On Constant-Factor Approximable Valued Constraint ... · On Constant-Factor Approximable Valued Constraint Satisfaction Problems Andrei Krokhin Durham University, UK Joint work with

Andrei Krokhin - On constant-factor approximable Min CSPs 1

On Constant-Factor Approximable

Valued Constraint Satisfaction Problems

Andrei Krokhin

Durham University, UK

Joint work with

Vıctor Dalmau (University Pompeu Fabra, Barcelona)

Rajsekar Manokaran (IIT Madras / KTH Stockholm)


Constraint Satisfaction Problems (CSPs)

• CSP(Γ): given R1(x1), . . . , Rq(xq) over V , all Ri ∈ Γ,

is there φ : V → A satisfying all constraints?

– Example: CSP({=2}) is 2-Colourability

• Max CSP(Γ): maximise∑q

i=1 wi ·Ri(xi)

– Example: Max CSP({=2}) is Max Cut

• Min CSP(Γ): minimise∑q

i=1 wi · (1−Ri(xi))

– Example: Min CSP({=2}) is MinUnCut

• complexity classification for finding optimal solutions

for Min CSP is known [Thapper, Zivny’12]

• In this talk: finding approximately optimal solutions


(Min/Max) CSP Instance Example

V = {x, y, z}, A = {a, b}, C = {x = y, y = z, x = z}.

x y

z

a

b b

a

a b


Min/Max CSP Solution Example

V = {x, y, z}, A = {a, b}, C = {x = y, y = z, x = z}.

x y

z

a

b b

a

a b


Approximation algorithms for Max CSP(Γ)

Definition 1 Call ALG a c-approximation algorithm for

Max CSP(Γ) if it runs in poly-time in |I| and for each I,

it finds a solution with value ALGVal(I) such that

OPT(I) ≤ c(|I|) · ALGVal(I).

Fact 1 Each Max CSP(Γ) belongs to APX, i.e. has a

c-approximation algorithm with constant c.

• The algorithm assigns values uniformly at random.

• Can be derandomized by a standard method.

• Much research into locating optimal c.


Approximation Algorithms for Min CSP(Γ)

Definition 2 Call ALG a c-approximation algorithm for

VCSP(Γ) if it runs in poly-time in |I|, and for each I,

it finds a solution with value ALGVal(I) such that

ALGVal(I) ≤ c(|I|) ·OPT(I).

Fact 2 c-approx algo for Min CSP(Γ) ⇒ CSP(Γ) ∈ P.

Problem 1 Which problems Min CSP(Γ) belong to

complexity class APX?

• Long-standing open problem: is MinUnCut there?

• Currently best answer: no, unless the UGC fails.


Some Known Results

k-HORN clauses: (x), (x1 ∨ . . .∨ x≤k), (x1 ∨ x2 ∨ . . .∨ x≤k).

k-IHBS clauses: (x), (x1 ∨ . . . x≤k), (x1 ∨ x2).

• Min CSP(k − IHBS) is in APX [Khanna et al’01]

• Min CSP(3− HORN) is NP-hard to constant-factor

approximate [Guruswami, Lee’14]

• MinUnCut has O(√log n)-approximation algorithm

[Agarwal et al’06]

• MinUnCut is not in APX unless the UGC fails

[Khot et al’07]

• Detailed classification for A = {0, 1} [Khanna et al’01]


Algebra Works

Min CSP(Γ) in APX - studied in (Dalmau, AK’13) as

“CSP(Γ) that are robustly tractable with linear loss”

• One class of problems Min CSP(Γ) in APX is found.

• Standard algebraic machinery works when Γ ⊇ {=}.– polymorphisms, algebras, idempotence, varieties

• Which algebraic properties lead to APX?


Fractional Solution Example

V = {x, y, z}, D = {a, b}, C = {x = y, y = z, x = z}.

.1

.5

.6 .4

.2

.2

.3

.7

.1

.6 .2

.1

.3

.7

.2 .1

.4

.3

x y

z

a

b b

a

a b


Consistent Marginals Example

V = {x, y, z}, D = {a, b}, C = {x = y, y = z, x = z}.

.1

.5

.6 .4

.2

.2

.3

.7

.1

.6 .2

.1

.3

.7

.2 .1

.4 .3

x y

z

a

b b

a

a b


Marginal Distributions Example

V = {x, y, z}, D = {a, b}, C = {x = y, y = z, x = z}.

