] (cid:2) R T R (cid:3) i,j i = E R hP i = j,i,j ∈ [ n ] P ℓ ∈ [ m ] R ℓ,i R ℓ,j i = n ( n − mp = Θ( n mp ), which goes to infinity as7 → ∞ , because np = Ω (cid:0)p nm (cid:1) = Ω(1) in the range of parameters that we consider. In particular,by (4), we have E R [ Max-Cut ( G )] = Θ( n mp ) . (5)By Chebyshev’s inequality, for any ǫ >
0, we havePr (cid:0) | Max-Cut ( G ) − E R [ Max-Cut ( G )] | ≥ ǫn mp (cid:1) ≤ Var R ( Max-Cut ( G )) ǫ n m p , (6)where Var R denotes variance with respect to R . To bound the variance on the right hand side of theabove inequality, we use the Efron-Stein inequality. In particular, we write Max-Cut ( G ) := f ( R ),i.e. we view Max-Cut ( G ) as a function of the label choices. For ℓ ∈ [ m ] , i ∈ [ n ], we also write R ( ℓ,i ) for the matrix R where entry ( ℓ, i ) has been replaced by an independent, identically distributed(i.i.d.) copy of R ℓ,i , which we denote by R ′ ℓ,i . By the Efron-Stein inequality, we now haveVar R ( Max-Cut ( G )) ≤ X ℓ ∈ [ m ] ,i ∈ [ n ] E (cid:20)(cid:16) f ( R ) − f (cid:16) R ( ℓ,i ) (cid:17)(cid:17) (cid:21) . (7)Notice now that, given all entries of R except R ℓ,i , the probability that f ( R ) is different from f (cid:0) R ( ℓ,i ) (cid:1) with probability at most Pr( R ℓ,i = R ′ ℓ,i ) = 2 p (1 − p ). Furthermore, if L ℓ \{ i } is the set ofvertices different from i which have selected ℓ , we then have that (cid:0) f ( R ) − f (cid:0) R ( ℓ,i ) (cid:1)(cid:1) ≤ | L ℓ \{ i }| ,because the intersection graph with representation matrix R differs by at most | L ℓ \{ i }| edges fromthe intersection graph with representation matrix R ( ℓ,i ) . Notice now that, by definition, | L ℓ \{ i }| follows the Binomial distribution B ( n − , p ). In particular, E (cid:2) | L ℓ \{ i }| (cid:3) = ( n − p ( np − p + 1),implying E h(cid:0) f ( R ) − f (cid:0) R ( ℓ,i ) (cid:1)(cid:1) i ≤ p (1 − p )( n − p ( np − p + 1), for any fixed ℓ ∈ [ m ] , i ∈ [ n ].Putting this all together, (7) becomesVar R ( Max-Cut ( G )) ≤ X ℓ ∈ [ m ] ,i ∈ [ n ] p (1 − p )( n − p ( np − p + 1)= nmp (1 − p )( n − p ( np − p + 1) = O ( n mp ) , (8)where the last equation comes from the fact that, in the range of values that we consider, we have p = o (1) and np = Ω(1). Therefore, by (6), we getPr (cid:0) | Max-Cut ( G ) − E R [ Max-Cut ( G )] | ≥ ǫn mp (cid:1) ≤ O ( n mp ) ǫ n m p = O (cid:18) ǫ nmp (cid:19) , which goes to 0 in the range of values that we consider. Together with (5), the above bound provesthat Max-Cut ( G ) is concentrated around its expected value, and the proof is completed. Max-Cut for small number of labels
Using Theorem 2, we can now show that, in the case m = n α , α <
