independence and chromatic number (and random k-sat): sparse case dimitris achlioptas microsoft
Post on 15-Jan-2016
217 views
TRANSCRIPT
![Page 1: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/1.jpg)
Independence and chromatic number (and random k-SAT):
Sparse Case
Dimitris AchlioptasMicrosoft
![Page 2: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/2.jpg)
Random graphs
W.h.p.: withprobability thattendsto 1 asn ! 1 .
![Page 3: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/3.jpg)
Let be the moment all vertices have degree 2¿2 ¸
Hamiltonian cycle
![Page 4: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/4.jpg)
Let be the moment all vertices have degree 2
W.h.p. has a Hamiltonian cycle
¿2 ¸
G(n;m = ¿2)
Hamiltonian cycle
![Page 5: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/5.jpg)
Hamiltonian cycle
Let be the moment all vertices have degree 2
W.h.p. has a Hamiltonian cycle
W.h.p. it can be found in time
¿2 ¸
G(n;m = ¿2)
O(n3 logn)
[Ajtai, Komlós, Szemerédi 85] [Bollobás, Fenner, Frieze 87]
![Page 6: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/6.jpg)
Let be the moment all vertices have degree 2
W.h.p. has a Hamiltonian cycle
W.h.p. it can be found in time
¿2 ¸
G(n;m = ¿2)
O(n3 logn)
[Ajtai, Komlós, Szemerédi 85] [Bollobás, Fenner, Frieze 87]
In Hamiltonicity can be decided in G(n;1=2) O(n)expected time.
[Gurevich, Shelah 84]
Hamiltonian cycle
![Page 7: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/7.jpg)
Cliques in random graphs
The largest clique in has size
[Bollobás, Erdős 75] [Matula 76]
2log2 n ¡ 2log2 log2 n§ 1
G(n;1=2)
![Page 8: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/8.jpg)
The largest clique in has size
No maximal clique of size <
2log2 n ¡ 2log2 log2 n§ 1
G(n;1=2)
log2 n
Cliques in random graphs
[Bollobás, Erdős 75] [Matula 76]
![Page 9: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/9.jpg)
The largest clique in has size
No maximal clique of size <
Can we find a clique of size ?
2log2 n ¡ 2log2 log2 n§ 1
G(n;1=2)
log2 n
(1+ ²) log2 n
[Karp 76]
Cliques in random graphs
[Bollobás, Erdős 75] [Matula 76]
![Page 10: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/10.jpg)
The largest clique in has size
No maximal clique of size <
Can we find a clique of size ?
2log2 n ¡ 2log2 log2 n§ 1
G(n;1=2)
log2 n
(1+ ²) log2 n
What if we “hide” a clique of size ?n1=2¡ ²
Cliques in random graphs
[Bollobás, Erdős 75] [Matula 76]
![Page 11: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/11.jpg)
Two problems for which we know much less.
Chromatic number of sparse random graphs Random k-SAT
![Page 12: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/12.jpg)
Two problems for which we know much less.
Chromatic number of sparse random graphs Random k-SAT
Canonical for random constraint satisfaction:– Binary constraints over k-ary domain
– k-ary constraints over binary domain
Studied in: AI, Math, Optimization, Physics,…
![Page 13: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/13.jpg)
A factor-graph representationof k-coloring
Each vertex is a variable with domain {1,2,…,k}.
Each edge is a constraint on two variables.
All constraints are “not-equal”.
Random graph = each constraint picks two variables at random.
Vertices
Edges
. . .
. . .
v1
e1
e2
v2
![Page 14: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/14.jpg)
SAT via factor-graphs(x12 _ x5 _ x9) ^(x34 _ x21 _ x5) ^ ¢¢¢¢¢¢̂ (x21 _ x9 _ x13)
![Page 15: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/15.jpg)
SAT via factor-graphs
Edge between x and c iff x occurs in clause c.
Edges are labeled +/- to indicate whether the literal is negated.
Constraints are “at least one literal must be satisfied”.
Random k-SAT = constraints pick k literals at random.
Clause nodes
Variable nodes
Clause nodes
. . .
. . .
xc
(x12 _ x5 _ x9) ^(x34 _ x21 _ x5) ^ ¢¢¢¢¢¢̂ (x21 _ x9 _ x13)
![Page 16: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/16.jpg)
Diluted mean-field spin glasses
Variables
Constraints
. . .
