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Graphs with tiny vector Graphs with tiny vector chromatic numbers and chromatic numbers and
huge chromatic numbershuge chromatic numbers
Michael Langberg
Weizmann Institute of Science
Joint work with U. Feige and G. Schechtman
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Two fundamental NP-Two fundamental NP-Hard problemsHard problems
•Minimum ColoringMinimum Coloring
•Maximum Independent Set Maximum Independent Set
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Minimum coloringMinimum coloring
•Vertex-coloringVertex-coloring: : Assignment of colors to Assignment of colors to VV s.t. endpoints of each s.t. endpoints of each edge have diff. colors.edge have diff. colors.
G=(V,E) (G)=3
•Chromatic numberChromatic number (G)(G): Minimum number : Minimum number of colors needed.of colors needed.
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Maximum independent Maximum independent setset
•ISIS: Set of vertices : Set of vertices that do not share that do not share any edges.any edges.
(G)(G): Size of : Size of maximum IS.maximum IS.
G=(V,E) (G)=3
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Coloring vs. ISColoring vs. ISEvery color class in a coloring of Every color class in a coloring of GG is an IS. is an IS.
•Coloring Coloring finding a cover of finding a cover of
GG with disjoint IS. with disjoint IS.
(G)(G)(G)(G) n n..
•Algorithms for IS Algorithms for IS algorithms for coloring. algorithms for coloring.
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Approximation Approximation algorithmsalgorithms•Not likely to find efficient algorithms.Not likely to find efficient algorithms.
•Settle on efficient Settle on efficient approximation algorithmsapproximation algorithms..
•Provide solutions whose value is guaranteed Provide solutions whose value is guaranteed to be within a ratio no worse than to be within a ratio no worse than rr from the from the value of the optimal solution. value of the optimal solution.
•App. ratio of algorithm ALG: App. ratio of algorithm ALG:
•r = ALG/OPT (min.), r = OPT/ALG (max.).
•rr 1 1, the smaller the better !, the smaller the better !
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This talkThis talk
•[KargerMotwaniSudan] introduce the notion [KargerMotwaniSudan] introduce the notion of of vector coloringvector coloring..
•Plays major role in approximation algorithms Plays major role in approximation algorithms for IS and Coloring.for IS and Coloring.
•Our work: present Our work: present tighttight results on the results on the limitationlimitation of vector coloring. of vector coloring.
•Structure:Structure:
•Background on IS and Coloring.Background on IS and Coloring.
•Vector coloring.Vector coloring.
•Our results.Our results.
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Approximating Approximating (G) & (G) & (G) (G)
•Good news:Good news:
•Both Both (G)(G) and and (G)(G) can be app. within can be app. within ratio ratio n(loglog n)n(loglog n)22/(log n)/(log n)33 [Haldorsson, Feige].[Haldorsson, Feige].
•Bad News:Bad News:
•Estimating both Estimating both (G)(G) and and (G)(G) up to a up to a factor of factor of nn1-1- is “hard” (unless NP is “hard” (unless NPis in is in random polynomial time). random polynomial time).
[Hastad, FeigeKilian ,EngebretsenHolmerin, Khot]. [Hastad, FeigeKilian ,EngebretsenHolmerin, Khot].
Relatively small gap.
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What about restricted What about restricted cases?cases?Consider a graph Consider a graph GG that is known to have that is known to have smallsmallchromatic number chromatic number (G)=k(G)=k..
•Good news:Good news:
•Can efficiently find coloring with:Can efficiently find coloring with:
•k=3 k=3 n n3/14 3/14 colors colors [KargerMotwaniSudan, BlumKarger].[KargerMotwaniSudan, BlumKarger].
•k=4 k=4 n n7/19 7/19 colors colors [HalperinNathanielZwick].[HalperinNathanielZwick].
•k k n nf(k) f(k) colors (colors (f(k)f(k)11 as as kk increases) increases) [KMS, HNZ].[KMS, HNZ].
•Bad news Bad news [KhannaLinialSafra, GuruswamiKhanna, Khot]:[KhannaLinialSafra, GuruswamiKhanna, Khot]:
•NP-hard to color NP-hard to color 33 colorable graphs with colorable graphs with 44 colors. colors.
