Satisfiability of Ordering CSPs Above Average Is Fixed-Parameter - - PowerPoint PPT Presentation

Satisfiability of Ordering CSPs Above Average Is Fixed-Parameter Tractable

Yury Makarychev, TTIC Konstantin Makarychev, Microsoft Research Yuan Zhou, MIT

Ordering CSP

Given a set of 𝑜 variables and 𝑛 constraints.

• Variables 𝑦1, … , 𝑦𝑜
• Constraints 𝜌1, … , 𝜌𝑛

Find a linear ordering of 𝑦1, … , 𝑦𝑛 that maximizes the number of satisfied constraints.

𝑦5 𝑦1 𝑦8 𝑦3 𝑦7 𝑦2 𝑦4 𝑦7

Ordering CSP

Given a set of 𝑜 variables and 𝑛 constraints.

• Variables 𝑦1, … , 𝑦𝑜
• Constraints 𝜌1, … , 𝜌𝑛

Find a linear ordering of 𝑦1, … , 𝑦𝑛 that maximizes the number of satisfied constraints.

• Each constraints 𝜌𝑠 has arity at most 𝑙.
• 𝜌𝑠(𝑦𝑗1, 𝑦𝑗2, … , 𝑦𝑗𝑙) specifies a list of orderings of

𝑦𝑗1, … , 𝑦𝑗𝑙.

• 𝜌𝑠 is satisfied if the relative ordering of 𝑦𝑗1 … , 𝑦𝑗𝑙 is in

the list.

Example 1: Max Acyclic Subgraph

• Given a directed graph 𝐻 on 𝑦1, … , 𝑦𝑜.
• Find a linear ordering of vertices so as to

maximize the number of edges going forward.

Example 1: Max Acyclic Subgraph

• Given a directed graph 𝐻 on 𝑦1, … , 𝑦𝑜.
• Find a linear ordering of vertices so as to

maximize the number of edges going forward. Each edge (𝑦𝑗, 𝑦𝑘) defines constraint 𝑦𝑗 < 𝑦𝑘 #forward edges = #satisfied constraints The problem is an ordering CSP of arity 2.

Example 2: Betweenness

• Given a set of vertices 𝑦1, … , 𝑦𝑜 and
• a set of betweenness constraints. Each constraint

is of the form “𝑦𝑗 lies between 𝑦𝑘 and 𝑦𝑙” 𝑦𝑘 < 𝑦𝑗 < 𝑦𝑙 or 𝑦𝑙 < 𝑦𝑗 < 𝑦𝑘 Find an ordering that maximizes the number of satisfied constraints.

𝑦5 𝑦1 𝑦8 𝑦3 𝑦7 𝑦2 𝑦4 𝑦7

NP-hardness

Max Acyclic Subgraph

• If all the constraints are satisfiable, the problem

can be easily solved.

• If 𝑃𝑄𝑈 = 1 − 𝜁 𝑛, the problem is NP-hard.

Betweenness

• The problem is NP-hard even when all the

constraints are satisfiable.

Random Assignment

There is a trivial approximation algorithm for

• rdering CSP:
• rder 𝑦1, … , 𝑦𝑜 randomly.

Max Acyclic Subgraph: each constraint is satisfied with probability ½. Satisfy 𝐵𝑊𝐻 = 𝑛/2 constraints in expectation. Betweenness: each constraint is satisfied with probability 1/3. Satisfy 𝐵𝑊𝐻 = 𝑛/3 constraints in expectation.

Hardness of Approximation (UGC)

[Guruswami, Håstad, Manokaran, Raghavendra, Charikar]

There is no non-trivial multiplicative approximation algorithm for ordering CSP of any arity 𝑙. For every 𝜁 > 0: No polynomial-time algorithm can find a solution satisfying at least 1 + 𝜁 𝐵𝑊𝐻 constraints if 𝑃𝑄𝑈 = 1 − 𝜁 𝑛.