. . .
xc
Small, discrete domains: spins
Conflicting, fixed constraints: quenched disorder
Random bipartite graph: lack of geometry, mean field
Sparse: diluted
Hypergraph coloring, random XOR-SAT, error-correcting codes…
![Page 17: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/17.jpg)
Random graph coloring:
Background
![Page 18: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/18.jpg)
A trivial lower bound
For any graph, the chromatic number is at least:
Number of vertices
Size of maximum independent set
![Page 19: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/19.jpg)
A trivial lower bound
For any graph, the chromatic number is at least:
Number of vertices
Size of maximum independent set
For random graphs, use upper bound for largest independent set.
µ ¶
µns
¶£ (1¡ p)(
s2) ! 0
![Page 20: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/20.jpg)
An algorithmic upper bound
RepeatPick a random uncolored vertexAssign it the lowest allowed number (color)
Uses 2 x trivial lower bound number of colors
![Page 21: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/21.jpg)
An algorithmic upper bound
RepeatPick a random uncolored vertexAssign it the lowest allowed number (color)
Uses 2 x trivial lower bound number of colors
No algorithm is known to do better
![Page 22: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/22.jpg)
The lower bound is asymptotically tight
As d grows, can be colored using
independent sets of essentially maximum size
[Bollobás 89]
[Łuczak 91]
G(n;d=n)
![Page 23: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/23.jpg)
The lower bound is asymptotically tight
As d grows, can be colored using
independent sets of essentially maximum size
[Bollobás 89]
[Łuczak 91]
G(n;d=n)
![Page 24: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/24.jpg)
Theorem.For everyd > 0, thereexistsan integerk = k(d) such thatw.h.p.thechromaticnumberofG(n;p= d=n)
Only two possible values
is eitherk or k + 1
[Łuczak 91]
![Page 25: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/25.jpg)
Theorem.For everyd > 0, thereexistsan integerk = k(d) such thatw.h.p.thechromaticnumberofG(n;p= d=n)
“The Values”
is eitherk or k + 1
wherek is thesmallestintegers.t. d < 2k logk.
![Page 26: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/26.jpg)
Examples
If , w.h.p. the chromatic number is or .d = 7 4 5
![Page 27: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/27.jpg)
Examples
If , w.h.p. the chromatic number is or .
If , w.h.p. the chromatic number is
or
d = 7
d = 1060
377145549067226075809014239493833600551612641764765068157
3771455490672260758090142394938336005516126417647650681576
5
4 5
![Page 28: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/28.jpg)
Theorem.If (2k ¡ 1)lnk < d < 2k lnk thenw.h.p.thechromaticnumberof G(n;d=n) is k + 1.
One value
![Page 29: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/29.jpg)
Theorem.If (2k ¡ 1)lnk < d < 2k lnk thenw.h.p.thechromaticnumberof G(n;d=n) is k + 1.
If , then w.h.p. the chromatic number isd = 10100
22306189349652674971022028151567123496495382061668383284446576222083269567040304449938600863080217
One value
![Page 30: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/30.jpg)
Random k-SAT:
Background
![Page 31: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/31.jpg)
Random k-SAT
Fix a set of n variables
Among all possible k-clauses select m
uniformly and independently. Typically .
