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Products
of Functions, Graphs, Games & Problems
Irit DinurWeizmann
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Products
• For fun: to “see what happens”• For “Hardness Amplification”
(holy grail = prove that things are hard)
Why would anyone want to multiply two functions ?graphs ?
problems ?
Given f that is a little hard construct f’ that is very hard
Circuit complexity, average case complexity, communication complexity,Hardness of approximation
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Products
• For fun: to “see what happens”• For “Hardness Amplification”
(holy grail = prove that things are hard)
Why would anyone want to multiply two functions ?graphs ?
problems ?
Given f that is a little hard construct f’ that is very hard
Circuit complexity, average case complexity, communication complexity,Hardness of approximation By taking
f’ = f x f x … x f
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P1 x P2
NumbersStringsFunctionsGraphsGamesComputational Problems
We can multiply many different objects
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1
1
0
1
1
0
0 1 1 0 1 1
10 11 11 10 11 11
10 11 11 10 11 11
00 01 01 00 01 01
10 11 11 10 11 11
10 11 11 10 11 11
00 01 01 00 01 01
For example, here is how to multiply two strings:
Direct Products of Strings / Functions
In the k-fold product of a string , for each we have a -bit substring corresponding to the restriction of to :
…
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For example, here is how to multiply two strings:
Direct Products of Strings / Functions
In the k-fold product of a string , for each we have a -bit substring corresponding to the restriction of to :
… …
sum
(the alphabet stays the same, but harder to analyze)
1
1
0
1
1
0
0 1 1 0 1 1
10 11 11 10 11 11
10 11 11 10 11 11
00 01 01 00 01 01
10 11 11 10 11 11
10 11 11 10 11 11
00 01 01 00 01 01
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Given a table of -substrings, , is there a local test that distinguishes between• is a direct product • is far from a direct product
In [GGR] terms: is the property of being a direct product locally testable ?
(answer: yes, with 2 queries)Testing Direct Products
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Given: a very large and difficult problem (e.g. 3sat)
On average, the local value is > On average, consistent with > fraction of neighbors
Question: is there a consistent global solution with value >
Local to Global
-sub-problem
We will solve it together, by splitting the work into many small sub-problems, each of (constant) size
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Testing Direct Products
Theorem [D.-Steurer 2013]Any collection of local solutions with pairwise consistency must be consistent with a global solution.
i.e. the property of being a direct product is testable with 2 queries.
[Goldreich-Safra,D.-Reingold, D.-Goldenberg, Impagliazzo-Kabanets-Wigderson]
k-substring
Theorem [David-D.-Goldenberg-Kindler-Shinkar 2013]The property of being a direct sum is testable with 3 queries.
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There are several natural graph productsIn the “strong direct product”:
1
2
3
1 2 3
u1u2 ~ v1v2 iff u1~v1 and u2 ~ v2
Multiplying Graphs
( u ~ v means u=v or u is adjacent to v )
11
21
31
12
22
32
13
23
33
V(G1 x G2) = V(G1) x V(G2)
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Basic question: how do natural graph properties (such as: chromatic number, max-clique, expansion, …) Behave wrt the product operation
If clique ( G1 ) = m1 and clique ( G2 ) = m2
then clique ( G1 x G2 ) = m1m2
Generally, the answer is easy if the maximizing solution is itself a product,but often this is not true. Then, the analysis is challenging
Multiplying Graphs
If independent-set ( G1 ) = m1 and independent-set ( G2 ) = m2
then independent-set ( G1 x G2 ) = ?
