Lower Bounds for Subgraph Isomorphism

Benjamin Rossman University of Toronto Outline

• Subgraph isomorphism problems • Two grand challenges in Circuit Complexity: P vs NP via the circuit size of k-CLIQUE NC1 vs P via the formula size of k-CYCLE • The average-case setting • Progress in restricted circuit classes Subgraph Isomorphism Problems k-CLIQUE Given a graph X, does it contain a k-clique (complete subgraph of size k)? k-CLIQUE Given a graph X, does it contain a k-clique (complete subgraph of size k)? k-CLIQUE Given a graph X, does it contain a k-clique (complete subgraph of size k)?

k is an arbitrary fixed constant n is the size of the input graph X k-CLIQUE Given a graph X, does it contain a or slow-growing function k-clique

(complete subgraph of size k(n) = O(log n)k)?

k is an arbitrary fixed constant n is the size of the input graph X k-CLIQUE Given a graph X, does it contain a k-clique (complete subgraph of size k)?

Time complexity of k-CLIQUE • exhaustive search O(nk)

• fastest known algorithm n0.79 k + O(1) k-CLIQUE Given a graph X, does it contain a k-clique (complete subgraph of size k)?

Time complexity of k-CLIQUE • exhaustive search O(nk)

• fastest known algorithm n0.79 k + O(1)

n(ω/3)k + O(1) where 2 ≤ ω < 2.37 is the exponent of matrix multiplication [Nesetril-Poljak ’85] k-CLIQUE Given a graph X, does it contain a k-clique (complete subgraph of size k)?

Time complexity of k-CLIQUE • exhaustive search O(nk)

• fastest known algorithm n0.79 k + O(1) • conjectured lower bound nΩ(k) (implies P ≠ NP) k-CYCLE (on layered graphs)

v0 v1 ⋯ vk–1 v0 k-CYCLE (on layered graphs)

v0 v1 ⋯ vk–1 v0

1 1 2 2 3 3 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ n n k-CYCLE (on layered graphs)

v0 v1 ⋯ vk–1 v0

1 1 2 2 3 3 ⋮ ⋮ X ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ n n k-CYCLE (on layered graphs)

v0 v1 ⋯ vk–1 v0

1 1 2 2 3 3 ⋮ ⋮ X ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ n n k-CYCLE (on layered graphs)

Space complexity of k-CYCLE • best upper bound O(log k · log n) • conjectured lower bound Ω(log k · log n) (implies L ≠ NL)

G-SUBGRAPH ISOMORPHISM

G G-SUBGRAPH ISOMORPHISM Given a graph X, does it contain a subgraph isomorphic to G?

G X COLORED G-SUBGRAPH ISOMORPHISM (SUB(G)) Given a graph X and a vertex-coloring V(X) → V(G), does X contain a properly colored G-subgraph?

• Includes k-CLIQUE and k-CYCLE as special cases COLORED G-SUBGRAPH ISOMORPHISM (SUB(G)) Given a graph X and a vertex-coloring V(X) → V(G), does X contain a properly colored G-subgraph?

• Includes k-CLIQUE and k-CYCLE as special cases

• Complexity is minor-monotone: if F is a minor of G, then SUB(F) reduces to SUB(G) COLORED G-SUBGRAPH ISOMORPHISM (SUB(G)) Given a graph X and a vertex-coloring V(X) → V(G), does X contain a properly colored G-subgraph?

• Includes k-CLIQUE and k-CYCLE as special cases

• Complexity is minor-monotone: if F is a minor of G, then SUB(F) reduces to SUB(G)

• Conjectured to be hard-on-average for natural input distributions: (G-colored) Erdős-Rényi random graphs at an appropriate threshold

COLORED G-SUBGRAPH ISOMORPHISM (SUB(G)) Given a graph X and a vertex-coloring V(X) → V(G), does X contain a properly colored G-subgraph?

Understanding the complexity of SUB(G) with respect to different computational resources (time, space, …) leads to separations of important complexity classes (P ≠ NP, L ≠ NL, …) COLORED G-SUBGRAPH ISOMORPHISM (SUB(G)) Given a graph X and a vertex-coloring V(X) → V(G), does X contain a properly colored G-subgraph?