Example ( ) :
X = f x1;x2; : : : ;xng
2kµ
nk
¶
k = 3
(x12 _ x5 _ x9) ^(x34 _ x21 _ x5) ^ ¢¢¢¢¢¢̂ (x21 _ x9 _ x13)
m = rn
![Page 32: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/32.jpg)
Generating hard 3-SAT instances
[Mitchell,
Selman,
Levesque 92]
![Page 33: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/33.jpg)
Generating hard 3-SAT instances
[Mitchell,
Selman,
Levesque 92]
The critical point appears to be around r ¼4:2
![Page 34: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/34.jpg)
The satisfiability threshold conjecture
For every , there is a constant such that
limn!1
Pr[F k(n;rn) is satis¯able] =½
1 if r = rk ¡ ²0 if r = rk + ²
k ¸ 3 rk
![Page 35: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/35.jpg)
The satisfiability threshold conjecture
For every , there is a constant such that
limn!1
Pr[F k(n;rn) is satis¯able] =½
1 if r = rk ¡ ²0 if r = rk + ²
k ¸ 3 rk
For every , k ¸ 3
2k
k< rk < 2k ln2
![Page 36: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/36.jpg)
Unit-clause propagation
Repeat– Pick a random unset variable and set it to 1– While there are unit-clauses satisfy them– If a 0-clause is generated fail
![Page 37: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/37.jpg)
Unit-clause propagation
Repeat– Pick a random unset variable and set it to 1– While there are unit-clauses satisfy them– If a 0-clause is generated fail
UC finds a satisfying truth assignment if
[Chao, Franco 86]
r <2k
k
![Page 38: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/38.jpg)
The probability of satisfiability it at most
An asymptotic gapµ ¶ · µ ¶ ¸
2nµ
1¡12k
¶m
=·2
µ1¡
12k
¶ r ¸n
! 0 for r ¸ 2k ln2µ ¶ · µ ¶ ¸
2nµ
1¡12k
¶m
=·2
µ1¡
12k
¶ r ¸n
! 0 for r ¸ 2k ln2
![Page 39: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/39.jpg)
An asymptotic gap
Since mid-80s, no asymptotic progress over
2k
k< rk < 2k ln2
![Page 40: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/40.jpg)
Getting to within a factor of 2
a randomk-CNF formulawith m = rnclausesw.h.p.hasa complementarypairof satisfyingtruthassignments.
r < 2k¡ 1 ln2¡ 1 :
Theorem:For all k ¸ 3 and
![Page 41: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/41.jpg)
The trivial upper bound is the truth!
r · 2k ln2¡ k2 ¡ 1
Theorem:For all k ¸ 3, a randomk-CNF formulawithm = rn clausesis w.h.p.satis¯able if
![Page 42: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/42.jpg)
k 3 4 5 7 20 21Upper bound 4:51 10:23 21:33 87:88 726;817 1;453;635
Our lower bound 2:68 7:91 18:79 84:82 726;809 1;453;626Algorithmic lower bound 3:52 5:54 9:63 33:23 95;263 181;453
Some explicit bounds for the k-SAT threshold
![Page 43: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/43.jpg)
The second moment method
Pr[X > 0]¸E[X ]2
E[X 2]
otherwise let = 0, so that
For any non-negative r.v. X,
Proof: Let Y = 1 if X > 0, andY = 0 otherwise.By Cauchy-Schwartz,
E[X ]2 = E[X Y ]2 · E[X 2]E[Y 2] = E[X 2]Pr[X > 0] :
![Page 44: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/44.jpg)
Ideal for sums
If X = X 1+X 2+¢¢¢ then
E[X ]2 =X
i;j
E[X i ]E[X j ]
E[X 2] =X
i;j
E[X i X j ]
X X
![Page 45: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/45.jpg)
Ideal for sums
If X = X 1+X 2+¢¢¢ then
E[X ]2 =X
i;j
E[X i ]E[X j ]
E[X 2] =X
i;j
E[X i X j ]
X X
Example:
¡ ¢
The X i correspondto the¡n
q
¢potential q-cliquesin G(n;1=2)
Dominant contribution fromnon-ovelappingcliques
![Page 46: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/46.jpg)
General observations
Method works well when the Xi are like
“needles in a haystack”
![Page 47: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/47.jpg)
General observations
Method works well when the Xi are like
“needles in a haystack”
Lack of correlations rapid drop in
influence around solutions
=)
![Page 48: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/48.jpg)
General observations
Method works well when the Xi are like
“needles in a haystack”
Lack of correlations rapid drop in
influence around solutions
Algorithms get no “hints”
=)
![Page 49: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/49.jpg)
Let X be the # of satisfying truth assignments
The second moment method for random k-SAT
![Page 50: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/50.jpg)
Let X be the # of satisfying truth assignments£
For everyclause-density r > 0, thereis ¯ = ¯ (r) > 0 such that
E[X ]2
E[X 2]< (1¡ ¯ )n
The second moment method for random k-SAT
![Page 51: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/51.jpg)
Let X be the # of satisfying truth assignments
The number of satisfying truth assignments has
huge variance.