.1

.5

.6 .4

.2

.2

.3

.7

.1

.6 .2

.1

.3

.7

.2 .1

.4

.3

x y

z

a

b b

a

a b

Prob distr pz

pz(a)=0.6

pz(b)=0.4


Basic LP Relaxation for Min CSP(Γ)

The basic LP relaxation for instance I with constraints C.The variables are

• pv(a) ∈ [0, 1] for each v ∈ V, a ∈ A;

• pC(t) ∈ [0, 1] for each constraint C in I and t ∈ Aar(C).

minimize∑

C=(x,R)∈C

wC ·∑

R(t)=0

pC(t) subject to:

• pv, pC - probability distributions for all v ∈ V,C ∈ C

• consistent marginals

Since Γ is fixed, this relaxation has polynomial size (in I).


Optimality of BLP

Rounding: converting fractional solution to proper solution

The integrality gap of BLP for Min CSP(Γ) is

α = supinstance I

OPT(I)

BLPVal(I)

Meaning: α is best poss approx factor from rounding BLP.

Theorem 1 (Ene, Vondrak, Wu’13)

For any Γ ⊇ {=}, if Min CSP(Γ) has a c-factor approx

algorithm with c < α then the UGC fails. In particular, if

α = ∞ then Min CSP(Γ) ∈ APX (unless the UGC fails).

Meaning: enough to consider BLP-based approx algorithms


The Standard Simplex

Let ∆(X) = {probability distributions on a set X}.

The standard (k − 1)-dimensional simplex where k = |X|

(1,0,0)=a

b=(0,1,0)

(0,0,1)=c

∆({a,b,c})


Simplex Discretized

Let ∆n(X) = {p ∈ ∆(X) | ∀x ∈ X p(x) ∈ n−1Z}.

∆4({a,b,c})

(1,0,0)= a

b c

(3/4, 0, 1/4)

(1/2, 1/2, 0)

=

(0,1,0) =

(0,0,1)


Rounding BLP Solution

• Let s be an optimal solution for BLP(I). Can assume

there is n such that s gives pv ∈ ∆n(A) for each v ∈ V .

• Any map g : ∆n(A) → A can be used to round s for I;

as follows: v 7→ g(pv). Good g ⇒ good approximation.

• ∆n(A) ↔ multisets on A of size n

– p ∈ ∆n(A) ↔ [a ∈ A appears p(a) · n times]

• An operation f : An → A is symmetric if, ∀π ∈ Sn,

f(x1, . . . , xn) = f(xπ(1), . . . , xπ(n)).

• n-ary symmetric operations ≡ mappings ∆n(A) → A.


Symmetric Operation Example

This is a 4-ary (idempotent) symmetric operation f

For example, f(a, c, a, a) = a and f(b, b, a, a) = a

∆4({a,b,c})

a

b c

(3/4, 0, 1/4)

(1/2, 1/2, 0)


Deciding CSP(Γ) by BLP

Theorem 2 (Kun et al ’11) For any Γ, TFAE

1. BLP decides CSP(Γ), i.e. BLPVal(I) = 0 ⇒ I is sat.

2. For each n, Γ has an n-ary symmetric polymorphism.

Let I be an instance of CSP(Γ) with BLPVal(I) = 0 and

let s be an optimal solution to BLP(I). Can assume ∃n

• s gives pv ∈ ∆n(A) for each v ∈ V .

• s gives pC ∈ ∆n(Aar(C)) for each C ∈ C.

If g ∈ SymPoln(Γ) then v 7→ g(pv) satisfies all constraints.


Proof of Satisfaction

• Pick a constraint C = R(x). Let x = (v1, . . . , vm).

• Know pC(t) > 0 ⇒ R(t) = 1. Recall: pC ∈ ∆n(Am).

• Take n · pC(t) copies of each tuple t with R(t) = 1.

• Call them a1 = (a11, . . . , a1m), . . . , an = (an1, . . . , anm).

g g g

R( a11 , . . . , a1m ) = 1...

......

...

R( an1 , . . . , anm ) = 1

R( g(pv1) , . . . , g(pvm) ) = 1


Stability and Integrality Gap

For d1, d2 ∈ ∆n(A), let dist(d1, d2) = maxa∈A |d1(a)− d2(a)|

Let ϕ be a probability distribution on SymPoln(Γ).

Say that ϕ is c-stable if, for all d1, d2 ∈ ∆n(A),

Prg∼ϕ

{g(d1) = g(d2)} ≤ c · dist(d1, d2).