1, and p = O (cid:16) √ m (cid:17) , a random cuthas asymptotically the same weight as Max-Cut ( G ), where G = G ( V, E, R T R ) is a random instanceof G n,m,p . In particular, let x ( rand ) be constructed as follows: for each i ∈ [ n ], set x ( rand ) i = − , and x ( rand ) i = +1 otherwise. The proof of the following Theoremcan be found in Section 6.3. 8 heorem 3. Let G ( V, E, R T R ) be a random instance of the G n,m,p model with m = n a , α < ,and C q nm ≤ p ≤ C √ m , for arbitrary positive constants C , C , and let R be its representationmatrix. Then the cut weight of the random 2-coloring x ( rand ) satisfies Cut ( G, x ( rand ) ) = (1 − o (1)) Max-Cut ( G ) with high probability over the choices of x ( rand ) and R . We note that the same analysis also holds when n = m and p is sufficiently large (e.g. p = ω ( ln nn )); more details can be found at the end of Section 6.3. In view of this, in the following sectionswe will only assume m = n (i.e. α = 1) and also p = cn , for some positive constant c . Besidesavoiding complicated formulae for p , the reason behind this assumption is that, in this range ofvalues, the expected weight of a fixed edge in G ( V, E, R T R ) is equal to mp = Θ(1 /n ), and thuswe hope that our work will serve as an intermediate step towards understanding and overcomingalgorithmic bottlenecks for finding maximum cuts in Erd˝os-R´enyi random graphs G n,c/n . In the following algorithm, the 2-coloring representing the bipartition of a cut is constructed asfollows: initially, a small constant fraction ǫ of vertices are randomly placed in the two partitions,and then in each subsequent step, one of the remaining vertices is placed in the partition thatmaximizes the weight of incident edges with endpoints in the opposite partition. Algorithm 1:
Majority Cut
Input: G ( V, E, R T R ) and its representation matrix R ∈ { , } m × n Output:
Large cut 2-coloring x ∈ {− , +1 } n Let v , . . . , v n an arbitrary ordering of vertices; for t = 0 to ǫn do Set x t to either − for t = ǫn + 1 to n do if P i ∈ [ t − [ R T R ] i,t x i ≥ then x t = − else x t = +1; return x ; Clearly the Majority Algorithm runs in polynomial time in n, m . Furthermore, the followingTheorem provides a lower bound on the expected weight of the cut constructed by the algorithmin the case m = n , p = cn , for constant c >
1, and ǫ →
0. The proof can be found in Section 6.4.
Theorem 4.
Let G ( V, E, R T R ) be a random instance of the G n,m,p model, with m = n , and p = cn , for some constant c > , and let R be its representation matrix. Then, with high probabilityover the choices of R , the majority algorithm constructs a cut with expected weight at least (1 + β ) E hP i = j,i,j ∈ [ n ] (cid:2) R T R (cid:3) i,j i , where β = β ( c ) ≥ q πc − o (1) is a constant, i.e. at least β times larger than the expected weight of a random cut. Intersection graph (weak) bipartization
Notice that we can view a weighted intersection graph G ( V, E, R T R ) as a multigraph, composedby m (possibly) overlapping cliques corresponding to the sets of vertices having chosen a certainlabel, namely L ℓ = { v : R ℓ,v } , ℓ ∈ [ m ]. In particular, let K ( ℓ ) denote the clique induced by label ℓ . Then G = ∪ + ℓ ∈ [ m ] K ( ℓ ) , where ∪ + denotes union that keeps multiple edges. In this section, wepresent an algorithm that takes as input an intersection graph G given as a union of overlappingcliques and outputs a subgraph that is “almost” bipartite. To this end, we first give the followingdefinition: Definition 3.