£
For everyclause-density r > 0, thereis ¯ = ¯ (r) > 0 such that
E[X ]2
E[X 2]< (1¡ ¯ )n
The second moment method for random k-SAT
![Page 52: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/52.jpg)
¾2 f0;1gn
Let X be the # of satisfying truth assignments
The number of satisfying truth assignments has
huge variance.
£
For everyclause-density r > 0, thereis ¯ = ¯ (r) > 0 such that
E[X ]2
E[X 2]< (1¡ ¯ )n
The satisfying truth assignments do not form a
“uniformly random mist” in
The second moment method for random k-SAT
![Page 53: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/53.jpg)
Let be the number of
satisfied literal occurrences in F under σ
H (¾;F )
where satisfies . (1+ °2)k¡ 1(1¡ °2) = 1° < 1
To prove 2k ln2 – k/2 - 1
![Page 54: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/54.jpg)
Let be the number of
satisfied literal occurrences in F under σ
H (¾;F )
Let be defined as
X (F ) =X
¾
1¾j=F °H (¾;F )
=X
¾
Y
c
1¾j=c °H (¾;c)
X = X (F )
where satisfies . (1+ °2)k¡ 1(1¡ °2) = 1° < 1
To prove 2k ln2 – k/2 - 1
![Page 55: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/55.jpg)
To prove 2k ln2 – k/2 - 1
Let be the number of
satisfied literal occurrences in F under σ
H (¾;F )
Let be defined as
X (F ) =X
¾
1¾j=F °H (¾;F )
=X
¾
Y
c
1¾j=c °H (¾;c)
X = X (F )
where satisfies . (1+ °2)k¡ 1(1¡ °2) = 1° < 1
![Page 56: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/56.jpg)
General functions
Given any t.a. σ and any k-clause c let
be the values of the literals in c under σ.
v = v(¾;c) 2 f¡ 1;+1gk
![Page 57: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/57.jpg)
General functions
Given any t.a. σ and any k-clause c let
be the values of the literals in c under σ.
We will study random variables of the form
where is an arbitrary functionf : f¡ 1;+1gk ! R
v = v(¾;c) 2 f¡ 1;+1gk
X =X
¾
Y
c
f (v(¾;c))
![Page 58: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/58.jpg)
X =X
¾
Y
c
f (v(¾;c))
f (v) = 1 for all v
f (v) =
(0 if v = (¡ 1; ¡ 1; : : : ;¡ 1)
1 otherwise
f (v) =
8><
>:
0 if v = (¡ 1;¡ 1; : : : ;¡ 1) orif v = (+1;+1; : : : ;+1)
1 otherwise
# of satisfying
truth assignments
2n=)
=)
=) # of “Not All Equal”
truth assignments
(NAE)
![Page 59: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/59.jpg)
X =X
¾
Y
c
f (v(¾;c))
f (v) = 1 for all v
f (v) =
(0 if v = (¡ 1; ¡ 1; : : : ;¡ 1)
1 otherwise
f (v) =
8><
>:
0 if v = (¡ 1;¡ 1; : : : ;¡ 1) orif v = (+1;+1; : : : ;+1)
1 otherwise
# of satisfying
truth assignments
# of satisfying
truth assignments
whose complement
is also satisfying
2n=)
=)
=)
![Page 60: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/60.jpg)
Overlap parameter = distance
Overlap parameter is Hamming distance
![Page 61: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/61.jpg)
Overlap parameter = distance
Overlap parameter is Hamming distance
For any f, if agree on variables¾;¿ z = n=2h i
Ehf (v(¾;c))f (v(¿;c))
i= E
£f (v(¾;c))
¤E
£f (v(¿;c))
¤
![Page 62: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/62.jpg)
Overlap parameter = distance
Overlap parameter is Hamming distance
For any f, if agree on variables¾;¿ z = n=2h i
Ehf (v(¾;c))f (v(¿;c))
i= E
£f (v(¾;c))
¤E
£f (v(¿;c))
¤
For any f , if agree on variables, let¾;¿ z = ®n
Cf (z=n) ´ Ehf (v(¾;c))f (v(¿;c))
i
![Page 63: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/63.jpg)
Independence
Contribution according to distance
![Page 64: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/64.jpg)
Entropy vs. correlation
E[X 2] = 2nnX
z=0
µnz
¶Cf (z=n)m
E[X ]2 = 2nnX
z=0
µnz
¶Cf (1=2)m
For every function f :
Recall: µ
n®n
¶=
µ1
®®(1¡ ®)1¡ ®
