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New hardness results for graph and hypergraph colorings Joshua Brakensiek , Venkatesan Guruswami Carnegie Mellon University CCC 2016 Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings Graph Coloring

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New hardness results for graph and hypergraph colorings

Joshua Brakensiek, Venkatesan Guruswami Carnegie Mellon University CCC 2016

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

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Graph Coloring

Source: Wikipedia

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

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Graph Coloring

Source: Wikipedia

Theorem (Karp, 1972) Determining if a graph can be colored with t colors is NP-complete when t ≥ 3.

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

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Graph Coloring

Source: Wikipedia

Theorem (Karp, 1972) Determining if a graph can be colored with t colors is NP-complete when t ≥ 3. If given that the graph is t-colorable, NP-hard to find t-coloring.

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

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Approximate Graph Coloring

Question (Approximate Coloring–search version) Can a t-colorable graph with n vertices be efficiently colored with c(n)-colors, where c(n) ≥ t ≥ 3?

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

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Approximate Graph Coloring

Question (Approximate Coloring–search version) Can a t-colorable graph with n vertices be efficiently colored with c(n)-colors, where c(n) ≥ t ≥ 3?

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

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Approximate Graph Coloring

Question (Approximate Coloring–search version) Can a t-colorable graph with n vertices be efficiently colored with c(n)-colors, where c(n) ≥ t ≥ 3?

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

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Approximate Graph Coloring

Question (Approximate Coloring–search version) Can a t-colorable graph with n vertices be efficiently colored with c(n)-colors, where c(n) ≥ t ≥ 3? Question (Approximate Coloring–decision version) Given a graph G on n vertices, Output YES if G can be colored with t colors, Output NO if G cannot be colored with c(n) colors.

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

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Approximate Graph Coloring

Question (Approximate Coloring) Can a t-colorable graph with n vertices be efficiently colored with c(n)-colors, where c(n) ≥ t ≥ 3?

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

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Approximate Graph Coloring

Question (Approximate Coloring) Can a t-colorable graph with n vertices be efficiently colored with c(n)-colors, where c(n) ≥ t ≥ 3? In P, t = 3, c = o(n1/5) (Kawarabayashi, Thorup, 2014).

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

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Approximate Graph Coloring

Question (Approximate Coloring) Can a t-colorable graph with n vertices be efficiently colored with c(n)-colors, where c(n) ≥ t ≥ 3? In P, t = 3, c = o(n1/5) (Kawarabayashi, Thorup, 2014). NP-hard

t = 3, c = 4. (Khanna, Linial, Safra, 1993; Guruswami, Khanna, 2000)

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

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Approximate Graph Coloring

Question (Approximate Coloring) Can a t-colorable graph with n vertices be efficiently colored with c(n)-colors, where c(n) ≥ t ≥ 3? In P, t = 3, c = o(n1/5) (Kawarabayashi, Thorup, 2014). NP-hard

t = 3, c = 4. (Khanna, Linial, Safra, 1993; Guruswami, Khanna, 2000) t ≥ 3 (small), c = max(2t − 5, ⌊5t/3⌋ − 1). (KLS, 1993; Garey and Johnson, 1976)

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

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Approximate Graph Coloring

Question (Approximate Coloring) Can a t-colorable graph with n vertices be efficiently colored with c(n)-colors, where c(n) ≥ t ≥ 3? In P, t = 3, c = o(n1/5) (Kawarabayashi, Thorup, 2014). NP-hard

t = 3, c = 4. (Khanna, Linial, Safra, 1993; Guruswami, Khanna, 2000) t ≥ 3 (small), c = max(2t − 5, ⌊5t/3⌋ − 1). (KLS, 1993; Garey and Johnson, 1976) t large, c = exp(Ω(t1/3)) (Huang, 2013)

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

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Approximate Graph Coloring

Question (Approximate Coloring) Can a t-colorable graph with n vertices be efficiently colored with c(n)-colors, where c(n) ≥ t ≥ 3? In P, t = 3, c = o(n1/5) (Kawarabayashi, Thorup, 2014). NP-hard

t = 3, c = 4. (Khanna, Linial, Safra, 1993; Guruswami, Khanna, 2000) t ≥ 3 (small), c = max(2t − 5, ⌊5t/3⌋ − 1). (KLS, 1993; Garey and Johnson, 1976) t large, c = exp(Ω(t1/3)) (Huang, 2013)

Theorem (Brakensiek, Guruswami) NP-hard to color a t = 4-colorable graph with c = 6 colors.