Given a weighted intersection graph G = G ( V, E, R T R ) and a subgraph G ( b ) ⊆ G , let C odd ( G ( b ) ) be the set of odd length closed vertex-label sequences σ := v , ℓ , v , ℓ , · · · , v k , ℓ k , v k +1 = v that additionally satisfy the following: (a) σ has distinct vertices (except the first and the last) and distinct labels. (b) v i is connected to v i +1 in G ( b ) , for all i ∈ [ | σ | ] . (c) σ is λ -strong, for some λ > . Algorithm 2 initially replaces each clique K ( ℓ ) by a random maximal matching M ( ℓ ) , and thusgets a subgraph G ( b ) ⊆ G . If C odd ( G ( b ) ) is not empty, then the algorithm selects σ ∈ C odd ( G ( b ) )and a strong label ℓ ∈ σ , and then replaces M ( ℓ ) in G ( b ) by a new random matching of K ( ℓ ) . Thealgorithm repeats until all odd cycles are destroyed (or runs forever trying to do so). Algorithm 2:
Intersection Graph Weak Bipartization
Input:
Weighted intersection graph G = ∪ + ℓ ∈ [ m ] K ( ℓ ) Output:
A subgraph of G ( b ) that has only 0-strong odd cycles for each ℓ ∈ [ m ] do Let M ( ℓ ) be a random maximal matching of K ( ℓ ) ; Set G ( b ) = ∪ + ℓ ∈ [ m ] M ( ℓ ) ; while C odd ( G ( b ) ) = ∅ do Let σ ∈ C odd ( G ( b ) ) and ℓ a label in σ with | L ℓ | ≥ Replace the part of G ( b ) corresponding to ℓ by a new random maximal matching M ( ℓ ) ; return G ( b ) ; The following results are the main technical tools that justify the use of the Weak BipartizationAlgorithm for
Weighted Max Cut . The proof details for Lemma 2 can be found in Section 6.5.
Lemma 2. If C odd ( G ( b ) ) is empty, then G ( b ) has only 0-strong odd cycles. We now prove the following:
Theorem 5.
Let G ( V, E, R T R ) be a random instance of the G n,m,p model, with n = m and p = cn , for some constant c > , and let R be its representation matrix. Let also Σ bea set system with incidence matrix R . With high probability over the choices of R , if Al-gorithm 2 for weak bipartization terminates on input G , its output can be used to constructa 2-coloring x ( disc ) ∈ arg min x ∈{± } n disc (Σ , x ) , which also gives a maximum cut in G , i.e. x ( disc ) ∈ arg max x ∈{± } n Cut ( G, x ) . roof. By construction, the output of Algorithm 2, namely G ( b ) , has only 0-strong odd cycles.Furthermore, by Lemma 1 these cycles correspond to vertex-label sequencies that are label-disjoint.Let H denote the subgraph of G ( b ) in which we have destroyed all 0-strong odd cycles by deletinga single (arbitrary) edge e C from each 0-strong odd cycle C (keeping all other edges intact), andnotice that e C corresponds to a weak label. In particular, H is a bipartite multi-graph and thus itsvertices can be partitioned into two independent sets A, B constructed as follows: In each connectedcomponent of H , start with an arbitrary vertex v and include in A (resp. in B ) the set of verticesreachable from v that are at an even (resp. odd) distance from v . Since H is bipartite, it does nothave odd cycles, and thus this construction is well-defined, i.e. no vertex can be placed in both A and B .We now define x ( disc ) by setting x ( disc ) i = +1 if i ∈ A and x ( disc ) i = +1 if i ∈ B . Let M denote the set of weak labels corresponding to the edges removed from G ( b ) in the constructionof H . We first note that, for each ℓ C ∈ M corresponding to the removal of an edge e C , wehave (cid:12)(cid:12)(cid:12)P i ∈ L ℓC x ( disc ) i (cid:12)(cid:12)(cid:12) = 2. Indeed, since e C belongs to an odd cycle in G ( b ) , its endpoints are ateven distance in H , which