¶n
£ poly(n)
![Page 65: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/65.jpg)
Independence
Contribution according to distance
![Page 66: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/66.jpg)
®=z/n
![Page 67: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/67.jpg)
The importance of being balanced
An analytic condition:
C0f (1=2)= 0 ( )
X
v2f¡ 1;+1gk
f (v)v = 0=) the s.m.m. fails
![Page 68: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/68.jpg)
®=z/n
![Page 69: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/69.jpg)
7< r < 11
NAE 5-SAT
![Page 70: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/70.jpg)
NAE 5-SAT7< r < 11
![Page 71: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/71.jpg)
The importance of being balanced
An analytic condition:
C0f (1=2)= 0 ( )
X
v2f¡ 1;+1gk
f (v)v = 0=) the s.m.m. fails
![Page 72: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/72.jpg)
The importance of being balanced
C0f (1=2)= 0 ( )
X
v2f¡ 1;+1gk
f (v)v = 0
An analytic condition:
C0f (1=2)= 0 ( )
X
v2f¡ 1;+1gk
f (v)v = 0=) the s.m.m. fails
A geometric criterion:
![Page 73: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/73.jpg)
The importance of being balanced
C0f (1=2)= 0 ( )
X
v2f¡ 1;+1gk
f (v)v = 0
(+1,+1,…,+1)(-1,-1,…,-1)
Constant
SAT
Complementary
![Page 74: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/74.jpg)
Balance & Information Theory
Want to balance vectors in “optimal” way
![Page 75: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/75.jpg)
Balance & Information Theory
Want to balance vectors in “optimal” way
Information theory
maximize the entropy of the subject to
=)f (v)
f (¡ 1;¡ 1; : : : ; ¡ 1)= 0andX
v2f¡ 1;+1gk
f (v)v = 0
![Page 76: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/76.jpg)
Balance & Information Theory
Want to balance vectors in “optimal” way
Information theory
maximize the entropy of the subject to
=)f (v)
f (¡ 1;¡ 1; : : : ; ¡ 1)= 0andX
v2f¡ 1;+1gk
f (v)v = 0
Lagrange multipliers the optimal f is
for the unique that satisfies the constraints
f (v) = ° # of +1sin v
°
=)
![Page 77: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/77.jpg)
Balance & Information Theory
Want to balance vectors in “optimal” way
Information theory
maximize the entropy of the subject tof (v)
f (¡ 1;¡ 1; : : : ; ¡ 1)= 0andX
v2f¡ 1;+1gk
f (v)v = 0
(+1,+1,…,+1)(-1,-1,…,-1)
Heroic
=)
![Page 78: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/78.jpg)
Random graph coloring
![Page 79: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/79.jpg)
Threshold formulation
Theorem.A randomgraphwithn verticesandm = cn edgesis w.h.p.k-colorableif
c · k logk ¡ logk ¡ 1 :
c ¸ k logk ¡ 12 logk.
and whp non-k-colorable if
![Page 80: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/80.jpg)
Main points
Non-k-colorability:
k-colorability:
knµ
1¡1k
¶cn
! 0
µ ¶
Proof. Applysecondmomentmethod to thenumber of balanced k-coloringsof G(n;m).
Proof. The¬probability that¬there exists¬any k-coloring¬is at¬most
![Page 81: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/81.jpg)
Setup
Let Xσ be the indicator that the balanced
k-partition σ is a proper k-coloring.
We will prove that if then for all
there is a constant
such that
This implies that is k-colorable w.h.p.
X =P
¾X ¾
Theorem.A randomgraphwithn verticesandm = cn edgesis w.h.p.k-colorableif
c · k logk ¡ logk ¡ 1 :D = D(k) > 0
E[X 2] < D E[X ]2
G(n;cn)
![Page 82: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/82.jpg)
Setup
sum over all of .
For any pair of balanced k-partitions
let aijn be the # of vertices having
color i in σ and color j in τ.
