1 Graph Sparsfication

Transcription

1 CME 305: Discrete Mathematics ad Algorithms 1 Graph Sparsficatio I this sectio we discuss the approximatio of a graph G(V, E) by a sparse graph H(V, F ) o the same vertex set. I particular, we cosider ay graph with E = Ω( 1+δ ) edges to be dese; we wish to fid sparse represetatios with F = O() edges that have proportioally the same umber of edges crossig ay cut. That is, if we scale the values of every edge i the sparse graph by E / F, the value of each cut will remai (approximately) the same. A commo applicatio of graph sparsificatio is iteret traffic routig. Cosider buildig a udirected etwor N o odes, ad suppose we would lie to route 1 uit of flow (directioless) betwee each pair of odes i N uder some capacity costraits o the edges. A complete graph K with c(e) = 1 e E would suffice, but for practical purposes it is udesirable to use so may (O( )) edges. If we require the umber of edges to be liear i, we might cosider a star graph ad scale up the capacities o the edges to 1. The drawbac i this case is that there is a sigle ode of degree 1 which has too much (O( )) traffic flowig through it if this ode were to fail it would brig dow the etire etwor. A reasoable goal, the, is to produce a graph o vertices ad O() edges that maitais the coectivity of the complete graph while also havig approximately uiform vertex degree. The expasio ρ(g) of a udirected graph G(V, E) is defied as the miimum cut value weighted by the size of the smaller cut partitio: ρ(g) = mi S V c(s, S) mi( S, S ) = mi S V, S c(s, S). S Taig all edge-values to be 1, our goal from above is equivalet to fidig a approximately regular graph G with O() edges ad ρ(g) = Ω(1). For compariso, ote that the complete graph has O( ) edges ad achieves ρ(k ) = /. We give two methods of costructig such G. 1.1 Erdős-Réyi Radom Graphs The Erdős-Réyi G(, p) model for costructig radom graphs deotes a graph o vertices where each of the ( 1)/ possible edges are icluded i the edge set idepedetly with probability p. To get m = O(), we may choose p = c/ for some costat c > 0, the we may compute c( 1) E[m] =, ( ) 1 E[d(v)] = c v V. The followig claim shows that we may sample from a Erdős-Réyi graph distributio ad obtai a suitable G with high probability. Claim 1 If we choose p = log ɛ, the G(, p) will have O( log ɛ ) edges ad with high probability, the size of every cut i G will be withi (1 ± ɛ) of its expected value, so ρ(g) = Ω(1). 1. Radom d-regular Graphs Recall that a d-regular graph is oe i which all vertices have the same degree d. Theorem 1 For all d 3, there exists a costat α > 0 such that with high probability, a radom d-regular graph G has expasio ρ(g) α.

2 CME 305: Discrete Mathematics ad Algorithms - Lecture 8 Proof: (To simplify the calculatios, we preset the proof for d sufficietly large istead of d 3. The proof for d = 3 is similar.) First we ote that we ca geerate d-regular graphs o vertices via the cofiguratio model: we split each vertex ito d mii-vertices, ad fid the edges by geeratig a radom perfect matchig o the miivertices. The whe we combie each vertex s mii-vertices, every vertex will have degree d. Note that a graph produced i this way may have multiple edges or self-loops. To prove the theorem we eed to show that for each set S V, S =, the probability that c(s, S) < α is sufficietly small. Assume that there exists such a S of size. For a give, there are ( ) possible choices for S. For a give S, there are ( )( d d d ) α α ways to choose the miivertices i S ad miivertices i S ivolved i a cut of size α. Let us start by calculatig the umber of d regular graphs o vertices, i.e., the umber of perfect matchigs of K d. ( )( ) ( ) d d 1 f(d) =... (d/)! (d)! = d/ (d/)!. We will use the followig Stirlig approximatio for factorials:! = ( ) ( ( )) 1 π 1 + O. e So f(d) = c(d) d/ e d/ (1 + O ( )) 1, for some costat c. The the probability of the evet that oly a certai α of miivertices match outside of their proper subset is at most f(d α)f(d d α)f(α). f(d) Therefore, the probability P that there is a subset S, with S = ad expasio less tha α may be bouded as ( )( )( ) d d d f(d α)f(d d α)f(α) P α. α α f(d) Usig the simple but useful iequality, we have ( ) ( ) ( e ) ( e ) ( ) α ed (d α) d α (d d α) (d d α)/ (α) α P cα α (d) d/ ( ) α ed ( ) ( ) +α (d α)/ cαe α ( ( ) ) α ( ) ((d ) 5α)/ ed cα e α cα(ce) (/) 3