means that either they both belong to A or they both belong to B .Therefore, their corresponding entries of x ( disc ) have the same sign, and so (taking into accountthat the endpoints of e C are the only vertices in L ℓ C ), we have (cid:12)(cid:12)(cid:12)P i ∈ L ℓC x ( disc ) i (cid:12)(cid:12)(cid:12) = 2. Second, weshow that, for all the other labels ℓ ∈ [ m ] \M , (cid:12)(cid:12)(cid:12)P i ∈ L ℓ x ( disc ) i (cid:12)(cid:12)(cid:12) will be equal to 1 if | L ℓ | is odd and 0otherwise. For any label ℓ ∈ [ m ] \M , let M ( ℓ ) denote the part of G ( b ) corresponding to a maximalmatching of K ( ℓ ) , and note that all edges of M ( ℓ ) are contained in H . Since H is bipartite, no edgein M ( ℓ ) can have both its endpoints in either A or B . Therefore, by construction, the contributionof entries of x ( disc ) corresponding to endpoints of edges in M ( ℓ ) to the sum P i ∈ L ℓ x ( disc ) i is 0. Inparticular, if | L ℓ | is even, then M ( ℓ ) is a perfect matching and (cid:12)(cid:12)(cid:12)P i ∈ L ℓ x ( disc ) i (cid:12)(cid:12)(cid:12) = 0, otherwise (i.e. if | L ℓ | is odd) there is a single vertex not matched in M ( ℓ ) and (cid:12)(cid:12)(cid:12)P i ∈ L ℓ x ( disc ) i (cid:12)(cid:12)(cid:12) = 1.To complete the proof of the theorem, we need to show that Cut( G, x ( disc ) ) is maximum. ByCorollary 1, this is equivalent to proving that k Rx ( disc ) k ≤ k Rx k for all x ∈ {− , +1 } n . Supposethat there is some x ( min ) ∈ {− , +1 } n such that k Rx ( disc ) k > k Rx ( min ) k . As mentioned above, forall ℓ ∈ [ m ] \M , we have [ Rx ( disc ) ] ℓ ≤
1, and so [ Rx ( disc ) ] ℓ ≤ [ Rx ( min ) ] ℓ . Therefore, the only labelswhere x ( min ) could do better are those corresponding to edges e C that are removed from G ( b ) inthe construction of H , i.e. ℓ C ∈ M , for which we have [ Rx ( disc ) ] ℓ C = 2. However, any such edge e C belongs to an odd cycle C , and thus any 2-coloring of the vertices of C will force at least one ofthe 0-strong labels corresponding to edges of C to be monochromatic. Taking into account the factthat, by Lemma 1, with high probability over the choices of R , all 0-strong odd cycles correspondto vertex-label sequences that are label-disjoint, we conclude that k Rx ( disc ) k ≤ k Rx ( min ) k , whichcompletes the proof.Notice that Theorem 5 does not follow from Theorem 1, because a random set system withincidence matrix R has discrepancy larger than 1 with (at least) constant probability when m = n and p = cn . Indeed, by a straightforward counting argument, we can see that the expected numberof 0-strong odd cycles is at least constant. Furthermore, in any 2-coloring of the vertices at leastone of the weak labels forming edges in a 0-strong odd cycle will be monochromatic. Therefore,with at least constant probability, for any x ∈ {− , +1 } n , there exists a weak label ℓ , such that x i x j = 1, for both i, j ∈ L ℓ , implying that disc( L ℓ ) = 2.11 Proof details
We first prove the following Lemma, by straightforward calculation from equation (1):
Lemma 3.
Let G ( V, E, W ) be a weighted graph such that W is symmetric and W i,j = 0 if { i, j } / ∈ E . Then Max-Cut ( G ) = 14 X i,j ∈ [ n ] W i,j − min x ∈{− , +1 } n x T Wx . (9) Proof.
For any x ∈ {− , +1 } n , we write X i,j ∈ [ n ] W i,j − x T Wx = X i,j ∈ [ n ] W i,j − X i,j ∈ [ n ] W i,j x i x j = 12 X i,j ∈ [ n ] W i,j (cid:0) x i + x j − x i x j (cid:1) = 12 X i,j ∈ [ n ] W i,j ( x i − x j ) = X { i,j }∈ E W i,j ( x i − x j ) . By (1), this completes the proof.