•NP-hard to color NP-hard to color kk col. graphs with col. graphs with 5k/3, k 5k/3, k(log k)(log k) colors.colors.
Gap is wide open.
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Vector coloring [KMS]Vector coloring [KMS]Plays a major role in approximation Plays a major role in approximation
algorithms for coloring and IS (restricted algorithms for coloring and IS (restricted cases).cases).
Definition:Definition: G=(V,E)G=(V,E) is is vector vector kk-colorable-colorable if one if one can assign unit vectors to its vertices, s.t. can assign unit vectors to its vertices, s.t. every two adjacent vertices are embedded every two adjacent vertices are embedded farfar apart. apart.
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Vector coloring cont.Vector coloring cont.
Definition:Definition: G=(V,E)G=(V,E) is is vector vector kk-colorable-colorable if one if one can assign unit vectors to its vertices, s.t. can assign unit vectors to its vertices, s.t. every two vectors corresponding to adjacent every two vectors corresponding to adjacent vertices have vertices have inner product at most inner product at most -1/(k-1)-1/(k-1). .
•k=3 <vi,vj> -1/(k-1) = -1/2 = cos(120o)
•k=11 <vi,vj> -1/(k-1) = -1/10 cos(95o)
120o
95o
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Vector coloring – Vector coloring – exampleexample
Definition:Definition: G=(V,E) is vector G=(V,E) is vector kk-colorable if one can assign unit -colorable if one can assign unit vectors to its vertices, s.t. every two vectors corresponding vectors to its vertices, s.t. every two vectors corresponding to adjacent vertices have inner product at most to adjacent vertices have inner product at most -1/(k-1)-1/(k-1). .
Vector 3-coloring:
•k=3 -1/(k-1) = -1/2 = cos(120o)
R2
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Vector coloring – Vector coloring – exampleexampleDefinition:Definition: G=(V,E) is vector G=(V,E) is vector kk-colorable if one can assign unit -colorable if one can assign unit
vectors to its vertices, s.t. every two vectors corr. to vectors to its vertices, s.t. every two vectors corr. to adjacent vertices have inner product at most adjacent vertices have inner product at most -1/(k-1)-1/(k-1). .
Vector 4-coloring:
•k=4 -1/(k-1) = -1/3 cos(109o)
R3
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Vector coloring Vector coloring (G) (G)
•Every Every kk colorable graph is also vector colorable graph is also vector kk-col.-col.
•Identify each color class with one vertex Identify each color class with one vertex
in a perfect in a perfect k-1k-1 dimensional simplex. dimensional simplex.
•k = 4k = 4::R3
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Vector coloring in PVector coloring in P
If If G=(V,E)G=(V,E) is vector is vector kk-colorable, such a vector-colorable, such a vector
coloring can be computed in polynomial timecoloring can be computed in polynomial time
(semidefinite programming).(semidefinite programming).
Min Min s.t. s.t.
•<v<vii,v,vjj> > for each edge (i,j) for each edge (i,j) E. E.
•<v<vii,v,vii> = 1 > = 1 for each node i for each node i V.V.
= -1/(k-1)= -1/(k-1)
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Algorithm of [KMS] Algorithm of [KMS]
Use vector coloring to color graphs with small Use vector coloring to color graphs with small
chromatic number.chromatic number.
•Input: Graph Input: Graph GG which satisfies which satisfies (G)=3(G)=3..
•Output: Coloring of Output: Coloring of GG with with fewfew colors. colors.
•ALG:ALG:(G)=3(G)=3 GG is vector is vector 33-colorable.-colorable.
•Find vector Find vector 33-coloring of -coloring of GG (SDP). (SDP).
•Use geometrical structure to Use geometrical structure to find find
good coloring of good coloring of GG..
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Algorithm of [KMS] cont.Algorithm of [KMS] cont.
•Pick random cap.
•Consider vertices corr. to vectors in cap.•Small cap small IS.•Large cap large set with many edges.•[KMS] optimize size of cap.
•Objective: find large IS in G.
Vector 3-coloring
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[KMS] results[KMS] results: maximum degree in : maximum degree in GG..
•Graphs which are vector Graphs which are vector 33-colorable can be -colorable can be colored eff. in colored eff. in 1/31/3 colors ( colors (+1+1 trivial). trivial).