Advantage over Random

[GHMRC] No algorithm performs considerably better than random. Can we get some additive advantage over random?

Advantage over Random

[GHMRC] No algorithm performs considerably better than random. Can we get some additive advantage over random? Conjecture of Gutin, van Iersel, Mnich, and Yeo. There a fixed-parameter algorithm that decides whether 𝑃𝑄𝑈 ≥ 𝐵𝑊𝐻 + 𝑢 or not.

Fixed Parameter Tractability

Conjecture of Gutin, van Iersel, Mnich, and Yeo. For every 𝑙, there a fixed-parameter tractable that decides whether 𝑃𝑄𝑈 ≥ 𝐵𝑊𝐻 + 𝑢 or not. The running time of the algorithm is 𝑔

𝑙 𝑢 𝑞𝑝𝑚𝑧𝑙(𝑛 + 𝑜)

Fixed Parameter Tractability

[Alon, Gutin, Kim, Szeider, Yeo] Satisfiability above average is fixed-parameter tractable for all “regular” (non-ordering) CSPs Conjecture was proved for: Gutin, Kim, Szeider, Yeo Max Acyclic Subgraph Gutin, Kim, Mnich, Yeo Betweenness Gutin, van Iersel, Mnich, Yeo Ordering CSPs of arity 3 [GIMY] “it appears technically very difficult to extend results obtained for arities 𝑠 = 2 and 3 to 𝑠 > 3”

Our Results

Prove the conjecture of Gutin et al. Prove that the satisfiability above average is fixed- parameter tractable for a large class of CSPs, which includes ordering CSPs.

Approach

Follow the high-level approach of Alon, Gutin, Kim, Szeider, and Yeo. Prove that there are two possibilities:

• 1. The instance depends on at most 𝑑𝑙𝑢2 variables.

Then try all possible orderings of these variables in time 2𝑃(𝑢2 log 𝑢) and find the optimal solution. 2. 𝑃𝑄𝑈 ≥ 𝐵𝑊𝐻 + 𝑢. (in case 1, there is a kernel on 𝑑𝑙𝑢2 variables)

Approach

Consider a random ordering of 𝑦1, … , 𝑦𝑜. Let r.v. 𝑎 be the number of constraints satisfied by the random ordering. E 𝑎 = 𝐵𝑊𝐻. Theorem 1 If the instance (non-trivially) depends

• n at least 𝑠 variables then Var 𝑎 ≥ 𝑏 𝑠.

Corollary If Var 𝑎 < 𝑏′ 𝑢2 then the instance depends on at most 𝑑𝑙𝑢2 variables. We are in case 1.

Approach

Consider a random ordering of 𝑦1, … , 𝑦𝑜. Let r.v. 𝑎 be the number of constraints satisfied by the random ordering. E 𝑎 = 𝐵𝑊𝐻. Theorem 2 If Var 𝑎 ≥ 𝑠 then 𝑃𝑄𝑈 ≥ 𝐵𝑊𝐻 + 𝑐 𝑠

Var 𝑎 ≥ 𝑠 implies that 𝑎 deviates by at least 𝑠 from 𝐵𝑊𝐻. For arbitrary r.v. 𝑎, it doesn’t follow that max 𝑎 ≥ 𝐵𝑊𝐻 + 𝑐 𝑠.

!

Consider a random ordering of 𝑦1, … , 𝑦𝑜. Let r.v. 𝑎 be the number of constraints satisfied by the random ordering. E 𝑎 = 𝐵𝑊𝐻. Theorem 2 If Var 𝑎 ≥ 𝑠 then 𝑃𝑄𝑈 ≥ 𝐵𝑊𝐻 + 𝑐 𝑠 Corollary If Var 𝑎 ≥ 𝑏 𝑢2 then 𝑃𝑄𝑈 ≥ 𝐵𝑊𝐻 + 𝑢. We are in case 2.