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

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Approximate Graph Coloring

Question (Approximate Coloring) Can a t-colorable graph with n vertices be efficiently colored with c(n)-colors, where c(n) ≥ t ≥ 3? In P, t = 3, c = o(n1/5) (Kawarabayashi, Thorup, 2014). NP-hard

t = 3, c = 4. (Khanna, Linial, Safra, 1993; Guruswami, Khanna, 2000) t ≥ 3 (small), c = max(2t − 5, ⌊5t/3⌋ − 1). (KLS, 1993; Garey and Johnson, 1976) t large, c = exp(Ω(t1/3)) (Huang, 2013)

Theorem (Brakensiek, Guruswami) NP-hard to color a t = 4-colorable graph with c = 6 colors. Result generalizes to c = 2t − 2.

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

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Hypergraph coloring

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

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Hypergraph coloring

c-colorability Hypergraph is c-colorable if each hyperedge is not monochromatic.

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

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Hypergraph coloring

c-colorability Hypergraph is c-colorable if each hyperedge is not monochromatic. k-uniformity Hypergraph is k-uniform if each hyperedge has exactly k vertices.

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

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Hypergraph coloring

c-colorability Hypergraph is c-colorable if each hyperedge is not monochromatic. k-uniformity Hypergraph is k-uniform if each hyperedge has exactly k vertices. Question (Approximate Hypergraph Coloring) Given a k-uniform 2-colorable hypergraph, can we efficiently color it with c colors so no hyperedge is monochromatic?

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

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Hypergraph coloring

Algorithms (k = 3)

c = ˜ O(n2/9) (Alon, Kelsen, Mahajan, Ramesh, 1996; Chen, Frieze, 1996) c = ˜ O(n9/41) (Krivelevich, 1997) c = ˜ O(n1/5) (Krivelevich, Nathaniel, Sudakov, 2000)

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

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Hypergraph coloring

Algorithms (k = 3)

c = ˜ O(n2/9) (Alon, Kelsen, Mahajan, Ramesh, 1996; Chen, Frieze, 1996) c = ˜ O(n9/41) (Krivelevich, 1997) c = ˜ O(n1/5) (Krivelevich, Nathaniel, Sudakov, 2000)

NP-hardness

k ≥ 4, c ≥ 2 (Guruswami, H˚ astad, Sudan, 2002) k = 3, c ≥ 2 (Dinur, Regev, Smyth, 2005) k = 8, c = 2(log n)1/10 (Huang, 2014)

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

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Hypergraph coloring

Algorithms (k = 3)

c = ˜ O(n2/9) (Alon, Kelsen, Mahajan, Ramesh, 1996; Chen, Frieze, 1996) c = ˜ O(n9/41) (Krivelevich, 1997) c = ˜ O(n1/5) (Krivelevich, Nathaniel, Sudakov, 2000)

NP-hardness

k ≥ 4, c ≥ 2 (Guruswami, H˚ astad, Sudan, 2002) k = 3, c ≥ 2 (Dinur, Regev, Smyth, 2005) k = 8, c = 2(log n)1/10 (Huang, 2014)

Since strong hardness results are known, we consider a problem with an even stronger promise.

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

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Strong hypergraph coloring

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

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Strong hypergraph coloring

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

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Strong hypergraph coloring

Hypergraph is t-partite if the vertices partition into t sets such that each hyperedge has at most one vertex in each set.