•As function of As function of n: n: obtain obtain nn1/4 1/4 [Wigderson][Wigderson]..
•Graphs which are vector Graphs which are vector kk-colorable can be -colorable can be colored efficiently in colored efficiently in min(min(1-2/k1-2/k,, nn1-3/(k+1)1-3/(k+1))) colors.colors.
Improving these results will yield Improving these results will yield improved improved
results inresults in [BlumKarger, AlonKahale, HalperinNathanielZwick].[BlumKarger, AlonKahale, HalperinNathanielZwick].
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Our resultsOur results
Negative in nature.Negative in nature.
Prove that the results of [KMS] are tight.Prove that the results of [KMS] are tight. [KMS]: Graphs which are vector [KMS]: Graphs which are vector kk-colorable can be -colorable can be
colored efficiently in colored efficiently in 1-2/k 1-2/k colors.colors.
Present vector Present vector kk-colorable graphs with -colorable graphs with chromatic number at least chromatic number at least 1-2/k-1-2/k- ..
•Will neglect Will neglect in remainder of talk. in remainder of talk.
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Previous work on Previous work on limitation of vector limitation of vector coloring. coloring. As a function on As a function on n n rather thanrather than ..
•Vector Vector 33-colorable graphs with -colorable graphs with n n0.050.05
[KMS,Alon,Szegedy].[KMS,Alon,Szegedy].
•Vector Vector kk-colorable graphs with -colorable graphs with n nf(k) f(k)
where where f(k) f(k) 1 1 [KMS,Charikar,Feige]. [KMS,Charikar,Feige].
Our results:Our results:
•For For k=3k=3 we obtain we obtain nn0.150.15..
•For other For other kk, improve, improve f(k) f(k)..
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How large is the gap? How large is the gap? How good of an app. is vector coloring to How good of an app. is vector coloring to (G)?(G)?
•There are graphs for which ratio between There are graphs for which ratio between (G)(G) and and vector chromatic number vector chromatic number n/2 n/2 [Feige].[Feige].
G is 2 vec. col. (G) n/2
large gap
Our results improve gap to n/polylog(n).n/polylog(n).•(k=log(n)/loglog(n)).(k=log(n)/loglog(n)).
Vector coloringVector coloring does not app.does not app. within within factor better than factor better than n/polylog(n).n/polylog(n).
O(log1/2(n))
O(log1/2(n))O(log1/2(n))
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Previous work – graphs Previous work – graphs usedusedAll previous work use similar graphs All previous work use similar graphs G=(V,E):G=(V,E):
•V:V: {0,1}{0,1}n n (“hypercube”).(“hypercube”).
•E:E: vertices vertices uu and and vv are connected iff are connected iff Hamming distance is large.Hamming distance is large.
•Natural embedding in unit sphere Natural embedding in unit sphere (ensures small vector chromatic number).(ensures small vector chromatic number).
•Known bounds on maximum IS Known bounds on maximum IS
(ensures large chromatic num).(ensures large chromatic num).
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00
01
10
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Our workOur work•Use different graphs.Use different graphs.
•We use graphs presented in We use graphs presented in [FeigeSchechtman] that that addresses a SDP relaxation of the Max-Cut problem addresses a SDP relaxation of the Max-Cut problem [GoemansWilliamson].[GoemansWilliamson].
•Goal:Goal: GG is vector is vector 33-colorable, -colorable, (G)(G) is large ( is large (k=3k=3). ).
•Our graph Our graph GG:: place place nn random points on the unit random points on the unit sphere, connect each two points that are far apart. sphere, connect each two points that are far apart. I.e. inner product at most I.e. inner product at most -1/2-1/2..
121200oo
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Main theoremMain theorem
•GG is vector is vector 33-colorable (by definition).-colorable (by definition).
•GG has chromatic number has chromatic number 1/31/3..
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Analyzing Analyzing (G)(G)
•Start with a continuous Start with a continuous graph.graph.
•V V = all points on unit sphere.= all points on unit sphere.
•E E = pairs of points = pairs of points farfar appart. appart.
•Analyze expansion properties.Analyze expansion properties.
•Switch to discrete version.Switch to discrete version.
•Take random sample.Take random sample.
