Complexity of maximum and minimum fixed point problem in Boolean - - PowerPoint PPT Presentation

Complexity of maximum and minimum fixed point problem in Boolean networks

Adrien Richard I3S laboratory, CNRS, Nice, France joint work with Florian Bridoux, Nicola Durbec & K´ evin Perrot LIS laboratory, CNRS, Marseille, France Workshop: Theory and applications of Boolean interaction networks Freie Universit¨ at, Berlin, September 12-13, 2019

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 1/15

A Boolean network (BN) with n components is a function f : {0, 1}n → {0, 1}n x = (x1, . . . , xn) → f(x) = (f1(x), . . . , fn(x))

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 2/15

A Boolean network (BN) with n components is a function f : {0, 1}n → {0, 1}n x = (x1, . . . , xn) → f(x) = (f1(x), . . . , fn(x)) Global transition function Locale transition functions

fi : {0, 1}n → {0, 1}

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 2/15

A Boolean network (BN) with n components is a function f : {0, 1}n → {0, 1}n x = (x1, . . . , xn) → f(x) = (f1(x), . . . , fn(x)) The synchronous dynamics is given by xt+1 = f(xt). The asynchronous dynamics is more realistic in many cases. Fixed points of f are stable states for both dynamics.

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 2/15

A Boolean network (BN) with n components is a function f : {0, 1}n → {0, 1}n x = (x1, . . . , xn) → f(x) = (f1(x), . . . , fn(x)) The interaction graph (IG) of f is the signed digraph defined by

• the vertex set is {1, . . . , n},
• there is a positive edge j → i if there is x ∈ {0, 1}n such that

fi(x1, . . . , xj−1, 0, xj+1, . . . , xn) = 0 fi(x1, . . . , xj−1, 1, xj+1, . . . , xn) = 1

• there is a negative edge j → i if there is x ∈ {0, 1}n such that

fi(x1, . . . , xj−1, 0, xj+1, . . . , xn) = 1 fi(x1, . . . , xj−1, 1, xj+1, . . . , xn) = 0

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 2/15

Example with n = 3    f1(x) = x2 ∨ x3 f2(x) = x1 ∧ x3 f3(x) = x3 ∧ (x1 ∨ x2) Synchronous dynamics 000 110 101 100 001 011 010 111 Interaction graph

1 2 3

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 3/15

BNs are classical models for gene networks. When biologists study a gene network, the interaction graph is often the first reliable data.

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 4/15

BNs are classical models for gene networks. When biologists study a gene network, the interaction graph is often the first reliable data. Interaction Graph Consistency Problem Input: An interaction graph G and a dynamical property P. Question: Is there a BN on G with a dynamics satisfying P?

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 4/15

BNs are classical models for gene networks. When biologists study a gene network, the interaction graph is often the first reliable data. Interaction Graph Consistency Problem Input: An interaction graph G and a dynamical property P. Question: Is there a BN on G with a dynamics satisfying P? We study this decision problem from a complexity point of view and for dynamical properties concerning the number of fixed points.

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 4/15

BNs are classical models for gene networks. When biologists study a gene network, the interaction graph is often the first reliable data. Interaction Graph Consistency Problem Input: An interaction graph G and a dynamical property P. Question: Is there a BN on G with a dynamics satisfying P? We study this decision problem from a complexity point of view and for dynamical properties concerning the number of fixed points. ֒ → Previous complexity results for BNs essentially concern the Boolean Network Consistency Problem Input: A Boolean network f and a dynamical property P. Question: Does the dynamics of f satisfies P?

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 4/15

BNs are classical models for gene networks. When biologists study a gene network, the interaction graph is often the first reliable data. Interaction Graph Consistency Problem Input: An interaction graph G and a dynamical property P. Question: Is there a BN on G with a dynamics satisfying P? We study this decision problem from a complexity point of view and for dynamical properties concerning the number of fixed points. ֒ → Previous complexity results for BNs essentially concern the Boolean Network Consistency Problem Input: A Boolean network f and a dynamical property P. Question: Does the dynamics of f satisfies P? Theorem [Kosub 2008] It is NP-complete to decide if a BN has a fixed point.