Proof of Corollary 1.
Notice that diagonal entries of the weight matrix in (9) cancel out, and so,for any x ∈ {− , +1 } n , we have X i,j ∈ [ n ] (cid:2) R T R (cid:3) i,j − k Rx k = X i = j,i,j ∈ [ n ] (cid:2) R T R (cid:3) i,j − X i = j,i,j ∈ [ n ] (cid:2) R T R (cid:3) i,j x i x j . Taking expectations with respect to x , the contribution of the second sum in the above expressionequals 0, which completes the proof. Proof.
We will use the first moment method and so we need to prove that the expectation of thenumber of pairs of distinct 0-strong closed vertex-label sequences in G that have at least one labelin common goes to 0. To this end, for j ∈ [min( k, k ′ ) − A j ( k, k ′ ) denote the number of suchsequences σ, σ ′ , with k = | σ | , k ′ = | σ ′ | , that have j labels in common. In particular, for integers k, k ′ , let σ := v , ℓ , v , ℓ , · · · , v k , ℓ k , v k +1 = v , and let σ ′ := v ′ , ℓ ′ , v ′ , ℓ ′ , · · · , v ′ k ′ , ℓ ′ k ′ , v ′ k ′ +1 = v .Notice that, any such fixed pair σ, σ ′ has the same probability to appear, namely p k + k ′ − j ) (1 − p ) ( n − k + k ′ − j ) ; indeed, p k (1 − p ) ( n − k is the probability that σ appears (recall that σ has k labelsand it is 0-strong, i.e. each label is only selected by two vertices) and p k ′ − j ) (1 − p ) ( n − k ′ − j ) isthe probability that σ ′ appears given that σ has appeared. Furthermore, the number of such pairsof sequences is dominated by the number of sequences that overlap in j consecutive labels (e.g. the12rst j ), which is at most n k m k n k ′ − j − m k ′ − j (notice that j common labels implies that there areat least j ′ + 1 common vertices). Overall, since n = m and p = cn , we have E [ A j ( k, k ′ )] ≤ (1 + o (1)) 1 n ( np ) k + k ′ − j ) (1 − p ) ( n − k + k ′ − j ) = (1 + o (1)) 1 n (cid:0) c (1 − p ) n − (cid:1) k + k ′ − j . Since n → ∞ and p = cn , by elementary calculus we have that c (1 − p ) n − bounded by a con-stant (which depends only on c ) strictly less than 1. Therefore, the above expectation is at most e − ln n − Θ(1)( k + k ′ − j ) . Therefore, summing over all choices of k, k ′ ∈ [ n ] and j ∈ [min( k, k ′ ) − X k,k ′ ∈ [ n ] X j ∈ [min( k,k ′ ) − e − ln n − Θ(1)( k + k ′ − j ) = o (1) , and the proof is completed by Markov’s inequality. Proof.