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

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Strong hypergraph coloring

Hypergraph is t-partite if the vertices partition into t sets such that each hyperedge has at most one vertex in each set. Question (Strong hypergraph coloring) Given a k-uniform t-partite hypergraph (t ≤ 2k − 2), can we efficiently color it with c colors so no hyperedge is monochromatic?

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

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Strong hypergraph coloring

Hypergraph is t-partite if the vertices partition into t sets such that each hyperedge has at most one vertex in each set. Question (Strong hypergraph coloring) Given a k-uniform t-partite hypergraph (t ≤ 2k − 2), can we efficiently color it with c colors so no hyperedge is monochromatic? Not told t-partite structure beforehand.

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

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Strong hypergraph coloring

Theorem (McDiarmid, 1993) A k-uniform k-partite hypergraph, can be colored in polynomial time with 2 colors.

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

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Strong hypergraph coloring

Theorem (McDiarmid, 1993) A k-uniform k-partite hypergraph, can be colored in polynomial time with 2 colors. What if the graph is k + 1-partite?

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

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Strong hypergraph coloring

Theorem (McDiarmid, 1993) A k-uniform k-partite hypergraph, can be colored in polynomial time with 2 colors. What if the graph is k + 1-partite? Conjecture It is NP-hard to color a k-uniform k + 1-partite hypergraph with c colors for all c ≥ 2.

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

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Strong hypergraph coloring

Theorem (McDiarmid, 1993) A k-uniform k-partite hypergraph, can be colored in polynomial time with 2 colors. What if the graph is k + 1-partite? Conjecture It is NP-hard to color a k-uniform k + 1-partite hypergraph with c colors for all c ≥ 2. If k = 2, this captures conjectures on graph coloring.

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

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Strong hypergraph coloring

Theorem (McDiarmid, 1993) A k-uniform k-partite hypergraph, can be colored in polynomial time with 2 colors. What if the graph is k + 1-partite? Conjecture It is NP-hard to color a k-uniform k + 1-partite hypergraph with c colors for all c ≥ 2. If k = 2, this captures conjectures on graph coloring. Theorem (Brakensiek, Guruswami) NP-hard to color a k-uniform, t-partite hypergraph with 2 colors when t = ⌈ 3k

2 ⌉.

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

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Proof Outline

Our approach adapts (Austrin, Guruswami, H˚ astad, FOCS 2014)

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

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Proof Outline

Our approach adapts (Austrin, Guruswami, H˚ astad, FOCS 2014)

1 Dictatorship test: weak polymorphism gadget. Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

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Proof Outline

Our approach adapts (Austrin, Guruswami, H˚ astad, FOCS 2014)

1 Dictatorship test: weak polymorphism gadget.

Motivated by theory of constraint satisfaction problems.

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

SLIDE 36

Proof Outline

Our approach adapts (Austrin, Guruswami, H˚ astad, FOCS 2014)

1 Dictatorship test: weak polymorphism gadget.

Motivated by theory of constraint satisfaction problems. Graph with ‘structured’ colorings.

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

SLIDE 37

Proof Outline

Our approach adapts (Austrin, Guruswami, H˚ astad, FOCS 2014)

1 Dictatorship test: weak polymorphism gadget.

Motivated by theory of constraint satisfaction problems. Graph with ‘structured’ colorings.

2 Label cover reduction Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

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Proof Outline

Our approach adapts (Austrin, Guruswami, H˚ astad, FOCS 2014)

1 Dictatorship test: weak polymorphism gadget.

Motivated by theory of constraint satisfaction problems. Graph with ‘structured’ colorings.

2 Label cover reduction

Often used in hardness of approximation reductions.

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

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Gadget Motivation: CSP theory

Consists of variables V and constraints R1, . . . , Rk over domain D.

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

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Gadget Motivation: CSP theory

Consists of variables V and constraints R1, . . . , Rk over domain D. Goal: find labeling σ : V → D satisfying all the constraints.