•Do not know how to analyze (G)(G) directly. directly.
•Follow ideas of Follow ideas of [FS]:[FS]: construct construct GG in three steps.
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Proof outline – Proof outline – (G) is (G) is large.large.•Step 1: continuous graph. Step 1: continuous graph.
•Continuous graph is vector Continuous graph is vector 33-col.-col.
•Continuous graph has nice expansion Continuous graph has nice expansion properties. properties.
•Step 2: discrete graph.Step 2: discrete graph.
•Discrete graph is vector Discrete graph is vector 33-col. -col.
•Inherits expansion properties.Inherits expansion properties.
•Step 3: random sample.Step 3: random sample.
•Random sample is vector Random sample is vector 33-col. -col.
•Expansion properties of discrete Expansion properties of discrete graph imply random sample has graph imply random sample has large large ..
A
B
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Remainder of this talkRemainder of this talk
•Step 1: continuous graph. Step 1: continuous graph.
•Continuous graph is vector Continuous graph is vector 33-col.-col.
•Continuous graph has nice Continuous graph has nice expansion properties expansion properties (isoperimetric inequalities on the (isoperimetric inequalities on the sphere)sphere)..
•Step 2: Step 2: discrete graph inherits discrete graph inherits expansion properties of continuous expansion properties of continuous graphgraph..
•Step 3: random sample.Step 3: random sample.
•Random sample is vector Random sample is vector 33-col. -col.
•Expansion prop. of discrete graph Expansion prop. of discrete graph imply random sample has large imply random sample has large (property testing)(property testing)..
A
B
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Wait a minute !Wait a minute !Continuous Continuous Discrete Discrete Random Random
Why do we need the random graph?Why do we need the random graph?
Doesn’t the discrete version suffice?Doesn’t the discrete version suffice?
Properties of discrete graph (easy to prove):Properties of discrete graph (easy to prove):
• Vector Vector 33-colorable.-colorable.
• Large chromatic number.Large chromatic number.
Problem:Problem: Discrete graph has large degree (Discrete graph has large degree (nn1-1-), can ), can not show not show 1/31/3..
Solution:Solution: Take random sample.Take random sample.
•Max. degree decreases.Max. degree decreases.
•Will show that Will show that remains large. remains large.
Expansion Expansion properties of properties of
continuous graph continuous graph
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The continuous graph GThe continuous graph Gcc
•Vertex set: all points in unit sphere Vertex set: all points in unit sphere SSd-1d-1..
•Edge set: Edge set: (v(vii,v,vjj) ) EE iff iff <v<vii,v,vjj> > -1/2 = -1/2 = cos(120cos(120oo)) (corresponds to vector (corresponds to vector 33-coloring). -coloring).
•Use natural measure for subsets of Use natural measure for subsets of V, EV, E..
121200oo
measure = 1/2
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Main theoremMain theoremLet Let dd = dimension of sphere, = dimension of sphere, (1- (1-))dd..
Let Let AA and and BB be two subsets of be two subsets of GGcc of measure of measure ..
Theorem:Theorem: The measure of edges between The measure of edges between AA and and BB is at least is at least 44|E|.|E|.
•Two random subsets of measure Two random subsets of measure are expected to are expected to share share 22|E||E| edges. edges.
A
B
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Proof outlineProof outline
Theorem:Theorem: Let Let AA and and BB be two subsets of be two subsets of GGcc of measure of measure .. The The measure of edges between measure of edges between AA and and BB is at least is at least 44|E|.|E|.
•Step 1: Subsets Step 1: Subsets A, BA, B which share the least which share the least measure of edges are caps (shifting).measure of edges are caps (shifting).
•Step 2: Analyze measure of edges between caps. Step 2: Analyze measure of edges between caps.
A
B
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Step 1: caps share few Step 1: caps share few edgesedges• Step 1: Subsets Step 1: Subsets A, BA, B which share the least measure of which share the least measure of
edges are caps (of same measure).edges are caps (of same measure).
Would like a Would like a shifting procedureshifting procedure that converts any two sets that converts any two sets AA and and
B B to caps while preserving measure and decreasing to caps while preserving measure and decreasing
the amount of edges between the amount of edges between AA and and BB..