Approach

Main Theorems

Theorem 1 If the instance (non-trivially) depends

• n at least 𝑠 variables then Var 𝑎 ≥ 𝑏 𝑠.

Theorem 2 If Var 𝑎 ≥ 𝑠 then 𝑃𝑄𝑈 ≥ 𝐵𝑊𝐻 + 𝑐 𝑠

Use the Fourier analysis: the Efron—Stein decomposition. Prove a Bonami-type lemma for the Efron—Stein decomposition.

Efron—Stein Decomposition

To use the Fourier analysis — want to work with a product space. The product space should be large, but shouldn’t depend on 𝑜.

• Assume that each 𝑦𝑗 ∈ [0,1].
• Each assignment (𝑦1, … , 𝑦𝑜) ∈ 0,1 𝑜 defines a

linear ordering of 𝑦1, … , 𝑦𝑜 a.s.

• Random assignment defines a random ordering.

Fourier Analysis on the Boolean Cube

ES decomposition is similar to Fourier decomposition of functions on −1,1 𝑜. For 𝑔: −1,1 𝑜 → ℝ

𝑔 =

𝑇⊂{1,…,𝑜}

𝑔

𝑇𝜓𝑇

• Function

𝑔

𝑇𝜓𝑇 depends only on variables in 𝑇.

• Functions

𝑔

𝑇𝜓𝑇 are mutually orthogonal.

• Var 𝑔 = Var[

𝑔

𝑇𝜓𝑇]

• Have only

𝑔

𝑇 with 𝑇 ≤ 𝑙 for CSPs of arity 𝑙.

Efron—Stein Decomposition

Consider 𝑔: [0,1]𝑜→ ℝ. There is a decomposition

𝑔 =

𝑇⊂{1,…,𝑜}

𝑔

𝑇

Such that

• Function 𝑔

𝑇 depends only on variables in 𝑇.

• Functions 𝑔

𝑇 are mutually orthogonal.

• Var 𝑔 = Var[𝑔

𝑇]

• Have only 𝑔

𝑇 with 𝑇 ≤ 𝑙 for CSPs of arity 𝑙.

Efron—Stein Decomposition

Consider 𝑜 = 1.Then 𝑔(𝑦1) = 𝑔

∅ + 𝑔 1(𝑦1)

where

• 𝑔

∅ = E 𝑔

• 𝑔

1 𝑦1 = 𝑔 𝑦1 − 𝑔 ∅

Efron—Stein Decomposition

Consider 𝑜 = 2. Assume

𝑔 = 𝑕 𝑦1 ⋅ ℎ(𝑦2)

Then

𝑔 = 𝑕∅ + 𝑕1 𝑦1 ℎ∅ + ℎ2 𝑦2 = 𝑕∅ℎ∅ + 𝑕1 𝑦1 ℎ∅ + 𝑕∅ℎ2 𝑦2 + 𝑕1 𝑦1 ℎ2 𝑦2

Let

• 𝑔

∅ = 𝑕∅ℎ∅

• 𝑔

{1} = 𝑕1(𝑦1)ℎ∅

• 𝑔

{2} = 𝑕∅ℎ2(𝑦2)

• 𝑔

{1,2} = 𝑕1(𝑦1)ℎ2(𝑦2)

Efron—Stein Decomposition

For 𝑜 > 2. Assume 𝑔 = 𝑕(1) 𝑦1 ⋅ 𝑕 2 𝑦2 … 𝑕 𝑜 (𝑦𝑜) Decompose each 𝑕(𝑗) 𝑕(𝑗) = 𝑕∅

𝑗 + 𝑕𝑗 𝑗 𝑦𝑗

Expand the expression for 𝑔, get 2𝑜 terms—one for each set 𝑇. Extend by linearity to all functions 𝑔: [0,1]𝑜→ ℝ.