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 4/15

Definitions max(G) := maximum number of fixed points in a BN on G min(G) := minimum number of fixed points in a BN on G

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 5/15

Definitions max(G) := maximum number of fixed points in a BN on G min(G) := minimum number of fixed points in a BN on G

1 2 3

max(G) = 3 min(G) = 1

(8 BNs)

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 5/15

Definitions max(G) := maximum number of fixed points in a BN on G min(G) := minimum number of fixed points in a BN on G

1 2 3

max(G) = 3 min(G) = 1

(8 BNs)

1 2 3

max(G) = 2 min(G) = 2

(8 BNs)

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 5/15

Definitions max(G) := maximum number of fixed points in a BN on G min(G) := minimum number of fixed points in a BN on G

1 2 3

max(G) = 3 min(G) = 1

(8 BNs)

1 2 3

max(G) = 2 min(G) = 2

(8 BNs)

k-MaxProblem: Given G, do we have max(G) ≥ k?

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 5/15

Definitions max(G) := maximum number of fixed points in a BN on G min(G) := minimum number of fixed points in a BN on G

1 2 3

max(G) = 3 min(G) = 1

(8 BNs)

1 2 3

max(G) = 2 min(G) = 2

(8 BNs)

k-MaxProblem: Given G, do we have max(G) ≥ k? k-MinProblem: Given G, do we have min(G) ≤ k?

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 5/15

max(G) ≥ 1?

Theorem max(G) ≥ 1 iff each initial strong component of G has a positive cycle.

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 6/15

max(G) ≥ 1?

Theorem max(G) ≥ 1 iff each initial strong component of G has a positive cycle. Theorem [Robertson, Seymour and Thomas 1999; McCuaig 2004] We can decide in polynomial time if G has a positive cycle.

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 6/15

max(G) ≥ 1?

Theorem max(G) ≥ 1 iff each initial strong component of G has a positive cycle. Theorem [Robertson, Seymour and Thomas 1999; McCuaig 2004] We can decide in polynomial time if G has a positive cycle. Corollary We can decide in polynomial time if max(G) ≥ 1.

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 6/15

max(G) ≥ 1?

Theorem max(G) ≥ 1 iff each initial strong component of G has a positive cycle. Theorem [Robertson, Seymour and Thomas 1999; McCuaig 2004] We can decide in polynomial time if G has a positive cycle. Corollary We can decide in polynomial time if max(G) ≥ 1. Recall that it is NP-complete to decide if a BN has a fixed point.

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 6/15

max(G) ≥ 2?

According to Thomas, max(G) ≥ 2 means that G can be the interaction graph of a gene network controlling a cell differentiation process.

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 7/15

max(G) ≥ 2?

According to Thomas, max(G) ≥ 2 means that G can be the interaction graph of a gene network controlling a cell differentiation process. Theorem [Aracena 2008]

• 1. If max(G) ≥ 2, then G has a positive cycle.

[Thomas’ 1st rule]

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 7/15

max(G) ≥ 2?

According to Thomas, max(G) ≥ 2 means that G can be the interaction graph of a gene network controlling a cell differentiation process. Theorem [Aracena 2008]

• 1. If max(G) ≥ 2, then G has a positive cycle.
• 2. If G has only positive cycles and no source, then min(G) ≥ 2.

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 7/15

max(G) ≥ 2?

According to Thomas, max(G) ≥ 2 means that G can be the interaction graph of a gene network controlling a cell differentiation process. Theorem [Aracena 2008]

• 1. If max(G) ≥ 2, then G has a positive cycle.
• 2. If G has only positive cycles and no source, then min(G) ≥ 2.

Can we hope for a simple characterization of max(G) ≥ 2?

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 7/15

max(G) ≥ 2?

According to Thomas, max(G) ≥ 2 means that G can be the interaction graph of a gene network controlling a cell differentiation process. Theorem [Aracena 2008]

• 1. If max(G) ≥ 2, then G has a positive cycle.
• 2. If G has only positive cycles and no source, then min(G) ≥ 2.

Can we hope for a simple characterization of max(G) ≥ 2? Theorem It is NP-complete to decide if max(G) ≥ 2.

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 7/15

max(G) ≥ 2?

According to Thomas, max(G) ≥ 2 means that G can be the interaction graph of a gene network controlling a cell differentiation process. Theorem [Aracena 2008]

• 1. If max(G) ≥ 2, then G has a positive cycle.
• 2. If G has only positive cycles and no source, then min(G) ≥ 2.

Can we hope for a simple characterization of max(G) ≥ 2? Theorem It is NP-complete to decide if max(G) ≥ 2. It is NP-complete to decide if max(G) ≥ k, for every fixed k ≥ 2.