Let G = G ( V, E, R T R ) be a weighted random intersection graph. By equation (2) ofCorollary 1, for any x ∈ {− , +1 } n , we have: Cut ( G, x ) = 14 X i,j ∈ [ n ] (cid:2) R T R (cid:3) i,j − k Rx k . Taking expectations with respect to random x and R , we get E x , R [ Cut ( G, x )] = 14 · E R X i,j ∈ [ n ] (cid:2) R T R (cid:3) i,j − X i ∈ [ n ] (cid:2) R T R (cid:3) i,i = 14 · E R X i = j,i,j ∈ [ n ] (cid:2) R T R (cid:3) i,j = 14 n ( n − mp . (10)To prove the Theorem, we will show that, with high probability over random x and R , wehave k Rx k = o (cid:16) E R h P i = j,i,j ∈ [ n ] (cid:2) R T R (cid:3) i,j i(cid:17) = o ( n mp ), in which case the theorem fol-lows by concentration of Max-Cut ( G ) around its expected value (Theorem 2), and the fact that Max-Cut ( G ) ≥ P i = j,i,j ∈ [ n ] (cid:2) R T R (cid:3) i,j .To this end, fix ℓ ∈ [ m ] and consider the random variable counting the number of ones in the ℓ -th row of R , namely Y ℓ = P i ∈ [ n ] R ℓ,i . By the multiplicative Chernoff bound, for any δ > Y ℓ > (1 + δ ) np ) ≤ (cid:18) e δ (1 + δ ) δ (cid:19) np . Since np ≥ C p nm = C n − α , taking any δ ≥
2, we getPr( Y ℓ > np ) ≤ (cid:18) e (cid:19) np = o (cid:18) m (cid:19) . (11)13herefore, by the union bound, Pr( ∃ ℓ ∈ [ m ] : Y ℓ > np ) = o (1) , (12)implying that, all rows of R have at most 3 np non-zero elements with high probability.Fix now ℓ and consider the random variable corresponding to the ℓ -th entry of Rx , namely Z ℓ = P i ∈ [ n ] R ℓ,i x i . In particular, given Y ℓ , notice that Z ℓ is equal to the sum of Y ℓ independentrandom variables x i ∈ {− , +1 } , for i such that R ℓ,i = 1. Therefore, since E x [ Z ℓ ] = E x [ Z ℓ | Y ℓ ] = 0,by Hoeffding’s inequality, for any λ ≥ | Z ℓ | > λ | Y ℓ ) ≤ e − λ Yℓ . Therefore, by the union bound, and taking λ ≥ √ np ln n ,Pr( | Z ℓ | > λ ) ≤ Pr( ∃ ℓ ∈ [ m ] : Y ℓ > np ) + me − λ np = o (1) + mn = o (1) , (13)implying that all entries of Rx have absolute value at most √ np ln n with high probability overthe choices of x and R . Consequently, with high probability over the choices of x and R , we have k Rx k = 6 mnp ln n , which is o ( n mp ), since np = ω (ln n ) in the range of parameters consideredin this theorem. This completes the proof.We note that the same analysis also holds when n = m and p is sufficiently large (e.g. p = ω ( ln nn )). In particular, similar probability bounds hold in equations (11), (12) and (13), for thesame choices of δ ≥ λ ≥ √ np ln n , implying that k Rx k = 6 mnp ln n = o ( n mp ) with highprobability. Proof.
Let G ( V, E, R T R ) (i.e. the input to the Majority Cut Algorithm 1) be a random instance ofthe G n,m,p model, with m = n , and p = cn , for some large enough constant c (in fact c > t ∈ [ n ], let M t denote the constructed cut size just after the consideration of a vertex v t , forsome t ≥ ǫn + 1. In particular, since the values x , . . . , x t − are already decided in previous steps,we have M t = 14 X i,j ∈ [ t ] (cid:2) R T R (cid:3) i,j − min x t ∈{− , +1 } (cid:13)(cid:13) R [ m ] , [ t ] x [ t ] (cid:13)(cid:13) (14)The first of the above terms is14 X i,j ∈ [ t ] (cid:2) R T R (cid:3) i,j = 14 X i,j ∈ [ t − (cid:2) R T R (cid:3) i,j + 2 X i ∈ [ t − (cid:2) R T R (cid:3) i,t + (cid:2) R T R (cid:3) t,t (15)14nd the second term is −
