2-to-2 Games Theorem via Expansion in the Grassmann Graph Based on joint works with Irit Dinur, Guy Kindler, Dor Minzer, Muli Safra

NP-hard Problems All equivalent to each other (in the exact version). Traveling Salesperson 3-SAT All equivalent to each other (in the exact version). P ≠ NP ≡ There is no fast (polynomial time) algorithm. Can we compute approximate solutions fast? (practice, theory, math).

NP-hard Problems How well can we approximate? Traveling Salesperson 3-SAT 7 8 but not better [Hȧstad 96] Within 1% [Arora, Mitchell 98] Focus of this talk: Hardness of approximation Amazing progress so far. Many challenges remain. Exact versus Approximate: a different ballgame.

Hardness of Approximation: Historically …… 1970 NP-hardness [Cook, Karp, Levin] 1990 PCP Theorem [Arora, Babai, Feige, Fortnow, Goldwasser] [Karloff, Lund, Motwani, Nisan, Safra, Shamir, Sudan, Szegedy] 1995- Recipe: 2-Prover-1-Round Games, [Bellare, Chan, Dinur, Feige, Goldreich, Guruswami, Hȧstad, K] Parallel Repetition, Fourier Analysis [Kindler, Moshkovitz, Mossel, O’Donnell, Raz, Regev] [Raghavendra, Safra, Samorodnitsky, Sudan, Steurer, Trevisan] 2018 2-to-2 Games Theorem [Dinur, K, Kindler, Minzer, Safra] ?? Unique Games Conjecture [K] ?? Small Set Expansion Conjecture [Raghavendra, Steurer]

PCP Theorem Gap-SAT Gap-reductions Examples ∃𝑐<1, s.t. given satisfiable 𝜙= 𝑥 1 ∨ 𝑥 2 ∨ 𝑥 3 ∧ 𝑥 1 ∨ 𝑥 4 ∨ 𝑥 2 ∧… NP-hard to satisfy ≥𝑐 fraction of clauses. Gap-reductions Use PCP Theorem as starting point for hardness of approx. results. Often challenging. [Hastad, Feige] Examples 3SAT hard to approx. within 7 8 +𝜀 Clique 𝑛 1−𝜀 , Set-Cover (1-𝜀) ln 𝑛 .

Hardness when clique size is Ω 𝑛 ? Limits Hardness of approx. 𝑛 0.998 Clique [Hastad] Given 𝐺, it is NP-hard to distinguish between two cases: YES case: 𝐺 has clique of size 𝑛 0.999 . NO case: 𝐺 has no clique of size 𝑛 0.001 . Promise formulation Equivalent 𝐺: promised to contain clique of size 𝑛 0.999 . Hard to find clique of size 𝑛 0.001 . Hardness when clique size is Ω 𝑛 ?

Gap-Clique Clique: 𝐺= 𝑉,𝐸 , 𝐶⊆𝑉 is a Clique if any 𝑢,𝑣∈𝐶 are adjacent. Promise: 𝐶𝑙𝑖𝑞𝑢𝑒 𝐺 ≥0.49𝑛 Goal: find a Clique of size 1 4 𝑛? of size 1 16 𝑛? of size 𝜀𝑛? Still seemingly hard. Related: Independent Set, Vertex Cover, Graph Coloring Towards hardness: stronger assumptions?

Unique Games Definition E.g: Max-cut Assignment Goal 1 A unique game[q]: variables 𝑋, equations Eq. Each equation of form 𝑥 𝑖 − 𝑥 𝑗 =𝑏 (𝑚𝑜𝑑 𝑞). Assignment is a mapping 𝐴:𝑋→{0,…,𝑞−1}. Satisfies the equation for 𝑖,𝑗 if 𝐴 𝑥 𝑖 −𝐴( 𝑥 𝑗 )=𝑏 Goal Find an assignment that satisfies as many equations as possible. Value = max fraction of satisfied equations by any assignment. E.g: Max-cut ∀ 𝑢,𝑣 ∈𝐸, 𝑥 𝑢 − 𝑥 𝑣 =1(𝑚𝑜𝑑 2) 1

Unique Games Conjecture [2002] Implications Evidence? ∀𝜀, ∃𝑞 s.t. given UG[q] instance -- NP-hard to distinguish between: Value ≥1−𝜀 Value ≤𝜀 Many Implications In the eyes of the beholder. Absence of evidence is not evidence of absence. Evidence? For: “basic” integrality gaps [K Vishnoi], candidate construction [K Moshkovitz]. Against: seemingly plenty.

Unique / 2-to-2 Games Conjecture Simplest Hard Problem ? Many other problems are hard to approximate. Many more ……. Max Acyclic Subgraph Unique / 2-to-2 Games Conjecture All CSPs Clique, Independent Set Max Cut Algorithms, Optimization. Computational complexity. (Boolean function) Analysis, Geometry.