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 7/15

max(G) ≥ k? is in NP

Theorem There is an algorithm with the following specifications: Input: G and k couples of states (x1, y1) . . . (xk, yk). Output: A BN f on G with f(xℓ) = yℓ for 1 ≤ ℓ ≤ k, if it exists. Running time: O(k2n2).

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 8/15

max(G) ≥ k? is in NP

Theorem There is an algorithm with the following specifications: Input: G and k couples of states (x1, y1) . . . (xk, yk). Output: A BN f on G with f(xℓ) = yℓ for 1 ≤ ℓ ≤ k, if it exists. Running time: O(k2n2). If max(G) ≥ k, there is a BN f on G with k fixed points x1, . . . , xk. Then (x1, . . . , xk) is a certificat of size O(kn) which can be checked in O(k2n2)-time by giving as input G and the couples (x1, x1), . . . , (xk, xk).

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 8/15

max(G) ≥ k? is in NP

Theorem There is an algorithm with the following specifications: Input: G and k couples of states (x1, y1) . . . (xk, yk). Output: A BN f on G with f(xℓ) = yℓ for 1 ≤ ℓ ≤ k, if it exists. Running time: O(k2n2). If max(G) ≥ k, there is a BN f on G with k fixed points x1, . . . , xk. Then (x1, . . . , xk) is a certificat of size O(kn) which can be checked in O(k2n2)-time by giving as input G and the couples (x1, x1), . . . , (xk, xk). Thus max(G) ≥ k? is in NP.

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 8/15

max(G) ≥ 2? is NP-hard

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 9/15

max(G) ≥ 2? is NP-hard

Theorem Given a SAT formula φ with n variables and m clauses, we can built in O(n + m)-time an interaction graph Gφ with O(n + m) vertices s.t. max(Gφ) ≥ 2 ⇐ ⇒ φ is satisfiable

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 9/15

max(G) ≥ 2? is NP-hard

Theorem Given a SAT formula φ with n variables and m clauses, we can built in O(n + m)-time an interaction graph Gφ with O(n + m) vertices s.t. max(Gφ) ≥ 2 ⇐ ⇒ φ is satisfiable Basic observation: 2 fixed points 1 fixed point

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 9/15

max(G) ≥ 2? is NP-hard

Theorem Given a SAT formula φ with n variables and m clauses, we can built in O(n + m)-time an interaction graph Gφ with O(n + m) vertices s.t. max(Gφ) ≥ 2 ⇐ ⇒ φ is satisfiable Basic observation: 2 fixed points 1 fixed point The idea is to “control” with φ the “effectiveness” of the negative chord, so that the chord can be “ineffective” if and only if φ is satisfiable.

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 9/15

max(G) ≥ 2? is NP-hard

Example with φ = (a ∨ ¯ b ∨ c) ∧ (¯ a ∨ ¯ c).

a ¯ a b ¯ b c ¯ c

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 10/15

max(G) ≥ 2? is NP-hard

Example with φ = (a ∨ ¯ b ∨ c) ∧ (¯ a ∨ ¯ c).

a ¯ a b ¯ b c ¯ c

2 fixed points

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 10/15

max(G) ≥ 2? is NP-hard

Example with φ = (a ∨ ¯ b ∨ c) ∧ (¯ a ∨ ¯ c).

a ¯ a b ¯ b c ¯ c C1

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 10/15

max(G) ≥ 2? is NP-hard

Example with φ = (a ∨ ¯ b ∨ c) ∧ (¯ a ∨ ¯ c).

a ¯ a b ¯ b c ¯ c C1 C2

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 10/15

max(G) ≥ 2? is NP-hard

Example with φ = (a ∨ ¯ b ∨ c) ∧ (¯ a ∨ ¯ c).

s a ¯ a b ¯ b c ¯ c C1 C2

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 10/15

max(G) ≥ 2? is NP-hard

Example with φ = (a ∨ ¯ b ∨ c) ∧ (¯ a ∨ ¯ c).

s a ¯ a b ¯ b c ¯ c C1 C2

Gφ

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 10/15

max(G) ≥ 2? is NP-hard

Example with φ = (a ∨ ¯ b ∨ c) ∧ (¯ a ∨ ¯ c).

