Testing k-colorability

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Presentation transcript:

Testing k-colorability Noga Alon and Michael Krivelevich Presented by Igor Shinkar

Graph property testing Let P be a graph property (bipartiteness, k- colorability, connectedness). Given a graph G=(V,E) we wish to distinguish between the following: G has the property P G is ε-far from having P The algorithm makes few queries to G If G has the property P, then Pr[Alg accepts] > 2/3 If G is ε-far from P, then Pr[Alg rejects] > 2/3

Testing k-colorability Given a dense graph G=(V,E) distinguish between G is k-colorable. G is ε-far from k-colorable, i.e. need to remove at least εn2 edges to make it k-colorable. Strategy: Pick a random subgraph S and accept if the G[S] is k-colorable.

Testing k-colorability The algorithm Pick a random subset of vertices S, of size O(klg(k)/ε2). Query all the edges in G[S]. Accept if and only if G[S] is k-colorable. Clearly if G is k-colorable then all its subgraphs are also k-colorable. So from now on we assume that G is ε-far from k-colorable.

G is ε-far from k-colorable Intuition: Let :S[k] be a k-coloring of S. If u∈V\S has no feasible colors w.r.t. S and , we say that it is colorless w.r.t. S and . If any coloring of u∈V\S restricts many vertices, we say that u is restricting w.r.t. S and . We show that for any small set S, either the set of colorless vertices or the set of restricting vertices must be large.

Notation: restricting vertices Given :S[k], denote by L(v) the feasible colors of v∈V\S w.r.t. (S,). Let U be the set of colorless vertices w.r.t. (S,). For v∈V\(SU) let (v,i) be the number of vertices that v restricts when colored with i. (v,i) = |{uN(v)\(SU):iL(u)}| By coloring v we shorten the lists of feasible colors for at least (v) = miniL(v) (v,i) vertices.

Illustration: restricting vertices L(w)={1,2,4} w L(x)={1,2,3} x L(v)={1,2,3,4} V (v,1)=|{w,x,z}| = 3 y L(y)={2,3,4,5,7} (v,2)=|{w,x,y}| = 3 z L(z)={1,4,5,6} (v,3)=|{x,y}| = 2 (v,4)=|{w,y,z}| = 3 By coloring v with 3 we restrict at most two vertices

Small U and   G is close to k-colorable Claim 1 For every S and a k-coloring :S[k], the graph G is at most M edges far from k- colorable, where M = (n-1)|SU| + ΣvV\(SU)(v) Proof: Remove all edges touching SU. For vSU remove (v) edges so that the coloring of v will restrict none of it neighbors.

G is ε-far from k-colorable Corollary 1 If G is ε-far from k-colorable, then ΣvV\(SU)(v) > εn2 – n(|S|+|U|) Corollary 2 Let W={v∈V\(SU) : (v) > εn/2} be the set of restricting vertices. Then |U|+|W|+|S| > εn/2

G is ε-far from k-colorable Corollary 2 Let W={v∈V\(SU) : (v) > εn/2} be the set of restricting vertices. Then |UW|> εn/2 -|S| That means that either there is a large set of colorless vertices or there is a large set of restricting vertices.

Defining the auxiliary k-tree The nodes of T are labeled with either v∈V or #. Nodes labeled with # are always leaves. The edges are colored with [k]. For a node t in T, let S(t) be the set of labels on the path from the root to t. Let (t):S(t)[k] be the coloring of S(t) defined according to the color of the edge along the path. The child of t is labeled # if the corresponding color is infeasible for t w.r.t. (S(t),(t)).

Defining the auxiliary k-tree b c # d # #  ={ac} # # ={abd} ={abd} Red is infeasible for d

Defining the auxiliary k-tree Start with a single unlabeled node. At the i’th step: Pick a random vertex vi=d. For each unlabeled node t consider (S(t),(t)). If d is in UW, call round i successful and set the label of t to d and update her children. a dUW b # c d # # # d d∈U d∈W # # # #

Bounding the depth of the tree Claim 3 The depth of T is bounded by 2k/ε. In particular all leaves on level 2k/ε are #. Proof: Fix a node t in T and let t1,t2,…tr=t be the path from the root to t. Each node ti is a restricting vertex w.r.t. (S(ti),(ti)). Thus removing at least εn/2 colors from the list of all feasible colors. Recall that the list of all colors is at most kn.

Two simple observations Observation 1: If a leaf t is labeled #, then the coloring (t):S(t)[k] is not proper. Observation 2: Suppose after s steps the sampled vertices of G are v1,v2,…,vs. If all leaves of T are labeled #, then the subgraph induced by v1,v2,…,vs is not k-colorable.

Proof of the theorem Claim 4 If G is ε-far from k-colorable, then w.h.p. after s = O(klg(k)/ε2) steps all leaves are labeled #, i.e. G[S] is not k-colorable. Proof: The number of leaves in T is≥ k2k/ε. At each step the probability of a node to be successful is >ε/2 For every path p=t1…tr (r<2k/ε) Pr[label(tr) ≠ #] < Pr[Bin(s, ε/2) < 2k/ε]

Lower bound Theorem: For every k≥3 and ε>0 small enough there are infinitely many graphs G such that 1) G is ε-far from being k-colorable. 2) Every Ω(1/ε) vertices induce a k-colorable subgraph.

Lower bound: Key lemma Lemma: For every k≥3 and sufficiently large m, there exists a graph H=Hk,m such that 1) Every subset on at most m/lg(k) vertices of H is k-colorable. 2) Every subset U of size |U|>m/2k span at least Ω(lg(k)|U|) edges. 3) H is Ω(lg(k)m) edges far from k-colorable.

So what did we show? If G is ε-far from k-colorable, then w.h.p. a subgarph of size O(klg(k)/ε2) is not k-colorable. There exist graphs ε-far from k-colorable, such that any subgraph of size Ω(1/ε) is k-colorable.

FIN