2-to-2 Games Definition Conjecture Theorem Evidence towards UGC Similar to Unique-Games. Equations of the form 𝑥 𝑖 − 𝑥 𝑗 ∈{𝑏, 𝑏 ′ }(𝑚𝑜𝑑 𝑞). Conjecture For every 𝜀>0, NP-hard to distinguish between: Value = 1 Value ≤𝜀 [K Minzer Safra, Dinur K Kindler MS, DKKMS, KMS] Theorem Value ≥1−𝜀 Evidence towards UGC [K, KS, Barak Kothari Steurer, KM Moshkovitz S] ∀𝜀>0, given a 2-to-2 Game it is NP-hard to distinguish: ⇒𝑈𝐺[ 1 2 −𝜀,𝜀] is NP-hard

Implications: New NP-Hardness Results Games 2-to-2-Games 1−𝜀,𝜀 , Unique-Games 1 2 −𝜀,𝜀 . Clique, Graph Coloring, Max-Cut Clique 1− 1 2 −𝜀,𝜀  Vertex-Cover 1 2 +𝜀,1−𝜀 . [Dinur Safra, K ] Coloring (almost) 4-colorable graphs with O(1) colors [Dinur Mossel Regev]. Max-Cut-Gain: Max-Cut 1 2 +𝜀, 1 2 +Ω 𝜀 log 1 𝜀 [K O’Donnell]. Intermediate CSP [Barak] Under standard complexity assumption, ∃𝛼<𝛽<1 such that UG 0.49,0.01 is solved in time 2 𝑛 𝛽 [Arora Barak Steurer], but not in time 2 𝑛 𝛼 .

“Evidence” Against Unique Games Conjecture No known distribution over hard instances. In contrast to 3SAT, Factoring, Clique( n1/3 ). No known counter-example to Sum-of-Squares algorithm. Strengthening of Lovasz Θ-function, Goemans-Williamson Semi-Definite Program. “Common” techniques “cannot” construct a counter-example. [ Arora Barak Brandão Harrow Kelner Kolla Makarychev2 Steurer Zhou ]

“Evidence” Against Unique Games Conjecture Improve Arora-Barak-Steurer algorithm? 2 𝑛 0.01 to 2 𝑛 1 log log log 𝑛 ? n – variable constraint satisfaction problem with intermediate complexity [ 2 𝑛 1/3 , 2 𝑛 1/2 ] would be counter-intuitive. 2-to-2 Games Conjecture correct nevertheless!

Cart before the Horse Run the reduction: 3SAT → 2-to-2 Games. Distribution over hard instances. Counter-example to Sum-of-Squares algorithm. “Logically” should precede NP-hardness reduction, but happens the other way around . [Barak, Windows on Theory] One way to resolve this conundrum is that while the unique games problem may well be hard in the worst case, it is extremely hard to come up with actual hard instances for it. ….. Trump …… While it is theoretically still possible for the unique games conjecture to be false (as I personally believed would be the case until this latest sequence of results) the most likely scenario is now that the UGC is true, ………

PCP’s and Codes PCP: how gap NP-hardness is obtained The core of a typical PCP contains: An error-correcting code (RM, Hadamard, Long-code) A query-efficient test: Query t entries in given word F. Accept/Reject Pr[Accept]>δ ⟹ F “close” to codeword From test to PCP? not clear. But, Properties of test determine gap-problem shown to be hard.

PCP’s and codes PCP: how gap NP-hardness is obtained The core of a typical PCP contains: An error-correcting code (RM, Hadamard, Long-code) A query-efficient test: Query t entries in given word F. Accept/Reject Pr[Accept]>δ ⟹ F “close” to codeword From test to PCP? not clear. But, Properties of test determine gap-problem shown to be hard. To get hardness for 2-to-2 Games, we need a new code and a new test: The Grassmann code/test

2-to-2 Games Hardness (~ A + B pages) Smooth Parallel Repetition Gap-Linear-LabelCover Encoding by Grassmann Code PCP Theorem Gap-3SAT Parallel Repetition Gap-LablelCover Subcode covering Proof complexity of random 3LIN(2) Encoding by Long-code Gap-3LIN(2) Zoom (decoding from next-to-nothing) Expansion of Grassmann graph Hardness of 2-to-2 Games

The Grassmann Graph 𝐺(𝑉,ℓ) Definition V is a linear space over F2, dimension 𝑘, 1 ≪ℓ≪𝑘. Vertices: {𝐿⊆𝑉 | dim 𝐿 =ℓ}. Edges : { 𝐿, 𝐿 ′ | dim 𝐿∩ 𝐿 ′ =ℓ−1} . Next Grassmann Code + Grassmann Test Expansion in the Grassmann Graph.