s a ¯ a b ¯ b c ¯ c C1 C2

Gφ φ is sat. ⇒ max(G) ≥ 2 Consider a true assignment: a = 1, b = 1, c = 0

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 11/15

max(G) ≥ 2? is NP-hard

Example with φ = (a ∨ ¯ b ∨ c) ∧ (¯ a ∨ ¯ c).

s a ¯ a b ¯ b c ¯ c C1 C2

Gφ φ is sat. ⇒ max(G) ≥ 2 Consider a true assignment: a = 1, b = 1, c = 0

fs = 1

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 11/15

max(G) ≥ 2? is NP-hard

Example with φ = (a ∨ ¯ b ∨ c) ∧ (¯ a ∨ ¯ c).

s a ¯ a b ¯ b c ¯ c C1 C2

Gφ φ is sat. ⇒ max(G) ≥ 2 Consider a true assignment: a = 1, b = 1, c = 0

fs = 1 OR AND

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 11/15

max(G) ≥ 2? is NP-hard

Example with φ = (a ∨ ¯ b ∨ c) ∧ (¯ a ∨ ¯ c).

s a ¯ a b ¯ b c ¯ c C1 C2

Gφ φ is sat. ⇒ max(G) ≥ 2 Consider a true assignment: a = 1, b = 1, c = 0

fs = 1 OR AND OR AND

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 11/15

max(G) ≥ 2? is NP-hard

Example with φ = (a ∨ ¯ b ∨ c) ∧ (¯ a ∨ ¯ c).

s a ¯ a b ¯ b c ¯ c C1 C2

Gφ φ is sat. ⇒ max(G) ≥ 2 Consider a true assignment: a = 1, b = 1, c = 0

fs = 1 OR AND OR AND AND OR

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 11/15

max(G) ≥ 2? is NP-hard

Example with φ = (a ∨ ¯ b ∨ c) ∧ (¯ a ∨ ¯ c).

s a ¯ a b ¯ b c ¯ c C1 C2

Gφ φ is sat. ⇒ max(G) ≥ 2 Consider a true assignment: a = 1, b = 1, c = 0

fs = 1 OR AND OR AND AND OR

AND AND AND AND Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 11/15

max(G) ≥ 2? is NP-hard

Example with φ = (a ∨ ¯ b ∨ c) ∧ (¯ a ∨ ¯ c).

s a ¯ a b ¯ b c ¯ c C1 C2

Gφ φ is sat. ⇒ max(G) ≥ 2 Consider a true assignment: a = 1, b = 1, c = 0

fs = 1 OR AND OR AND AND OR

AND AND AND AND

OR OR

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 11/15

max(G) ≥ 2? is NP-hard

Example with φ = (a ∨ ¯ b ∨ c) ∧ (¯ a ∨ ¯ c).

s a ¯ a b ¯ b c ¯ c C1 C2

Gφ φ is sat. ⇒ max(G) ≥ 2 Consider a true assignment: a = 1, b = 1, c = 0

fs = 1 OR AND OR AND AND OR

AND AND AND AND

OR OR OR OR

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 11/15

max(G) ≥ 2? is NP-hard

Example with φ = (a ∨ ¯ b ∨ c) ∧ (¯ a ∨ ¯ c).

1 a ¯ a b ¯ b c ¯ c C1 C2

Gφ φ is sat. ⇒ max(G) ≥ 2 Consider a true assignment: a = 1, b = 1, c = 0

fs = 1 OR AND OR AND AND OR

AND AND AND AND

OR OR OR OR

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 11/15

max(G) ≥ 2? is NP-hard

Example with φ = (a ∨ ¯ b ∨ c) ∧ (¯ a ∨ ¯ c).

1 1 ¯ a b ¯ b c ¯ c C1 C2

Gφ φ is sat. ⇒ max(G) ≥ 2 Consider a true assignment: a = 1, b = 1, c = 0

fs = 1 OR AND OR AND AND OR

AND AND AND AND

OR OR OR OR

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 11/15

max(G) ≥ 2? is NP-hard

Example with φ = (a ∨ ¯ b ∨ c) ∧ (¯ a ∨ ¯ c).

1 1 ¯ a 1 ¯ b c ¯ c C1 C2

Gφ φ is sat. ⇒ max(G) ≥ 2 Consider a true assignment: a = 1, b = 1, c = 0

fs = 1 OR AND OR AND AND OR

AND AND AND AND

OR OR OR OR

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 11/15

max(G) ≥ 2? is NP-hard

Example with φ = (a ∨ ¯ b ∨ c) ∧ (¯ a ∨ ¯ c).