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Definition : The Shannon capacity of G is the limit of ( a(Gk) )1/k as k infty [Shannon 1956]
Lovasz 1979 computed the Shannon capacity of several graphs, e.g. C5, by introducing the theta function
C7 is still open – (one of the most notorious problems in extremal
combinatorics)
Consider a transmission scheme of one symbol at a time, and draw a graph with an edge between each pair of symbols that might be confusable in transmission.
a(G) = number of symbols transmittable with zero errora(Gk) = set of such words of length k(a(Gk))1/k = effective alphabet size
a(G) – stands for maximum independent set
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Multiplying Games
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Games (2-player 1-round)
uAlice
U
…
v
Bob
V
…
Alice Bob
Referee: random u v
u v
A(u) B(v)
𝐴 :𝑈 Σ 𝐵 :𝑉 Σ𝜋𝑢𝑣 :ΣΣ
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Games (2-player 1-round)
𝐴 :𝑈 Σ 𝐵 :𝑉 ΣuAlice
U
…
v
Bob
V
…
Value ( G ) = maximal success probability, over all possible strategies
𝜋𝑢𝑣 :ΣΣ
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𝜋𝑢𝑣 : [7 ] [2 ]U = set of variables V = set of 3sat clauses
uAlice
U
…
v
Bob
V
…
Label-Cover Problem : Given a game G, find value ( G )
Value ( G ) = maximal success probability, over all possible strategies
Strong PCP Theorem: Label Cover is NP-hard to approximate[AS, ALMSS 1991] + [Raz 1995]
FGLSS
Games (2-player 1-round) The 3SAT game
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PCP theorem: “gap-3SAT is NP-hard”Proof: By reduction from small gap to large gap,
aka amplification
Start with and end up with , s.t.If then
How? • by algebraic encoding [AS, ALMSS 1991]; or• by “multiplying” with itself,
repeatedly [D. 2007]
The PCP Theorem [AS, ALMSS]
If then
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Multiplying Games A game is specified by its constraint-graph,so a product of two games can be defined by a product of two constraint graphs
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X =𝐺1 𝐺2
𝐺1⊗𝐺2
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X =
𝐺1⊗𝐺2u1
U1
…
v1
V1
…Π1 : Σ
1 Σ1
u2
U2
…
v2
V2
…Π2 : Σ
2 Σ2
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X =
A : U1 x U2 Σ1 x Σ2
Alice Bob
B : V1 x V2 Σ1 x Σ2
u1
U1
…
v1
V1
…Π1 : Σ
1 Σ1
u2
U2
…
v2
V2
…Π2 : Σ
2 Σ2
u1u2
U1 x U2
…v1v2
V1 x V2
…Π1 Π
2
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u1u2…uk
U x … x U
…
v1v2…vk
V x … x V
…A : Uk Σk
Alice Bob
B : Vk ΣkΠ
1 Π2 … Π
k
k-fold product of a game
Also called: the k-fold parallel repetition of a game
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Q1: If and
then what is ?
Q2: If , then what is for ?
One obvious candidate is the direct product strategy.
But it is not, in general, the best strategy.
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If thenTheorem [D.-Steurer 2013]: Let be a projection game.
𝑣𝑎𝑙 (𝐺⊗𝑘 )≤( 2√ 𝜌1+𝜌 )𝑘/2
If (close to 1), then (known; we just improve the constants of [ Rao, Holenstein, Raz ])
If (close to 0), then (new; implies new hardness results for label-cover & optimal NP-hardness results for set-cover)
Also: short proof for “strong PCP theorem” or “hardness of label-cover”
Ideas extend to give a parallel repetition theorem for entangled games, i.e. when the two players share a quantum state [with Vidick & Steurer]
BGLR “sliding scale”conjecture
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val+¿ (G )≔¿𝐻 ¿|𝐺⊗𝐻|∨ ¿
||𝐻|| ¿¿
Think of as an “environmental value” of : how much harder is it to play in parallel with environment , compared to playing alone
( is the collision value of , closely related to )
Multiplicativity: Approximation:
One slide about the new proof
Approximation is proven by expressing as an “eigenvalue”, enabled by factoring out H; easy for expanders
2. Define:
3. Show:
1. View a game as a linear operator acting on (Bob)-assignments
So:
The game value a natural norm of this operator
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Summary• Direct product of strings & functions
and a related local-to-global lifting theorem
• Direct product of gamesand new parallel repetition theorem
• Direct products of computational problems ??e.g. for graph problems (max-cut, vertex-cover, ... )