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

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Gadget Motivation: CSP theory

Consists of variables V and constraints R1, . . . , Rk over domain D. Goal: find labeling σ : V → D satisfying all the constraints. Polymorphism f : DL → D such that if σ1, . . . , σL satisfy constraints then so does σ(x) = f (σ1(x), . . . , σL(x)).

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

SLIDE 42

Gadget Motivation: CSP theory

Consists of variables V and constraints R1, . . . , Rk over domain D. Goal: find labeling σ : V → D satisfying all the constraints. Polymorphism f : DL → D such that if σ1, . . . , σL satisfy constraints then so does σ(x) = f (σ1(x), . . . , σL(x)). Example: Linear equations over F2, f (x1, x2, x3) = x1 ⊕ x2 ⊕ x3.

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

SLIDE 43

Gadget Motivation: CSP theory

Consists of variables V and constraints R1, . . . , Rk over domain D. Goal: find labeling σ : V → D satisfying all the constraints. Polymorphism f : DL → D such that if σ1, . . . , σL satisfy constraints then so does σ(x) = f (σ1(x), . . . , σL(x)). Example: Linear equations over F2, f (x1, x2, x3) = x1 ⊕ x2 ⊕ x3. Fact (Galois correspondence) CSPs with the same polymorphisms have the same complexity!

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

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Gadget Motivation: CSP theory

Consists of variables V and constraints R1, . . . , Rk over domain D. Goal: find labeling σ : V → D satisfying all the constraints. Polymorphism f : DL → D such that if σ1, . . . , σL satisfy constraints then so does σ(x) = f (σ1(x), . . . , σL(x)). Example: Linear equations over F2, f (x1, x2, x3) = x1 ⊕ x2 ⊕ x3. Fact (Galois correspondence) CSPs with the same polymorphisms have the same complexity! For promise problems, need weak polymorphisms (Austrin, Guruswami, H˚ astad; 2014)

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

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Gadget: Weak Polymorphisms

f

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

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Graph Coloring: Dictatorship Gadget

Recall: Show it is NP-hard to c-color t-colorable graphs.

f

f : Z2

3 → Z4

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

SLIDE 47

Graph Coloring: Dictatorship Gadget

Recall: Show it is NP-hard to c-color t-colorable graphs.

f

f : Z2

3 → Z4

Dictatorship Gadget Graph on vertex set ZL

t = {0, 1, . . . , t − 1}L.

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

SLIDE 48

Graph Coloring: Dictatorship Gadget

Recall: Show it is NP-hard to c-color t-colorable graphs.

f

f : Z2

3 → Z4

Dictatorship Gadget Graph on vertex set ZL

t = {0, 1, . . . , t − 1}L.

x, y ∈ ZL

t connected iff xi = yi for all i.

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

SLIDE 49

Graph Coloring: Dictatorship Gadget

Recall: Show it is NP-hard to c-color t-colorable graphs.

f

f : Z2

3 → Z4

Dictatorship Gadget Graph on vertex set ZL

t = {0, 1, . . . , t − 1}L.

x, y ∈ ZL

t connected iff xi = yi for all i.

Coloring: f : ZL

t → Zc

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

SLIDE 50

Graph Coloring: Dictatorship Gadget

Recall: Show it is NP-hard to c-color t-colorable graphs.

f

f : Z2

3 → Z4

Dictatorship Gadget Graph on vertex set ZL

t = {0, 1, . . . , t − 1}L.

x, y ∈ ZL

t connected iff xi = yi for all i.

Coloring: f : ZL

t → Zc

Infer the label by decoding f .

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

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Graph Coloring: Dictatorship Gadget

f : ZL

t → Zc, f (x) = f (y) if xi = yi for all i ∈ {1, . . . , L}.

f

f : Z2

3 → Z4

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

SLIDE 52

Graph Coloring: Dictatorship Gadget

f : ZL

t → Zc, f (x) = f (y) if xi = yi for all i ∈ {1, . . . , L}.

f

f : Z2

3 → Z4

f is a dictator in the ith coordinate if f (x) = g(xi) for some g : Zt → Zc.