3 CME 305: Discrete Mathematics ad Algorithms - Lecture 8 3 for d 15, α < 1/100. Therefore, / P = O(1/ ) =1 The Probabilistic Method Theorem 1 gives a affirmative aswer to the questio of whether there exists a graph with m = O() edges ad expasio ρ(g) = Ω(1). It is iterestig to ote, however, that the proof is o-costructive. We oly give a distributio of graphs from which a radom sampled istace is liely to have the properties we wat. This simple idea is the premise of a combiatorial aalysis techique ow as the probabilistic method: i order to prove the existece of a structure, we merely eed to show that there is a positive probability that the structure exists..1 A Simple Example: Moochromatic Colorig We let S 1,..., S m be subsets of a larger set S such that each subset S i cotais exactly l elemets from S. Is it possible to color the elemets of S with two colors say, red ad blue such that o set S i is moochromatic? Lemma 1 If the umber of subsets m < l 1, the such a colorig is always possible Proof: We use the probabilistic method. Toss a coi for each vertex ad color the vertex red if the coi lads heads, blue for tails, so the probability that a vertex is red is 1/, idepedet of the color of ay other vertex. The the probability that a give set S i is etirely red or etirely blue is l, so the probability p i moo that a S i is moochromatic is p i moo = l = l+1. Recall from basic probability the uio boud or subadditivity property of probabilities. That is, for ay (arbitrary, ot ecessarily disjoit) evets E 1, E,..., E j, P r( j E i) j P r(e i ). Usig the uio boud, the probability p moo that some set is moochromatic obeys for m < l 1. m m p moo p i moo = l+1 = m l+1 < 1 Therefore there is a positive probability that o set is moochromatic, ad so there must exist some assigmet of colors to vertices such that o set is moochromatic. Note agai that this proof is ocostructive. We ve created a distributio over all possible color assigmets (amely, the uiform distributio) ad used this to show a positive probability that a graph with the desired property exists. By explicitly givig a distributio over the space of all possible colorigs, we tur a exhaustive search algorithm ito a simple probability calculatio.

4 4 CME 305: Discrete Mathematics ad Algorithms - Lecture 8. Chromatic Number of a Graph A proper vertex -colorig of a graph G(V, E) is a assigmet of colors to vertices such that o two vertices of the same color share a edge. The chromatic umber χ(g) is equal to the smallest umber of colors eeded to have a proper vertex colorig. Chromatic umber might appear to be based mostly o the local structure of a graph. For example, it is simple to see that if a graph G cotais K as a subgraph, the χ(g). I geeral, very tightly coected subregios of graphs eed may colors for proper colorig. A reasoable questio to as is whether there exist graphs with high chromatic umber that do ot have ay particularly dese subregios; their global structure is what maes them require may colors. As a measure of local coectivity, we defie the girth of a graph G, g(g), to be the legth of the smallest cycle i G. If g(g) > 3, we say that G is triagle-free. I 1954, B. Descartes was the first to show that triagle-free graphs ca have arbitrarily high chromatic umbers, but this costructio was complicated ad cotaied may short cycles. I 1959, Paul Erdős used the probabilistic method to prove the existece of graphs with arbitrarily high girth ad chromatic umber. Theorem (Erdős, 1959) For every g, > 0, there exists a graph G with χ(g) ad g(g) g. Proof: A idepedet set i a graph G is U V such that o two vertices i U are coected by a edge. If a graph has chromatic umber, the there must exist at least oe idepedet set of size at least, sice each color i a proper colorig correspods to a idepedet set. We cosider Erdős-Réyi radom graphs G(, p). I order to show that χ(g), it suffices to prove that with high probability the size of ay idepedet set i G is at most. We will prove that with high probability for a suitable selectio of p, the graph does t have ay idepedet set of size. We use the uio boud. The probability that ay set of / vertices is a idepedet set is (1 p) (/ ). There are ( /) possibilities for vertex sets of size /. By the uio boud, the probability of G(, p) havig such a idepedet set is therefore at most [ P r G,p has a idepedet set of size ] ( ) (1 p) (/ ) / e p /8 e log p /8 e log ɛ+1 /8. where we set p = ɛ 1 for some ɛ < 1/g. The above expressio teds to zero as, so is therefore smaller tha 1/4 for sufficietly large. Let X be the radom variable coutig the umber of cycles of legth g ad smaller. expectatio, By liearity of E[X] = g ( ) (i 1)! p i i g(p) g = g ɛg. For 0 < ɛ < 1/g, the above expressio is o(). Thus, for sufficietly large, E(X) < /4. By Marov s iequality, P r(x > /) P r(x > E(X)) < 1/.

5 CME 305: Discrete Mathematics ad Algorithms - Lecture 8 5 Therefore, if we choose sufficietly large, ad p = ɛ 1 for 0 < ɛ < 1/g, the probability that G(, p) has a idepedet set of size or that the umber of cycles of legth at most g is is less tha 1 by the uio boud. Cosiderig the complemet of that evet, we see that there must exist a graph G with o idepedet set of size ad with at most cycles of legth at most g. Now, we ca costruct a graph G by removig a vertex from each short cycle of G. The umber of vertices i G is at least, the size of the maximum idepedet set i G is o more that, ad there are o cycles of legth less tha g. This implies that χ(g ) > ad g(g ) > g.