Understanding the complexity of SUB(G) with respect to different computational resources (time, space, …) leads to separations of important complexity classes (P ≠ NP, L ≠ NL, …)

We focus on complexity measures defined by Boolean circuits Circuit Complexity Circuit Complexity

• A concrete, combinatorial approach to the major questions of Complexity Theory including P vs. NP Circuit Complexity

• Studies the complexity of specific problems (e.g. k-CLIQUE, MATRIX MULTIPLICATION, etc.) in combinatorial models of computation, most importantly Boolean circuits Circuit Complexity

• Studies the complexity of specific problems (e.g. k-CLIQUE, MATRIX MULTIPLICATION, etc.) in combinatorial models of computation, most importantly Boolean circuits

• Goal is to prove unconditional lower bounds, which do not rely on any unproven assumptions Circuit Complexity

• Studies the complexity of specific problems (e.g. k-CLIQUE, MATRIX MULTIPLICATION, etc.) in combinatorial models of computation, most importantly Boolean circuits

• Goal is to prove unconditional lower bounds which a problem (i.e. decision problem) is do not reply on any unproven assumptions represented by a sequence of n boolean functions fn : {0,1} → {0,1} Circuit Complexity

• Studies the complexity of specific problems (e.g. k-CLIQUE, MATRIX MULTIPLICATION, etc.) in combinatorial models of computation, most importantly Boolean circuits

• Goal is to prove unconditional lower bounds which do not reply on any unproven assumptions represented by a seq. of functions ( )n {0,1} 2 (≈ {n-vertex graphs}) → {0,1}

Boolean Circuits

∧

¬ ∨

∨ ∧ ¬

¬ ∨ ∧

∧ ∧ ¬ ∨

x1 x2 x3 x4 x5 Boolean Circuits

• Every n-variable Boolean circuit computes an n- variable Boolean function {0,1}n → {0,1} Boolean Circuits

• Every n-variable Boolean circuit computes an n- variable Boolean function {0,1}n → {0,1}

• We say that a problem is “solved” by a sequence of Boolean circuits C1, C2, …, Cn, … if Cn computes the appropriate function {0,1}n → {0,1}

Boolean Circuits

• Every n-variable Boolean circuit computes an n- variable Boolean function {0,1}n → {0,1}

• We say that a problem is “solved” by a sequence of Boolean circuits C1, C2, …, Cn, … if Cn computes the appropriate function {0,1}n → {0,1}

in contrast to uniform models of computation (e.g. Turing machines) where a single algorithm solves the problem on all instances Circuit Size

• The circuit size of a function f : {0,1}n → {0,1} is the minimum number of gates in a circuit computing f Circuit Size

• The circuit size of a function f : {0,1}n → {0,1} is the minimum number of gates in a circuit computing f

• Theorem [Shannon 1949] Almost all Boolean functions have circuit size Ω(2n/n) Circuit Size

• The circuit size of a function f : {0,1}n → {0,1} is the minimum number of gates in a circuit computing f

• Theorem [Shannon 1949] Almost all Boolean functions have circuit size Ω(2n/n)

• The goal in Circuit Complexity is proving lower bounds for explicit Boolean functions (e.g. k-CLIQUE) Circuit Size

• Theorem [Schnorr 1976, Fischer-Pippenger 1979] Turing mach. time T(n) ⟹ circuit size O(T(n)*log T(n))

• Corollary A super-polynomial lower bound on the circuit size of any function in NP implies P ≠ NP Circuit Size

• Theorem [Schnorr 1976, Fischer-Pippenger 1979] Turing mach. time T(n) ⟹ circuit size O(T(n)*log T(n))

• Corollary A super-polynomial lower bound on the circuit size of any function in NP implies P ≠ NP

Circuit Complexity is widely believed to be the most viable approach to P ≠ NP Circuit Size

• Grand challenge (P ≠ NP) Prove a super-polynomial lower bound on the circuit size of any problem in NP