¾;¿ E[X ¾X ¿ ]
¾;¿
E[X 2] =
Pr[¾and¿ areproper] =
0
@1¡2k
+X
ij
a2ij
1
A
cn
![Page 83: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/83.jpg)
Examples
When are uncorrelated, is
the flat matrix
¾;¿
1k
A
Balance is doubly-stochastic.=) A
1=k
![Page 84: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/84.jpg)
Examples
When are uncorrelated, is
the flat matrix
As align,
tends to the
identity matrix
¾;¿
1k
A
Balance is doubly-stochastic.=) A
I
¾;¿
1=kA
![Page 85: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/85.jpg)
A matrix-valued overlap
So,
X
A 2B k
µ ¶ µ X ¶
E[X 2] =X
A 2B k
µn
An
¶ µ1¡
2k
+1k2
Xa2
ij
¶cn
![Page 86: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/86.jpg)
A matrix-valued overlap
over doubly-stochastic matrices .k £ k A = (aij )
So,
which is controlled by the maximizer of
X
A 2B k
µ ¶ µ X ¶
E[X 2] =X
A 2B k
µn
An
¶ µ1¡
2k
+1k2
Xa2
ij
¶cn
X µ X ¶
¡X
aij logaij + c logµ
1¡2k
+1k2
Xa2
ij
¶
![Page 87: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/87.jpg)
X µ X ¶
¡X
aij logaij + c logµ
1¡2k
+1k2
Xa2
ij
¶
A matrix-valued overlap
over doubly-stochastic matrices .k £ k A = (aij )
So,
which is controlled by the maximizer of
X
A 2B k
µ ¶ µ X ¶
E[X 2] =X
A 2B k
µn
An
¶ µ1¡
2k
+1k2
Xa2
ij
¶cn
![Page 88: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/88.jpg)
Entropy decreases away from the flat matrix
– For small , this loss overwhelms the sum of squares gain– But for large enough ,….
The maximizer jumps instantaneously from flat, to a matrix where vertices capture the majority of mass
Proved this happens only for
¡X
i;j
aij logaij + c¢1k
X
i;j
a2ij
1=k
cc
k
c > k logk ¡ logk ¡ 1
2
![Page 89: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/89.jpg)
Entropy decreases away from the flat matrix
– For small , this loss overwhelms the sum of squares gain– But for large enough ,….
The maximizer jumps instantaneously from flat, to a matrix where vertices capture the majority of mass
Proved this happens only for
¡X
i;j
aij logaij + c¢1k
X
i;j
a2ij
1=k
c
k
c > k logk ¡ logk ¡ 1
c
2
![Page 90: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/90.jpg)
Entropy decreases away from the flat matrix
– For small , this loss overwhelms the sum of squares gain– But for large enough ,….
The maximizer jumps instantaneously from flat, to a matrix where vertices capture the majority of mass
Proved this happens only for
¡X
i;j
aij logaij + c¢1k
X
i;j
a2ij
1=k
cc
k
c > k logk ¡ logk ¡ 1
2
![Page 91: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/91.jpg)
Entropy decreases away from the flat matrix
– For small , this loss overwhelms the sum of squares gain– But for large enough ,….
The maximizer jumps instantaneously from flat, to a matrix where entries capture majority of mass
¡X
i;j
aij logaij + c¢1k
X
i;j
a2ij
1=k
k
cc
2
![Page 92: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/92.jpg)
Entropy decreases away from the flat matrix
– For small , this loss overwhelms the sum of squares gain– But for large enough ,….
The maximizer jumps instantaneously from flat, to a matrix where entries capture majority of mass
¡X
i;j
aij logaij + c¢1k
X
i;j
a2ij
1=k
k
c > k logk ¡ logk ¡ 1 This jump happens only after
cc
2
![Page 93: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/93.jpg)
Proof overview
Proof. Compare the value at the flat matrix with upper bound for everywhere else derived by:
![Page 94: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/94.jpg)
Proof overview
Proof. Compare the value at the flat matrix with upper bound for everywhere else derived by:
1. Relax to singly stochastic matrices.
![Page 95: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/95.jpg)
Proof overview
Proof. Compare the value at the flat matrix with upper bound for everywhere else derived by:
1. Relax to singly stochastic matrices.
2. Prescribe the L2 norm of each row ρi.
![Page 96: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/96.jpg)
Proof overview
Proof. Compare the value at the flat matrix with upper bound for everywhere else derived by:
1. Relax to singly stochastic matrices.
2. Prescribe the L2 norm of each row ρi.
3. Find max-entropy, f(ρi), of each row given ρi.
![Page 97: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/97.jpg)
Proof overview
Proof. Compare the value at the flat matrix with upper bound for everywhere else derived by:
1. Relax to singly stochastic matrices.
2. Prescribe the L2 norm of each row ρi.
3. Find max-entropy, f(ρi), of each row given ρi.
4. Prove that f’’’ > 0.
![Page 98: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/98.jpg)
Proof overview
Proof. Compare the value at the flat matrix with upper bound for everywhere else derived by:
1. Relax to singly stochastic matrices.
2. Prescribe the L2 norm of each row ρi.
3. Find max-entropy, f(ρi), of each row given ρi.
4. Prove that f’’’ > 0.
5. Use (4) to determine the optimal distribution of the ρi given their total ρ.
![Page 99: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/99.jpg)
Proof overview
Proof. Compare the value at the flat matrix with upper bound for everywhere else derived by:
1. Relax to singly stochastic matrices.
2. Prescribe the L2 norm of each row ρi.
3. Find max-entropy, f(ρi), of each row given ρi.
4. Prove that f’’’ > 0.
5. Use (4) to determine the optimal distribution of the ρi given their total ρ.
6. Optimize over ρ.
![Page 100: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/100.jpg)
Theorem.For everyintegerd > 0, w.h.p.thechromaticnumberof a randomd-regulargraph
is eitherk, k + 1,or k + 2
wherek is thesmallestintegers.t. d < 2k logk.
Random regular graphs
![Page 101: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/101.jpg)
¡kX
i=1
ai logai
kX
i=1
ai = 1
kX
i=1
a2i = ½
Maximize
subject to
for some 1=k< ½< 1
A vector analogue(optimizing a single row)
![Page 102: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/102.jpg)
¡kX
i=1
ai logai
kX
i=1
ai = 1
kX
i=1
a2i = ½
subject to
for some 1=k< ½< 1
k
Maximize
A vector analogue(optimizing a single row)
![Page 103: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/103.jpg)
¡kX
i=1
ai logai
kX
i=1
ai = 1
kX
i=1
a2i = ½
subject to
for some 1=k< ½< 1
For k = 3the maximizer is(x;y;y) wherex > y
Maximize
A vector analogue(optimizing a single row)
![Page 104: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/104.jpg)
¡kX
i=1
ai logai
kX
i=1
ai = 1
kX
i=1
a2i = ½
subject to
for some 1=k< ½< 1
For k = 3the maximizer is(x;y;y) wherex > y
Maximize
A vector analogue(optimizing a single row)
![Page 105: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/105.jpg)
¡kX
i=1
ai logai
kX
i=1
ai = 1
kX
i=1
a2i = ½
subject to
for some 1=k< ½< 1
For k = 3the maximizer is(x;y;y) wherex > y
Maximize
A vector analogue(optimizing a single row)
![Page 106: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/106.jpg)
¡kX
i=1
ai logai
kX
i=1
ai = 1
kX
i=1
a2i = ½
subject to
for some 1=k< ½< 1
For k = 3the maximizer is(x;y;y) wherex > y
Maximize
A vector analogue(optimizing a single row)
![Page 107: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/107.jpg)
¡kX
i=1
ai logai
kX
i=1
ai = 1
kX
i=1
a2i = ½
subject to
for some 1=k< ½< 1
For k = 3the maximizer is(x;y;y) wherex > y
For the maximizer isk > 3
(x;y;: : : ;y)
Maximize
A vector analogue(optimizing a single row)
![Page 108: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/108.jpg)
Create a composite image of an object that:
– Minimizes “empirical error” Typically, least-squares error over luminance
– Maximizes “plausibility” Typically, maximum entropy
Maximum entropy image restoration
![Page 109: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/109.jpg)
Structure of maximizer helps detect stars in astronomy
Maximum entropy image restoration
![Page 110: Independence and chromatic number (and random k-SAT): Sparse Case Dimitris Achlioptas Microsoft](https://reader035.vdocuments.us/reader035/viewer/2022062423/56649d625503460f94a45322/html5/thumbnails/110.jpg)
The End