•Use Use [BaernsteinTaylor][BaernsteinTaylor] two point two point
symmetrization procedure:symmetrization procedure:
•Choose arbitrary hyperplane.Choose arbitrary hyperplane.
•Consider each point and its Consider each point and its mirror image. mirror image.
•““Shift up” if possible.Shift up” if possible.
•Procedure converges into cap.Procedure converges into cap.
AA*
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Measure of edges Measure of edges decreasesdecreasesAt each step measure of edges between At each step measure of edges between AA
and and BB does not increase. does not increase.
•Consider two vetrices and their mirror image.
•Vertices may be in A or B in both or not in any.
•Check number of edges before and after shifting.
•Case analysis.
•Step 1: OK
A
B
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Step 2: edges between Step 2: edges between capscaps
•First show that caps First show that caps AA and and BB that share that share minimal measure of edges satisfy minimal measure of edges satisfy A = BA = B..
•Then compute measure of edges in cap. Then compute measure of edges in cap.
•Good estimates are known.Good estimates are known.
A
B
A=B
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Theorem restatedTheorem restated
Let Let AA and and BB be two subsets of be two subsets of GGcc of measure of measure
(( (1- (1-))dd).).
Theorem:Theorem: The measure of edges between The measure of edges between AA and and BB is at least is at least 44|E|.|E|.
Continuous Continuous graph to discrete graph to discrete
graphgraph
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Discrete graphDiscrete graph
•Partition Partition GGcc into many small cells each of into many small cells each of small diameter and equal measure.small diameter and equal measure.
•Discrete graph Discrete graph GGdd
•VV = cells. = cells.
•EE = pairs of cells which = pairs of cells which
share edges in share edges in GGcc..
•Vector Vector 33-colorable.-colorable.
•Inherits expansion properties of GInherits expansion properties of Gcc..
A
B
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Theorem – discrete Theorem – discrete graphgraph•Let Let GGdd = (V,E) = (V,E) be the discrete graph. be the discrete graph.
•Let Let AA and and BB be two subsets of be two subsets of GGdd of size of size
|V| = |V| = n (n ( n n1-1-).).
Theorem:Theorem: The number of edges between The number of edges between AA and and BB is at least is at least 44|E|.|E|.
A
B
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RecallRecall
Goal: Goal: vector vector 33-colorable graphs with -colorable graphs with chromatic number at least chromatic number at least 1/3 1/3 ((k=3k=3)) ..
Shown:Shown: Discrete graph Discrete graph GGdd: Every two subsets : Every two subsets A, BA, B of size of size nn n n1-1- share many edges. share many edges.
IS of IS of GGdd is less than is less than n n (G(Gdd) ) 1/ 1/ nn..
Problem:Problem: dd (max. degree in (max. degree in GGdd) is very large, ) is very large, will not imply desired bounds.will not imply desired bounds.
Solution:Solution: Take random sample Take random sample RR of of GGdd..
Random SampleRandom Sample
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Expansion Expansion bounds on bounds on
•Discrete graph Discrete graph GGdd: :
•Nice expansion on setsNice expansion on sets of size of size |G|Gdd|.|.
• (G(Gdd) ) 1/ 1/..
•Random sample Random sample RR::(R)(R)= = (1/(1/).).
•The smaller the sample the The smaller the sample the better (better relation better (better relation vs. vs. ). ).
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Property testing [GGR]Property testing [GGR]GG is is farfar from having property from having property PP small random small random
sample of sample of GG will not have property will not have property PP..
•Use property testing on discrete graph to prove Use property testing on discrete graph to prove that small random sample has large that small random sample has large ..
•Use property testing on discrete graph to prove Use property testing on discrete graph to prove that random sample does not have large ISthat random sample does not have large IS..
•Consider property Consider property PP: “having large IS”.: “having large IS”.
““having small having small ”.”.
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Property testing [GGR]Property testing [GGR]
GG is is farfar from having property from having property PP small random small random
sample of sample of GG will not have property will not have property PP..
•Theorem Theorem [GoldreichGoldwasserRon][GoldreichGoldwasserRon]:: Let Let GG be a graph in be a graph in which each subset of size which each subset of size nn induces at least induces at least nn22
edges. W.h.p. a random subset edges. W.h.p. a random subset RR of size of size ss//44 will not will not have IS of size have IS of size s s (R) > 1/(R) > 1/..