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

SLIDE 53

Graph Coloring: Dictatorship Gadget

f : ZL

t → Zc, f (x) = f (y) if xi = yi for all i ∈ {1, . . . , L}.

f

f : Z2

3 → Z4

f is a dictator in the ith coordinate if f (x) = g(xi) for some g : Zt → Zc. If t = c, f must be a dictator.

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

SLIDE 54

Graph Coloring: Dictatorship Gadget

f : ZL

t → Zc, f (x) = f (y) if xi = yi for all i ∈ {1, . . . , L}.

f

f : Z2

3 → Z4

f is a dictator in the ith coordinate if f (x) = g(xi) for some g : Zt → Zc. If t = c, f must be a dictator. Expected by algebraic theory for CSPs.

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

SLIDE 55

Graph Coloring: Dictatorship Gadget

f : ZL

t → Zc, f (x) = f (y) if xi = yi for all i ∈ {1, . . . , L}.

f

f : Z2

3 → Z4

f is a dictator in the ith coordinate if f (x) = g(xi) for some g : Zt → Zc. If t = c, f must be a dictator. Expected by algebraic theory for CSPs. If t < c ≤ 2t − 2, f might not be a dictator, but is close to one.

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

SLIDE 56

Graph Coloring: Dictatorship Gadget

f : ZL

t → Zc, f (x) = f (y) if xi = yi for all i ∈ {1, . . . , L}.

f

f : Z2

3 → Z4

f is a dictator in the ith coordinate if f (x) = g(xi) for some g : Zt → Zc. If t = c, f must be a dictator. Expected by algebraic theory for CSPs. If t < c ≤ 2t − 2, f might not be a dictator, but is close to one. Lemma (Near-dictatorship) If c ≤ 2t − 2, there exists unique i ∈ {1, . . . , L} and S ⊆ Zc of size t such that if f (x) = f (y) ∈ S then xi = yi

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

SLIDE 57

Outer Verifier: Label Cover

U V π(u,v) π(u,v) : [L] → [L]

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

SLIDE 58

Outer Verifier: Label Cover

U V π(u,v) π(u,v) : [L] → [L]

Bipartite graph Ψ = (U, V , E).

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

SLIDE 59

Outer Verifier: Label Cover

U V π(u,v) π(u,v) : [L] → [L]

Bipartite graph Ψ = (U, V , E). Label set [L] = {1, . . . , L}.

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

SLIDE 60

Outer Verifier: Label Cover

U V π(u,v) π(u,v) : [L] → [L]

Bipartite graph Ψ = (U, V , E). Label set [L] = {1, . . . , L}. For each e ∈ E, projection πe : [L] → [L]

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

SLIDE 61

Outer Verifier: Label Cover

U V π(u,v) π(u,v) : [L] → [L]

Bipartite graph Ψ = (U, V , E). Label set [L] = {1, . . . , L}. For each e ∈ E, projection πe : [L] → [L] Goal: Find labelings σ1 : U → [L], σ2 : V → [L] so that for all (u, v) ∈ E π(u,v)(σ1(u)) = σ2(v).

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

SLIDE 62

Outer Verifier: Label Cover

U V π(u,v) π(u,v) : [L] → [L]

Bipartite graph Ψ = (U, V , E). Label set [L] = {1, . . . , L}. For each e ∈ E, projection πe : [L] → [L] Goal: Find labelings σ1 : U → [L], σ2 : V → [L] so that for all (u, v) ∈ E π(u,v)(σ1(u)) = σ2(v). Theorem (Label Cover) For all ǫ > 0, exists L where distinguishing YES: exists σ1, σ2 satisfying all edges. NO: all σ1, σ2 satisfy at most ǫ of the edges. is NP-hard.