1 1 ¯ a 1 ¯ b c 1 C1 C2

Gφ φ is sat. ⇒ max(G) ≥ 2 Consider a true assignment: a = 1, b = 1, c = 0

fs = 1 OR AND OR AND AND OR

AND AND AND AND

OR OR OR OR

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 11/15

max(G) ≥ 2? is NP-hard

Example with φ = (a ∨ ¯ b ∨ c) ∧ (¯ a ∨ ¯ c).

1 1 ¯ a 1 ¯ b c 1 1 C2

Gφ φ is sat. ⇒ max(G) ≥ 2 Consider a true assignment: a = 1, b = 1, c = 0

fs = 1 OR AND OR AND AND OR

AND AND AND AND

OR OR OR OR

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 11/15

max(G) ≥ 2? is NP-hard

Example with φ = (a ∨ ¯ b ∨ c) ∧ (¯ a ∨ ¯ c).

1 1 ¯ a 1 ¯ b c 1 1 1

Gφ φ is sat. ⇒ max(G) ≥ 2 Consider a true assignment: a = 1, b = 1, c = 0

fs = 1 OR AND OR AND AND OR

AND AND AND AND

OR OR OR OR

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 11/15

max(G) ≥ 2? is NP-hard

Example with φ = (a ∨ ¯ b ∨ c) ∧ (¯ a ∨ ¯ c).

1 1 ¯ a 1 ¯ b c 1 1 1

Gφ φ is sat. ⇒ max(G) ≥ 2 Consider a true assignment: a = 1, b = 1, c = 0

fs = 1 OR AND OR AND AND OR

AND AND AND AND

OR OR OR OR

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 11/15

max(G) ≥ 2? is NP-hard

Example with φ = (a ∨ ¯ b ∨ c) ∧ (¯ a ∨ ¯ c).

1 1 ¯ a 1 ¯ b c 1 1 1

Gφ φ is sat. ⇒ max(G) ≥ 2 Consider a true assignment: a = 1, b = 1, c = 0 Isolated positive cycle ⇓ 2 fixed points

fs = 1 OR AND OR AND AND OR

AND AND AND AND

OR OR OR OR

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 11/15

max(G) ≥ 2? is NP-hard

Example with φ = (a ∨ ¯ b ∨ c) ∧ (¯ a ∨ ¯ c).

s a ¯ a b ¯ b c ¯ c C1 C2

Gφ max(G) ≥ 2 ⇒ φ is sat. Let f be a BN on G with two fixed points: x and y

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 12/15

max(G) ≥ 2? is NP-hard

Example with φ = (a ∨ ¯ b ∨ c) ∧ (¯ a ∨ ¯ c).

s a ¯ a b ¯ b c ¯ c C1 C2

Gφ max(G) ≥ 2 ⇒ φ is sat. Let f be a BN on G with two fixed points: x and y

i i i i

xi < yi xi > yi xi = yi xi ≤ yi

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 12/15

max(G) ≥ 2? is NP-hard

Example with φ = (a ∨ ¯ b ∨ c) ∧ (¯ a ∨ ¯ c).

s a ¯ a b ¯ b c ¯ c C1 C2

Gφ max(G) ≥ 2 ⇒ φ is sat. Let f be a BN on G with two fixed points: x and y Since all the positive cycles are full-positive, by a thm

• f Aracena there is a positive

cycle where vertices are all • or all •

i i i i

xi < yi xi > yi xi = yi xi ≤ yi

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 12/15

max(G) ≥ 2? is NP-hard

Example with φ = (a ∨ ¯ b ∨ c) ∧ (¯ a ∨ ¯ c).

s a ¯ a b ¯ b c ¯ c C1 C2

Gφ max(G) ≥ 2 ⇒ φ is sat. Let f be a BN on G with two fixed points: x and y Since all the positive cycles are full-positive, by a thm

• f Aracena there is a positive

cycle where vertices are all • or all •

i i i i

xi < yi xi > yi xi = yi xi ≤ yi

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 12/15

max(G) ≥ 2? is NP-hard

Example with φ = (a ∨ ¯ b ∨ c) ∧ (¯ a ∨ ¯ c).