14 min x t ∈{− , +1 } (cid:13)(cid:13) R [ m ] , [ t ] x [ t ] (cid:13)(cid:13) = −
14 min x t ∈{− , +1 } (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) R [ m ] ,t x t + X i ∈ [ t − R [ m ] ,i x i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = −
14 min x t ∈{− , +1 } X i,j ∈ [ t ] (cid:2) R T R (cid:3) i,j x i x j = − X i,j ∈ [ t − (cid:2) R T R (cid:3) i,j x i x j + 2 min x t ∈{− , +1 } X i ∈ [ t − (cid:2) R T R (cid:3) i,t x i x t + (cid:2) R T R (cid:3) t,t (16)By (14), (15) and (16), we have M t = M t − + 12 X i ∈ [ t − (cid:2) R T R (cid:3) i,t −
12 min x t ∈{− , +1 } X i ∈ [ t − (cid:2) R T R (cid:3) i,t x i x t = M t − + 12 X i ∈ [ t − (cid:2) R T R (cid:3) i,t + 12 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X i ∈ [ t − (cid:2) R T R (cid:3) i,t x i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (17)Define now the random variable Z t = Z t ( x , R ) = X i ∈ [ t − (cid:2) R T R (cid:3) i,t x i = X ℓ ∈ [ m ] R ℓ,t X i ∈ [ t − R ℓ,i x i , so that M t = M t − + P i ∈ [ t − (cid:2) R T R (cid:3) i,t + | Z t | . Notice also that, given x [ t − = { x i , i ∈ [ t − } ,and R [ m ] , [ t − = { R ℓ,i , ℓ ∈ [ m ] , i ∈ [ t − } , Z t is the sum of m independent random variables.Furthermore, E [ Z t | x [ t − , R [ m ] , [ t − ] = p P ℓ ∈ [ m ] P i ∈ [ t − R ℓ,i x i and Var( Z t | x [ t − , R [ m ] , [ t − ) = p (1 − p ) P ℓ ∈ [ m ] (cid:16)P i ∈ [ t − R ℓ,i x i (cid:17) .Before applying the Central Limit Theorem to approximate the distribution of Z t , we need toshow that, for a sufficient number of ℓ ∈ [ m ], say Y t = Y t ( R , x ), we have P i ∈ [ t − R ℓ,i x i = 0,because this will affect the approximation error of the CLT, which is O ( Y − / t ) [21]. To showthis, we note that Y t is lower bounded by the number of ℓ ∈ [ m ] for which P i ∈ [ t − R ℓ,i is an oddnumber. But, for any fixed i and t , we have Pr( P i ∈ [ t − R ℓ,i is odd) = P j odd (cid:0) t − j (cid:1) p j (1 − p ) t − − j ,which is bounded below by a constant as long as c is large enough (e.g. c > Y t stochastically dominates a binomial random variable B ( t − , ). Therefore, by the multiplicativeChernoff (upper) bound, for any δ > (cid:18) Y t < (1 − δ ) t − (cid:19) < (cid:18) e − δ (1 − δ ) − δ (cid:19) t − . Taking δ = and noting that t ≥ ǫn + 1, we havePr (cid:18) Y t < t − (cid:19) < (cid:16) e (cid:17) − ǫn , o ( n ), for any constant ǫ >
0. By the union bound,Pr (cid:18) ∃ t : t ≥ ǫn + 1 , Y t < t − (cid:19) = o (1) , implying that, with high probability over the choices of R , we have Y t ≥ t − ≥ ǫn , for any t ≥ ǫn +1.In view of the above, by the Central Limit Theorem, given x [ t − , R [ m ] , [ t − , the distributionof Z t is approximately Normal with expectation E [ Z t | x [ t − , R [ m ] , [ t − ] = p P ℓ ∈ [ m ] P i ∈ [ t − R ℓ,i x i ,variance Var( Z t | x [ t − , R [ m ] , [ t − ) = p (1 − p ) P ℓ ∈ [ m ] (cid:16)P i ∈ [ t − R ℓ,i x i (cid:17) and approximation error O (cid:16)q ǫn (cid:17) with high probability over the choices of R . Furthermore, given x [ t − , R [ m ] , [ t − , | Z t | follows approximately (i.e. with approximation error O (cid:16)q ǫn (cid:17) ) the folded normal distributionwith mean value