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

SLIDE 63

Outer Verifier + Inner Verifier

U V π(u,v) π(u,v) : [L] → [L]

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

SLIDE 64

Outer Verifier + Inner Verifier

U V π(u,v) π(u,v) : [L] → [L]

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

SLIDE 65

Graph Coloring: Reduction

Lemma (Near-dictatorship) If c ≤ 2t − 2, there exists unique i ∈ {1, . . . , L} and S ⊆ Zc of size t such that if f (x) = f (y) ∈ S then xi = yi

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

SLIDE 66

Graph Coloring: Reduction

Lemma (Near-dictatorship) If c ≤ 2t − 2, there exists unique i ∈ {1, . . . , L} and S ⊆ Zc of size t such that if f (x) = f (y) ∈ S then xi = yi Ψ = (U, V , E) Label Cover instance. e = (u, v) ∈ E, πe : [L] → [L]. fu, gv : ZL

t → Zc gadgets, near-dictators in iu, iv, respectively.

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

SLIDE 67

Graph Coloring: Reduction

Lemma (Near-dictatorship) If c ≤ 2t − 2, there exists unique i ∈ {1, . . . , L} and S ⊆ Zc of size t such that if f (x) = f (y) ∈ S then xi = yi Ψ = (U, V , E) Label Cover instance. e = (u, v) ∈ E, πe : [L] → [L]. fu, gv : ZL

t → Zc gadgets, near-dictators in iu, iv, respectively.

Edge Constraint (“co-gadget”) For all x, y ∈ DL such that yπ(i) = xi for all i ∈ [L], let gv(y) = fu(x) (pretend variables are the same)

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

SLIDE 68

Graph Coloring: Reduction

Lemma (Near-dictatorship) If c ≤ 2t − 2, there exists unique i ∈ {1, . . . , L} and S ⊆ Zc of size t such that if f (x) = f (y) ∈ S then xi = yi Ψ = (U, V , E) Label Cover instance. e = (u, v) ∈ E, πe : [L] → [L]. fu, gv : ZL

t → Zc gadgets, near-dictators in iu, iv, respectively.

Edge Constraint (“co-gadget”) For all x, y ∈ DL such that yπ(i) = xi for all i ∈ [L], let gv(y) = fu(x) (pretend variables are the same) Can prove π(iu) = iv, so c-coloring perfectly labels Ψ.

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

SLIDE 69

Graph Coloring: Reduction

Lemma (Near-dictatorship) If c ≤ 2t − 2, there exists unique i ∈ {1, . . . , L} and S ⊆ Zc of size t such that if f (x) = f (y) ∈ S then xi = yi Ψ = (U, V , E) Label Cover instance. e = (u, v) ∈ E, πe : [L] → [L]. fu, gv : ZL

t → Zc gadgets, near-dictators in iu, iv, respectively.

Edge Constraint (“co-gadget”) For all x, y ∈ DL such that yπ(i) = xi for all i ∈ [L], let gv(y) = fu(x) (pretend variables are the same) Can prove π(iu) = iv, so c-coloring perfectly labels Ψ. Thus, c-coloring a t-colorable graph is NP-hard if c ≤ 2t − 2.

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

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Hypergraph coloring

Hardness of c-coloring a k-uniform, t-partite hypergraph.

f

t = 5, k = 4, c = 2 f : Z2

5 → {0, 1}

k-uniform hypergraph on ZL

t .

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

SLIDE 71

Hypergraph coloring

Hardness of c-coloring a k-uniform, t-partite hypergraph.

f

t = 5, k = 4, c = 2 f : Z2

5 → {0, 1}

k-uniform hypergraph on ZL

t .

Hyperedges: all S ⊂ ZL

t of size k such

that for all distinct x, y ∈ S, xi = yi.

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

SLIDE 72

Hypergraph coloring

Hardness of c-coloring a k-uniform, t-partite hypergraph.

f

t = 5, k = 4, c = 2 f : Z2

5 → {0, 1}

k-uniform hypergraph on ZL

t .

Hyperedges: all S ⊂ ZL

t of size k such

that for all distinct x, y ∈ S, xi = yi. Coloring: f : ZL

t → Zc.