Explicit Formulas

Define 𝑔

⊂𝑈 = E 𝑔 𝑦𝑗 with 𝑗 ∈ 𝑈]

Then 𝑔

𝑇 = 𝑈⊆𝑇

−1 |𝑇∖𝑈| 𝑔

⊂𝑈

ES decomposition of 𝑎

• 𝑎 is a sum of indicators 𝐽of elementary events of

the form 𝑦1 < 𝑦2 < ⋯ < 𝑦𝑙.

• Use explicit formulas to compute the ES

decomposition of 𝐽. 𝐽⊂𝑇 = 1 𝐵𝑙 𝑞𝜌 𝑦𝑇 𝐽{𝑦𝜌 𝑡1 < 𝑦𝜌 𝑡2 < ⋯ } where 𝑞𝜌 are polynomials with integer coefficients

• f degree at most 𝑙.
• By linearity, 𝐽𝑇 and 𝑎𝑇 are of the same form.

ES decomposition of 𝑎

𝑎S = 1 𝐵𝑙 𝑞′𝜌 𝑦𝑇 𝐽{𝑦𝜌 𝑡1 < 𝑦𝜌 𝑡2 < ⋯ } Thus if 𝑎𝑇 ≠ 0 Var 𝑎𝑇 ≥ 𝐶𝑙 > 0

Proof of Theorem 1

Theorem 1 If the instance (non-trivially) depends

• n at least 𝑠 variables then Var 𝑎 ≥ 𝑏 𝑠.

Proof Idea:

• Consider the ES decomposition of 𝑎
• Each 𝑎𝑇 depends on at most k variables
• There are at least 𝑠/𝑙 non-zero terms.
• For each of them, Var 𝑎𝑇 ≥ 𝐶𝑙
• Thus Var 𝑎 = Var [𝑎𝑇] ≥ 𝑠𝐶𝑙/𝑙

Proof of Theorem 2

Theorem 2 If Var 𝑎 ≥ 𝑠 then 𝑃𝑄𝑈 ≥ 𝐵𝑊𝐻 + 𝑐 𝑠 Need to show that E 𝑎 − 𝐵𝑊𝐻 4 < 𝐷𝑙Var 𝑎 2 Then by [Alon et al] 𝑎 > 𝐵𝑊𝐻 + 𝑐 𝑠 with positive probability.

Bonami Lemma

Bonami Lemma. Let 𝑔 be a polynomial of degree 𝑙 on −1, 1 𝑜 E 𝑔4 ≤ 9𝑙E 𝑔2 2 Bonami Lemma for ES. Let 𝑔 be a function with ES decomposition of degree 𝑙. Then E 𝑔4 ≤ 81𝑙𝐷𝑙 E 𝑔2 2 If 𝐹 𝑔

𝑇1𝑔 𝑇2𝑔 𝑇3𝑔 𝑇4 ≤ 𝐷𝑙 𝑗

Var 𝑔

𝑇𝑗 1/2

Proof of Theorem 2

Theorem 2 If Var 𝑎 ≥ 𝑠 then 𝑃𝑄𝑈 ≥ 𝐵𝑊𝐻 + 𝑐 𝑠 Letting 𝑔 = 𝑎 − 𝐵𝑊𝐻, get E 𝑎 − 𝐵𝑊𝐻 4 < 𝐷𝑙81𝑙 Var 𝑎 2 Then 𝑎 > 𝐵𝑊𝐻 + 𝑐 𝑠 with positive probability.

Other Results

Result applies to CSPs on domain [0, 1] with constraints defined by linear or bounded degree inequalities with bounded integer coefficients. E.g., constraints of the form

• 𝑦𝑗 is greater than the average of 𝑦𝑘 and 𝑦𝑙, or
• 𝑦𝑗 is closer to 𝑦𝑘 than to 𝑦𝑙

Summary

• Proved the conjecture of Gutin, van Iersel, Mnich,

and Yeo

• Proved a Bonami-type Inequality for the Efron—

Stein decomposition.

• Proved that the satisfiability beyond average is

fixed parameter tractable for a new class of CSPs defined by LPs & low degree polynomials.