that is at least q π Var( Z t | x [ t − , R [ m ] , [ t − ), since in the worst case we have E [ Z t | x [ t − , R [ m ] , [ t − ] = 0 (i.e. vertices are equally divided between the two partitions). Noticenow that, by definition of Y t and the equation for the conditional variance of Z t , we haveVar( Z t | x [ t − , R [ m ] , [ t − ) ≥ p (1 − p ) Y t . Since Y t ≥ t − ≥ ǫn with high probability, and also p = cn , we get thatVar( Z t | x [ t − , R [ m ] , [ t − ) ≥ c ( t − n + o (1), with high probability, where the o (1) comes from theapproximation error of the CLT. Consequently, with high probability (which is 1 − o (1 /n )), E [ | Z t | ] = E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X i ∈ [ t − (cid:2) R T R (cid:3) i,t x i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ r c ( t − πn + o (1) . Summing over all t ≥ ǫn + 1, we get X t ≥ ǫn +1 E [ | Z t | ] ≥ r c πn X t ≥ ǫn √ t + o ( n ) = r c πn X t ≥ √ t − ǫn √ ǫn + o ( n ) . Using the fact that P t ≥ √ t = n / + o ( n ), we thus have that X t ≥ ǫn +1 E [ | Z t | ] ≥ r c π (cid:18) − ǫ / (cid:19) n + o ( n ) . On the other hand, we have that the expected weight of a random cut is equal to n ( n − mp = c n + o ( n ) (see e.g. equation (10). The proof is completed by taking ǫ → Proof.
For the sake of contradiction, assume C odd ( G ( b ) ) = ∅ , but G ( b ) = ∪ + ℓ ∈ [ m ] M ( ℓ ) has an oddcycle C k that is not 0-strong and has minimum length. Notice that C k corresponds to a closedvertex-label sequence, say σ := v , ℓ , v , ℓ , · · · , v k , ℓ k , v k +1 = v , where { v i , v i +1 } ∈ M ( ℓ i ) , for all i ∈ [ k ]. Furthermore, by assumption, conditions (b) and (c) of Definition 3 are satisfied by σ (indeed16 v i , v i +1 } ∈ M ( ℓ i ) , for all i ∈ [ k ], and σ is λ -strong, for some λ > σ does not belong to C odd ( G ( b ) ) is that condition (a) of Definition 3 is not satisfied, i.e.there are distinct indices i > i ′ ∈ [ k ] such that ℓ i = ℓ i ′ . Clearly, such indices are not consecutive(i.e. i ′ = i + 1), because ℓ i is strong and step 6 of our algorithm implies that M ( ℓ i ) is a matchingof K ( ℓ i ) . But then either the vertex-label sequence v , . . . , v i , ℓ i , v i ′ +1 , ℓ i ′ +1 , v i ′ +2 , . . . , v k +1 = v orthe vertex-label sequence v i +1 , ℓ i +1 , v i +2 , . . . , v i ′ , ℓ i , v i +1 corresponds to a shorter odd cycle, whichis a contradiction on the minimality of C k . In this paper, we introduced the model of weighted random intersection graphs and we studied theaverage case analysis of
Weighted Max Cut through the prism of discrepancy of random setsystems. One of the main problems left open in our work concerns the termination of our WeakBipartization Algorithm. We conjecture the following:
Conjecture 1.
Let G ( V, E, R T R ) be a random instance of the G n,m,p model, with m = n , and p = cn , for some constant c > . With high probability over the choices of R , on input G , Algorithm2 for weak bipartization terminates in polynomial time. Towards strengthening the connection between
Weighted Max Cut under the G n,m,p model,and the problem of discrepancy minimization in random set systems, we conjecture the following: Conjecture 2.
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