•TheoremTheorem [AlonKrivelevich]:[AlonKrivelevich]: Let Let GG be a graph in which at be a graph in which at least least nn22 edges need to be removed in order to color edges need to be removed in order to color GG with with 1/1/ colors. W.h.p. a random subset colors. W.h.p. a random subset RR of size of size ss1/1/22 will satisfy will satisfy (R) > 1/(R) > 1/..
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Naïve approach Naïve approach
•Theorem Theorem [GoldreichGoldwasserRon][GoldreichGoldwasserRon]:: Let Let GG be a graph in be a graph in which each subset of size which each subset of size nn induces at least induces at least nn22
edges. W.h.p. a random subset edges. W.h.p. a random subset RR of size of size ss//44 will not will not have IS of size have IS of size s s (R) > 1/(R) > 1/..
•Our case Our case GGdd satisfies satisfies nn22==44|E|.|E|.
•ss//44 does not suffice (for our proof).does not suffice (for our proof).
•Will yield graphs which are vector Will yield graphs which are vector 33-colorable -colorable and have chromatic number and have chromatic number 0.038 0.038 ((<<<< 1/31/3).).
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Naïve approach #2 Naïve approach #2
•TheoremTheorem [AlonKrivelevich]:[AlonKrivelevich]: Let Let GG be a graph in which be a graph in which at least at least nn22 edges need to be removed in order to color edges need to be removed in order to color GG with with 1/1/ colors. W.h.p. a random subset colors. W.h.p. a random subset RR of size of size ss1/1/22 will satisfy will satisfy (R) > 1/(R) > 1/..
•Can prove: Can prove: GGdd satisfies satisfies nn22==33|E|.|E|.
•ss1/1/22 does not suffice (for our proof).does not suffice (for our proof).
•Will yield graphs which are vector Will yield graphs which are vector 33-colorable and -colorable and have chromatic number have chromatic number 0.0870.087..
•Need Need ss to be much smaller ( to be much smaller (s s //)) ..
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Main theoremMain theorem
Let Let GG be a graph in which each two subsets be a graph in which each two subsets AA and and BB of size of size nn share at least share at least nn22 edges (as edges (as GGdd). ).
Theorem:Theorem: W.h.p. a random subset W.h.p. a random subset RR of size of size ss// will satisfy will satisfy (R) = (R) = (1/(1/))..
Theorem:Theorem: W.h.p. a random subset W.h.p. a random subset RR of size of size ss// will satisfy will satisfy (R) = O((R) = O(s)s)..
•Properties of our graphs Properties of our graphs GG are are stronger.stronger.
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Proof outline: Proof outline: (R) (R) = = O(O(|R|) |R|) •Use ideas appearing in [GGR,AK].Use ideas appearing in [GGR,AK].
•Let Let RR be random sample of be random sample of GG..
•Goal:Goal: Every subset of size Every subset of size |R||R|
has at least one edge.has at least one edge.
•Consider one such subset.Consider one such subset.
•Choose vertices one by one.Choose vertices one by one.
•Each vertex defines set of neighbors Each vertex defines set of neighbors
(forbidden vertices).(forbidden vertices).
•Once set of neighboors is very large, few additional Once set of neighboors is very large, few additional vertices suffice.vertices suffice.
•We show that properties of We show that properties of GG imply set imply set of neighbors grows fast of neighbors grows fast |R| is small. |R| is small.
R
xx x x xxxxxx
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Additional resultAdditional result
Applying our proof tech. (extension of [AK]) on Applying our proof tech. (extension of [AK]) on graphs graphs GG with properties as defined in [GGR] with properties as defined in [GGR] yields improved results.yields improved results.
Theorem [GGR]:Theorem [GGR]: Let Let GG be a graph in which each subset of size be a graph in which each subset of size nn induces at least induces at least nn22 edges. W.h.p. a random subset of edges. W.h.p. a random subset of
size size ss//44 will not have IS of size will not have IS of size ss..
Theorem:Theorem: Let Let GG be a graph in which each subset of size be a graph in which each subset of size nn induces at least induces at least nn22 edges. W.h.p. a random subset of size edges. W.h.p. a random subset of size
ss44//33 will not have IS of size will not have IS of size ss..