Circuit Size

• Grand challenge (P ≠ NP) Prove a super-polynomial lower bound on the circuit size of any problem in NP

Concretely, show that k-CLIQUE requires circuit size nΩ(k) Circuit Size

• Grand challenge (P ≠ NP) Prove a super-polynomial lower bound on the circuit size of any problem in NP

• Best known lower bound

Circuit Size

• Grand challenge (P ≠ NP) Prove a super-polynomial lower bound on the circuit size of any problem in NP

• Best known lower bound

3n – O(1) 1976 [Schnorr] 4n – O(1) 1991 [Zwick] 4.5n – o(n) 2001 [Lachish-Raz] 5n – o(n) 2002 - today [Iwama-Morizumi] Boolean Formulas

Formulas are circuits with ∧ the structure of a tree ¬ ∨

∨ ∧ ∧

¬ ∨ x4 ¬ ¬ ∧

∧ ∧ ∨ x2 ¬ x3 x4

x3 x1 x2 x4 x5 x1 x5 Formulas lack “memory”: the result of each sub-computation is only used once Boolean Formulas

Formulas are circuits with ∧ the structure of a tree ¬ ∨

∨ ∧ ∧

¬ ∨ x4 ¬ ¬ ∧

∧ ∧ ∨ x2 ¬ x3 x4

x3 x1 x2 x4 x5 x1 x5 Formulas lack “memory”: the result of each sub-computation is only used once Boolean Formulas

Formulas are circuits with ∧ the structure of a tree ¬ ∨ Open: Are circuits more

powerful than formulas? ∨ ∧ ∧

¬ ∨ x4 ¬ ¬ ∧

∧ ∧ ∨ x2 ¬ x3 x4

x3 x1 x2 x4 x5 x1 x5 Formulas vs. Circuits

• Grand challenge #2 (NC1 ≠ P) Prove that polynomial-size circuits are strictly more powerful than polynomial-size formulas Formulas vs. Circuits

• Grand challenge #2 (NC1 ≠ P) Prove that polynomial-size circuits are strictly more powerful than polynomial-size formulas

NC1 = the class of problems solvable by polynomial-size formulas Formulas vs. Circuits

• Grand challenge #2 (NC1 ≠ P) Prove that polynomial-size circuits are strictly more powerful than polynomial-size formulas

Concretely, show that k-CYCLE requires formula size nΩ(log k) Formulas vs. Circuits

• Grand challenge #2 (NC1 ≠ P) Prove that polynomial-size circuits are strictly more powerful than polynomial-size formulas

• Best known formula size lower bound n1.5 – o(1) 1961 [Subbotovskaya] n2 1971 [Khrapchenko] n2.5 – o(1) 1991 [Andreev] n3 – o(1) 1998 - today [Hastad] Restricted Circuit Classes Restricted Circuit Classes

• Most research in Circuit Complexity studies restricted classes of circuits and formulas in the hope of developing stronger techniques Restricted Circuit Classes

• Most research in Circuit Complexity studies restricted classes of circuits and formulas in the hope of developing stronger techniques

• The two most natural and well-studied restrictions: AC0 circuits constant-depth, unbounded fan-in Boolean circuits monotone circuits Boolean circuits without negation (no NOT gates) AC0 Circuits

∧

∨ ∨ ∨ ∨ ∨

∧ ∧ ∧ ∧ ∧ ∧

∨ ∨ ∨ ∨ ∨ ∨

x1 ¬x1 x2 ¬x2 x3 ¬x3 x4 ¬x4 x5 ¬x5 AC0 Circuits

∧ depth is bounded by a constant d

∨(or slow-growing function d(n) << log n) ∨ ∨ ∨ ∨

∧ ∧ ∧ ∧ ∧ ∧

∨ ∨ ∨ ∨ ∨ ∨

x1 ¬x1 x2 ¬x2 x3 ¬x3 x4 ¬x4 x5 ¬x5 AC0 Formulas

∧

∨ ∨ ∨

∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧

… x5 ¬x8 AC0 Lower Bounds