s a ¯ a b ¯ b c ¯ c C1 C2

Gφ max(G) ≥ 2 ⇒ φ is sat. Let f be a BN on G with two fixed points: x and y Since all the positive cycles are full-positive, by a thm

• f Aracena there is a positive

cycle where vertices are all • or all •

i i i i

xi < yi xi > yi xi = yi xi ≤ yi

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 12/15

max(G) ≥ 2? is NP-hard

Example with φ = (a ∨ ¯ b ∨ c) ∧ (¯ a ∨ ¯ c).

s a ¯ a b ¯ b c ¯ c C1 C2

Gφ max(G) ≥ 2 ⇒ φ is sat. Let f be a BN on G with two fixed points: x and y Since all the positive cycles are full-positive, by a thm

• f Aracena there is a positive

cycle where vertices are all • or all •

i i i i

xi < yi xi > yi xi = yi xi ≤ yi

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 12/15

max(G) ≥ 2? is NP-hard

Example with φ = (a ∨ ¯ b ∨ c) ∧ (¯ a ∨ ¯ c).

s a ¯ a b ¯ b c ¯ c C1 C2

Gφ max(G) ≥ 2 ⇒ φ is sat. Let f be a BN on G with two fixed points: x and y Since all the positive cycles are full-positive, by a thm

• f Aracena there is a positive

cycle where vertices are all • or all •

i i i i

xi < yi xi > yi xi = yi xi ≤ yi

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 12/15

max(G) ≥ 2? is NP-hard

Example with φ = (a ∨ ¯ b ∨ c) ∧ (¯ a ∨ ¯ c).

s a ¯ a b ¯ b c ¯ c C1 C2

Gφ max(G) ≥ 2 ⇒ φ is sat. Let f be a BN on G with two fixed points: x and y Since all the positive cycles are full-positive, by a thm

• f Aracena there is a positive

cycle where vertices are all • or all •

i i i i

xi < yi xi > yi xi = yi xi ≤ yi

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 12/15

max(G) ≥ 2? is NP-hard

Example with φ = (a ∨ ¯ b ∨ c) ∧ (¯ a ∨ ¯ c).

s a ¯ a b ¯ b c ¯ c C1 C2

Gφ max(G) ≥ 2 ⇒ φ is sat. Let f be a BN on G with two fixed points: x and y Since all the positive cycles are full-positive, by a thm

• f Aracena there is a positive

cycle where vertices are all • or all •

i i i i

xi < yi xi > yi xi = yi xi ≤ yi

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 12/15

max(G) ≥ 2? is NP-hard

Example with φ = (a ∨ ¯ b ∨ c) ∧ (¯ a ∨ ¯ c).

s a ¯ a b ¯ b c ¯ c C1 C2

Gφ max(G) ≥ 2 ⇒ φ is sat. Let f be a BN on G with two fixed points: x and y Since all the positive cycles are full-positive, by a thm

• f Aracena there is a positive

cycle where vertices are all • or all •

i i i i

xi < yi xi > yi xi = yi xi ≤ yi

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 12/15

max(G) ≥ 2? is NP-hard

Example with φ = (a ∨ ¯ b ∨ c) ∧ (¯ a ∨ ¯ c).

s a ¯ a b ¯ b c ¯ c C1 C2

Gφ max(G) ≥ 2 ⇒ φ is sat. Let f be a BN on G with two fixed points: x and y

i i i i

xi < yi xi > yi xi = yi xi ≤ yi ⇒

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 12/15

max(G) ≥ 2? is NP-hard

Example with φ = (a ∨ ¯ b ∨ c) ∧ (¯ a ∨ ¯ c).

s a ¯ a b ¯ b c ¯ c C1 C2

Gφ max(G) ≥ 2 ⇒ φ is sat. Let f be a BN on G with two fixed points: x and y

i i i i

xi < yi xi > yi xi = yi xi ≤ yi ⇒

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 12/15

max(G) ≥ 2? is NP-hard

Example with φ = (a ∨ ¯ b ∨ c) ∧ (¯ a ∨ ¯ c).

s a ¯ a b ¯ b c ¯ c C1 C2

Gφ max(G) ≥ 2 ⇒ φ is sat. Let f be a BN on G with two fixed points: x and y

i i i i

xi < yi xi > yi xi = yi xi ≤ yi ⇒

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 12/15

max(G) ≥ 2? is NP-hard

Example with φ = (a ∨ ¯ b ∨ c) ∧ (¯ a ∨ ¯ c).