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Putting things togetherPutting things together
•GGdd: Every two subsets : Every two subsets AA and and BB of size of size nn share share nn22 edges. edges. 44|E|/n|E|/n22 44((/n)./n).
•R R G Gdd is of size is of size s s // (n/ (n/)1/)1/33 satisfies:satisfies: (R) (R) s s (R) (R) 1/ 1/..RR ( (/n)s /n)s 1/ 1/33..
(R) (R) ( (RR))1/31/3
51
Concluding remarksConcluding remarks
•Present Present tighttight bounds on the chromatic bounds on the chromatic number of vector number of vector kk-colorable graphs.-colorable graphs.
•Open problems:Open problems:•Consider stronger relaxations (strict vector Consider stronger relaxations (strict vector
coloring coloring Lovasz Lovasz function). function).•Do they improve [KMS] ?Do they improve [KMS] ?
•Do our negative results extend ?Do our negative results extend ?
•Prove similar expansion on “hypercube”.Prove similar expansion on “hypercube”.
•Further improve sample size in theorem of Further improve sample size in theorem of [GGR] (property testing framework) to [GGR] (property testing framework) to 1/1/22..Thank you.
Expansion Expansion properties of properties of
continuous graphcontinuous graph
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Main theoremMain theoremLet Let AA and and BB be two subsets of be two subsets of GGcc of measure of measure (( < 1). < 1).
Theorem:Theorem: The measure of edges between The measure of edges between AA and and BB is at least is at least 44|E|.|E|.
•Two random subsets of measure Two random subsets of measure are expected to are expected to share share 22|E||E| edges. edges.
A
B
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Symmetrization procedure Symmetrization procedure [BT][BT]
•Choose arbitrary hyperplane.Choose arbitrary hyperplane.
•Consider each point x and its Consider each point x and its
mirror image.mirror image.
•Shift up if possible.Shift up if possible.
•Procedure converges into capProcedure converges into cap..
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Proof sketchProof sketch
Definition: Definition: E(A,B) = E(A,B) = measure of edges between measure of edges between AA and and BB..
Definition:Definition: 22 = all pairs of closed subsets in = all pairs of closed subsets in SSd-1d-1 (compact (compact w.r.t. Hausdorff metric).w.r.t. Hausdorff metric).
Theorem:Theorem: If If AA and and BB are of measure are of measure then then E(A,B) ≥ E(C,C)E(A,B) ≥ E(C,C) where where CC is a cap of measure is a cap of measure ..
Proof:Proof: Let Let 22 be all pairs be all pairs ((,,))22 that satisfy that satisfy
(A) = (A) = ((), ), (B) = (B) = (().).
(A(A) ≥ ) ≥ ((), ), (B(B) ≥ ) ≥ (().).
•E(A,B) ≥ E(E(A,B) ≥ E(,,). ).
Will show that Will show that (C,C)(C,C)22..
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Proof outlineProof outline
Let Let 22 be all pairs be all pairs ((,,))22 that satisfy that satisfy
(A) = (A) = ((), ), (B) = (B) = (().).
(A(A) ≥ ) ≥ ((), ), (B(B) ≥ ) ≥ (().).
•E(A,B) ≥ E(E(A,B) ≥ E(,,). ).
Claim:Claim: (C,C) (C,C)22 : :
2 2 is closed under symmetrization.is closed under symmetrization.
2 2 is a closed subset of is a closed subset of 22..
• (C,C)(C,C)2 2 (two steps: first show(two steps: first show (C,*) (C,*) 22))..
Property testingProperty testing
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Main theoremMain theorem
Let Let GG be a graph in which each subsets be a graph in which each subsets AA and and BB of size of size nn share at least share at least nn22 edges. edges.
Theorem:Theorem: W.h.p. a random subset W.h.p. a random subset RR of size of size ss// will will satisfy satisfy (R) < 2(R) < 2s.s.
Will show:Will show:
Let Let GG be a graph in which each subset be a graph in which each subset AA size size nn induces induces at least at least nn22 edges. edges.
Theorem:Theorem: W.h.p. a random subset W.h.p. a random subset RR of size of size ss1/1/ will will satisfy satisfy (R) < 2(R) < 2s.s.