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

SLIDE 73

Hypergraph coloring

Hardness of c-coloring a k-uniform, t-partite hypergraph.

f

t = 5, k = 4, c = 2 f : Z2

5 → {0, 1}

k-uniform hypergraph on ZL

t .

Hyperedges: all S ⊂ ZL

t of size k such

that for all distinct x, y ∈ S, xi = yi. Coloring: f : ZL

t → Zc.

For all hyperedges S, |{f (x) : x ∈ S}| ≥ 2.

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

SLIDE 74

Hypergraph coloring

Hardness of c-coloring a k-uniform, t-partite hypergraph.

f

t = 5, k = 4, c = 2 f : Z2

5 → {0, 1}

k-uniform hypergraph on ZL

t .

Hyperedges: all S ⊂ ZL

t of size k such

that for all distinct x, y ∈ S, xi = yi. Coloring: f : ZL

t → Zc.

For all hyperedges S, |{f (x) : x ∈ S}| ≥ 2. Lemma (Brakensiek, Guruswami) If c = 2 and t = ⌈ 3k

2 ⌉, there are injective

maps gi : Z2⌈ k

2 ⌉+2 → Zt (i ∈ L) and a

dictator g : ZL

2⌈ k

2 ⌉+2 → Zt such that

g(x1, . . . , xL) = f (g1(x1), g2(x2), . . . , gL(xL)).

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

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Limitations

f

f : Z2

3 → Z5

When c > 2t − 2, ‘juntas’ are possible. Would need to extend methods of (Austrin, Guruswami, H˚ astad; 2014).

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

SLIDE 76

Limitations

f

f : Z2

3 → Z5

When c > 2t − 2, ‘juntas’ are possible. Would need to extend methods of (Austrin, Guruswami, H˚ astad; 2014). Combinatorial arguments become increasingly more tedious.

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

SLIDE 77

Limitations

f

f : Z2

3 → Z5

When c > 2t − 2, ‘juntas’ are possible. Would need to extend methods of (Austrin, Guruswami, H˚ astad; 2014). Combinatorial arguments become increasingly more tedious. Unlikely to beat state-of-the-art for large t without new techniques.

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

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Open Questions

Question Approximate graph coloring (t = 3): can we close the gap between c ≥ 4 and c ≤ o(n1/5)?

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

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Open Questions

Question Approximate graph coloring (t = 3): can we close the gap between c ≥ 4 and c ≤ o(n1/5)? Characterize colorings of ZL

3 → Z5.

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

SLIDE 80

Open Questions

Question Approximate graph coloring (t = 3): can we close the gap between c ≥ 4 and c ≤ o(n1/5)? Characterize colorings of ZL

3 → Z5.

Do (Dinur, Mossel, Regev, 2008) and (Huang, 2013) shed insight on the weak polymorphisms (Galois correspondence)?

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

SLIDE 81

Open Questions

Question Approximate graph coloring (t = 3): can we close the gap between c ≥ 4 and c ≤ o(n1/5)? Characterize colorings of ZL

3 → Z5.

Do (Dinur, Mossel, Regev, 2008) and (Huang, 2013) shed insight on the weak polymorphisms (Galois correspondence)? Question Is it NP-hard to c-color a k-uniform (k + 1)-partite hypergraph?

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

SLIDE 82

Open Questions

Question Approximate graph coloring (t = 3): can we close the gap between c ≥ 4 and c ≤ o(n1/5)? Characterize colorings of ZL

3 → Z5.

Do (Dinur, Mossel, Regev, 2008) and (Huang, 2013) shed insight on the weak polymorphisms (Galois correspondence)? Question Is it NP-hard to c-color a k-uniform (k + 1)-partite hypergraph? Case c = 2 may be in reach with current techniques.

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings

SLIDE 83

Thank you!

f

Joshua Brakensiek, Venkatesan Guruswami New hardness results for graph and hypergraph colorings