s a ¯ a b ¯ b c ¯ c C1 C2

Gφ max(G) ≥ 2 ⇒ φ is sat. Let f be a BN on G with two fixed points: x and y

i i i i

xi < yi xi > yi xi = yi xi ≤ yi ⇒

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 12/15

max(G) ≥ 2? is NP-hard

Example with φ = (a ∨ ¯ b ∨ c) ∧ (¯ a ∨ ¯ c).

s a ¯ a b ¯ b c ¯ c C1 C2

Gφ max(G) ≥ 2 ⇒ φ is sat. Let f be a BN on G with two fixed points: x and y

i i i i

xi < yi xi > yi xi = yi xi ≤ yi ⇒

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 12/15

max(G) ≥ 2? is NP-hard

Example with φ = (a ∨ ¯ b ∨ c) ∧ (¯ a ∨ ¯ c).

s a ¯ a b ¯ b c ¯ c C1 C2

Gφ max(G) ≥ 2 ⇒ φ is sat. Let f be a BN on G with two fixed points: x and y

i i i i

xi < yi xi > yi xi = yi xi ≤ yi ⇒

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 12/15

max(G) ≥ 2? is NP-hard

Example with φ = (a ∨ ¯ b ∨ c) ∧ (¯ a ∨ ¯ c).

s a ¯ a b ¯ b c ¯ c C1 C2

Gφ max(G) ≥ 2 ⇒ φ is sat. Let f be a BN on G with two fixed points: x and y

i i i i

xi < yi xi > yi xi = yi xi ≤ yi a = 1, b = 0, c = 0 a = 1, b = 1, c = 0 are true assignments of φ a = 1 ¯ c = 1

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 12/15

k-MaxProblem: Given G, do we have max(G) ≥ k? Theorem k-MaxProblem is in P if k ≤ 1 and NP-complete if k ≥ 2.

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 13/15

k-MaxProblem: Given G, do we have max(G) ≥ k? Theorem k-MaxProblem is in P if k ≤ 1 and NP-complete if k ≥ 2. k-MinProblem: Given G, do we have min(G) ≤ k?

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 13/15

k-MaxProblem: Given G, do we have max(G) ≥ k? Theorem k-MaxProblem is in P if k ≤ 1 and NP-complete if k ≥ 2. k-MinProblem: Given G, do we have min(G) ≤ k? This problem is much more difficult: Theorem k-MinProblem is NEXPTIME-complete for every k.

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 13/15

k-MaxProblem: Given G, do we have max(G) ≥ k? Theorem k-MaxProblem is in P if k ≤ 1 and NP-complete if k ≥ 2. k-MinProblem: Given G, do we have min(G) ≤ k? This problem is much more difficult: Theorem k-MinProblem is NEXPTIME-complete for every k. With a construction very similar to Gφ, we can prove that min(G) ≤ k? is NP-hard. But to prove the NEXPTIME-hardness, we use a much more technical reduction from SuccintSAT.

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 13/15

MaxProblem: Given G and k, do we have max(G) ≥ k? MinProblem: Given G and k, do we have min(G) ≤ k?

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 14/15

MaxProblem: Given G and k, do we have max(G) ≥ k? MinProblem: Given G and k, do we have min(G) ≤ k? Theorem MaxProblem and MinProblem are NEXPTIME-complete.

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 14/15

Conclusion

We study, from a complexity point of view, a natural class of problems. Interaction Graph Consistency Problem Input: An interaction graph G and a dynamical property P. Question: Is there a BN on G with a dynamics satisfying P? We obtain exact classes of complexity for this problem when P = “to have at least/most k fixed points” Our main result is about bistability: It is NP-complete to decide if there is a BN on G with two fixed points.

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 15/15

Conclusion

We study, from a complexity point of view, a natural class of problems. Interaction Graph Consistency Problem Input: An interaction graph G and a dynamical property P. Question: Is there a BN on G with a dynamics satisfying P? We obtain exact classes of complexity for this problem when P = “to have at least/most k fixed points” Our main result is about bistability: It is NP-complete to decide if there is a BN on G with two fixed points.

Perspectives

• 1. Other dynamical properties.

֒ → number/size of cyclic attractors in the (a)synchronous case.

• 2. Non-Boolean case and unsigned case.

Adrien RICHARD Maximum/Minimum Fixed Point Problem Freie Universit¨ at, Berlin 15/15