Grassmann Test Grassmann-code 2-to-2 Test Local to Global Encoding of linear 𝑓:𝑉→ 𝐹 2 is G 𝐿 =𝑓 ​ 𝐿 . 2-to-2 Test Agreeing assignments align in pairs – according to restrictions to 𝐿∩ 𝐿 ′ . Any function on 𝐿∩𝐿′ has two extension to 𝐿 and to 𝐿′ Local to Global Completeness: codeword G passes test w.p. 1. Soundness: If G passes >𝜀 of the tests, then G corresponds to a global 𝑓:𝑉→ 𝐹 2 ? Not in the most obvious sense… Test: Each edge - local consistency check: edge 𝐿, 𝐿 ′ tests whether G 𝐿 , G 𝐿′ agree on 𝐿∩ 𝐿 ′ . 𝐿′ 𝐿 𝐿∩𝐿′

Graph Expansion G=(V,E) d-regular Definition Expander In Grassmann The expansion of 𝑆⊆𝑉 is Φ 𝑆 = |𝐸 𝑆, 𝑆 | 𝑑 𝑆 . 𝐸 𝑆, 𝑆 is the set of edges from S to its complement. Expander Usually: Φ 𝑆 ≥0.2 for all 𝑆 ≤ 𝑛 2 , often constant degree. For us: concern is: expansion ≈1 for small sets, e.g. 𝑆 = 𝑛 log 𝑛 In Grassmann Fact: A random small set 𝑆⊆𝐺(𝑉,ℓ) has expansion 1 – o(1). S Are there sets with Φ 𝑆 ≤1−𝛿 ?

Yes! Subgraphs that are Grassmann-- non expanding 𝐿′ For any 𝑥∈𝑉∖{0}, consider 𝑆 𝑥 ={𝐿⊆𝑉:𝑥∈𝐿}. Φ 𝑆 𝑥 ≈ 1 2 : for all 𝐿∈ 𝑆 𝑥 , a random neighbor 𝐿′ in S 𝑥 w.p.≈1/2. Example: zoom-in Take 𝑊⊆𝑉 of co-dimension 1. 𝑆 𝑊 ={𝐿⊆𝑉:𝐿⊆𝑊}. Φ 𝑆 𝑊 ≈ 1 2 : ∀𝐿∈ 𝑆 𝑊 , a random neighbor 𝐿 ′ is in 𝑊 w.p≈1/2. Dual example: zoom-out Induced subgraph is isomorphic to smaller Grassmann Graph. “zooming” into spaces contain ing-𝑥 / ed-in-𝑊 gives constant density Note 𝐿 𝑥∈𝐿∩𝐿′

Expansion statement 𝑆⊆𝐺 𝑉,ℓ is (𝑟,𝜀)-pseudorandom if density of 𝑆 after every zoom-in/out combination of dimension 𝑟 remains ≤𝜀. Definition Random small set is pseudorandom. Previous two examples: not (1, 0.5)-pseudorandom. Note If 𝑆⊆𝐺 𝑉,ℓ has expansion ≤ 1−𝛿, then S is not-pseudorandom. Expansion Statement: I.e. ∃zoom in/out, increase density significantly

[DKKMS]: Statement suffices to prove 2-to-2 Theorem If 𝑆⊆𝐺 𝑉,ℓ is pseudorandom, then near-perfect expansion. Expansion Statement (contra-positive) [proposed in DKKMS]: [DKKMS] hoped studying this question would help analyze “local to global” Grassmann Test. Turns out we were right (and blind). Motivation? Expansion Statement → “local to global” test works! Theorem [Barak Kothari Steurer] [DKKMS]: Statement suffices to prove 2-to-2 Theorem

Results Expansion Statement [KMS] 𝑆⊆𝐺 𝑉,ℓ is (𝑟,𝜀)-pseudorandom if density of 𝑆 after every zoom-in/out combination of dimension 𝑟 remains ≤𝜀. Definition ∀ 𝛿 ∃ 𝑟, 𝜀 s.t. if 𝑆 𝑖𝑠 (𝑟,𝜀)-pseudorandom, then Φ 𝑆 ≥1−𝛿. Expansion Statement [KMS] If S is 1, 𝜀 − pseudorandom, then Φ 𝑆 ≥ 3 4 −𝑂( 𝜀 ) If S is 2, 𝜀 − pseudorandom, then Φ 𝑆 ≥ 7 8 −𝑂( 𝜀 1/3 ) Earlier [DKKMS]

Tools in the proof Lots of Fourier Analysis. Ingredients: 𝑓: −1,1 𝑛 →𝑅 can be represented as 𝑓 𝑥 = 𝑆⊆[𝑛] 𝑓 𝑆 𝜒 𝑆 (𝑥) . 𝜒 𝑆 𝑥 = 𝑖∈𝑆 𝑥 𝑖 monomial basis. On the Boolean hypercube: No such canonical basis… need to use block decomposition. Or on different but related graph… Fourier analysis on Grassmann

Refuting UG must exploit near perfect completeness. Summary On Grassmann: Pseudorandomness ⇒ near-perfect expansion. Also on Johnson Graphs [KM Moshkovitz S] Theorems 2-to-2 Games, 𝑈𝐺[ 1 2 −𝜀,𝜀] NP-hard New NP-hardness for Clique, Vertex Cover, (almost) Graph Coloring, Max-Cut Mysteries finally unraveling. Implications Greater mysteries ahead. Future research Refuting UG must exploit near perfect completeness.