Proof used ideas from Proof used ideas from [AlonKrivelevich].[AlonKrivelevich].
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Proof outline: Proof outline: (R) < 2(R) < 2s s
•Let R be random sample of Let R be random sample of G.G.
•Goal:Goal: Every subset of size Every subset of size 22ss
has at least one edge.has at least one edge.
•Consider one such subset.Consider one such subset.
•Choose vertices one by one.Choose vertices one by one.
•Each vertex defines set of neighbors Each vertex defines set of neighbors
(forbidden vertices).(forbidden vertices).
•Once set of neighboors is very large, few additional Once set of neighboors is very large, few additional vertices suffice.vertices suffice.
•We show that properties of We show that properties of GG imply set imply set of neighbors grows fast of neighbors grows fast s is small. s is small.
R
xx x x xxxxxx
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NotationNotation
Every subset in Every subset in G G has few vertices of low degree.has few vertices of low degree.
Definition:Definition: For each subset For each subset II of of GG, let , let N(I)N(I) be the vertices be the vertices adjacent to a vertex in adjacent to a vertex in II and and F(I)F(I) be the remaining vertices. be the remaining vertices.
Definition:Definition: A vertex A vertex vv is is GOODGOOD w.r.t. a subset w.r.t. a subset II if if
• vv N(I). N(I).
• vv F(I), F(I), and and F(I) >F(I) > n, n, andand v v has at least has at least ((//)n )n neighbors inneighbors in F(I).F(I).
Lemma:Lemma: For every small For every small II, the probability that a random vertex , the probability that a random vertex in in V-IV-I is is GOODGOOD (w.r.t. (w.r.t. II) is at least ) is at least 1-1-..
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Tree of independent setsTree of independent sets
Consider the tree defined by choosing Consider the tree defined by choosing R={rR={r11,…,r,…,rss}} one by one. one by one.
Node is indexed by a couple Node is indexed by a couple (X,Y),(X,Y), where where XX and and YY are subsets of are subsets of RR..
If If XX is an IS then node is OPEN, otherwise CLOSED. is an IS then node is OPEN, otherwise CLOSED.
• Root = Root = ((,,) ) (OPEN).(OPEN).
• Inductively define:Inductively define:
•(X,Y)(X,Y) is OPEN and is OPEN and rrii RR is GOOD w.r.t. is GOOD w.r.t. XX::
•If If |Y|<s-2|Y|<s-2ss define two sons: define two sons: (X (X {r {rii},Y)},Y) and and (X, Y (X, Y {r {rii}).}).
•Otherwise define one son Otherwise define one son (X (X {r {rii},Y).},Y).
•(X,Y)(X,Y) is CLOSED, do nothing. is CLOSED, do nothing.
Properties: for each node Properties: for each node XX and and YY are disjoint, are disjoint, |Y| |Y| s-2 s-2s.s.
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Properties of treeProperties of tree
•Depth: bounded by Depth: bounded by
s - 2s - 2s +s +//..
•All leaves CLOSED All leaves CLOSED
no IS of size no IS of size 22s.s.
•For For s s 1/ 1/, w.h.p. all , w.h.p. all leaves are CLOSED.leaves are CLOSED.
…… … …
((,,))
({r({r11},},)) ((,{r,{r11})})
((,{r,{r11,r,r22})})({r({r11,r,r33},},))
(X,Y)(X,Y)
({r({r22},{r},{r11})})
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Proof sketchProof sketch
Claim: Claim: For For s s 1/ 1/, w.h.p. all leaves are , w.h.p. all leaves are CLOSED.CLOSED.
• Consider potential leaf (left right choices).Consider potential leaf (left right choices).
• Compute the probability that the path Compute the probability that the path
remains OPEN after remains OPEN after ss random vertices. random vertices.
• The path is surely CLOSED if we hit The path is surely CLOSED if we hit s-s-22s+s+// GOOD vertices out of the possible GOOD vertices out of the possible ss..
• Pr[path OPEN] Pr[path OPEN] Pr[did not hit Pr[did not hit s-2s-2s+s+//].].
• This probability is This probability is << 1/#of leaves<< 1/#of leaves if if ss is is 1/1/..
…… … …
((,,))
(X,Y)(X,Y)