Theorem 3 (Dalmau, AK, Manokaran)

For any Γ ⊇ {=}, TFAE

1. BLP has finite integrality gap for Min CSP(Γ).

2. There is c ≥ 1 such that, for all n, Γ admits a c-stable

probability distribution on SymPoln(Γ).


Fractional Symmetric Operation Example

1/6 1/6 1/8

1/8 1/6 1/4


Examples

• Non-example: Take 3− HORN.

– Only one n-ary symmetric polym g(x) =∧xi.

– Take d1 = (1, 1, . . . , 1) and d2 = (0, 1, . . . , 1)

– Easy: dist(d1, d2) = 1/n, but Pr[g(d1) = g(d2)] = 1.

– Hence infinite integrality gap and UG-hardness

• Example: Take Γ = {≤, 0, 1} on A = {0, 1}.– For 1 ≤ j ≤ n, let gn,j(x) = 1 iff |{xi : xi = 1}| ≥ j.

– Each gn,j is monotone, so polymorphism

– If dist(d1, d2) = r/n then Pr[g(d1) = g(d2)] ≤ r/n.

– Hence 1-stability and finite integrality gap.


Rounding from Stable Distributions

• Let I be an instance of CSP(Γ), take an optimal

solution to BLP(I), obtain pv ∈ ∆n(A) for each v ∈ V

and pC ∈ ∆n(Aar(C)) for each C ∈ C.

• Draw g from the c-stable distribution ϕn; v 7→ g(pv).

• This is a randomized (2 ·maxar · c)-approx algorithm.

• Pick C = R(x) and estimate Prg∼ϕn {R(g(x)) = 0}.

• Modify pC to qC such that qC(t) > 0 ⇒ R(t) = 1.

• For marginals qi’s of qC , have R(g(q1), . . . , g(qm)) = 1.

• Marginals of pC and qC are close, use c-stability of ϕn.

• Get Prg∼ϕn {R(g(x)) = 0} ≤ 2 ·m · c · (1− pC(R)).


A Positive Result

Theorem 4 (Dalmau,AK’13)

Assume that Γ is hom-equivalent to a CL Γ′ on some set L

(of subsets) s.t. Γ′ has polymorphism x ∩ (y ∪ z) where

(L,∩,∪) is a distrib lattice. Then Min CSP(Γ) ∈ APX.

• There are other problems Min CSP(Γ) in APX.


NP-hardness Result

• Let Γ have c-stable distributions ϕn on its symmetric

polymorphisms g. Wlog assume ∀x g(x, . . . , x) = x.

• For n-tuples d1 = (b, a, . . . , a) and d2 = (a, . . . , a), have

Prg∼ϕn

{g(d1) = a} = Prg∼ϕn

{g(d1) = g(d2)} ≤ c·dist(d1, d2) =c

n.

• So, for n > c · |A|2, supp(ϕn) contains NU operations:

∀x, y ∀i f(x, . . . , x, yi, x, . . . , x) = x.

Theorem 5 (Dalmau, AK, Manokaran)

If Γ has no NU polymorphism then it is NP-hard to

constant-factor approximate Min CSP(Γ).


From VCSP to Min CSP

• Valued constraint: f(x) where f : Am → [0, 1]

• VCSP(Γ): minimise∑q

i=1 wi · fi(xi) where all fi ∈ Γ

• Min CSP is a special case of VCSP

Lemma 1 (Dalmau, AK, Manokaran)

For each valued CL Γ, there is a (non-valued) CL Γ′ such

that VCSP(Γ) is in APX iff Min CSP(Γ′) is in APX.


Open Problems

• Use c-stability to get an efficient rounding algorithm.

• Improve more UG-hardness to NP-hardness.

• Get rid of the {=} ⊆ Γ assumption (if possible).

• Study algebras with many symmetric operations.

• Decidability issues for symmetric polymorphisms.

• Link c-stability with Prague-like strategies.

• Extend results to non-constant c and/or SDP.


The Unique Games Conjecture (UGC)

For a permutation σ on A, let σ◦ = {(x, y) | y = σ(x)}.For A = {0, 1, . . . , k − 1}, let Γk = {σ◦ | σ a perm on A}.

Conjecture 1 (Khot’02)

For each ϵ > 0, there is k = k(ϵ) such that it is NP-hard to

tell (1− ϵ)-satisfiable from at most ϵ-satisfiable instances of

Max CSP(Γk) (aka Unique Games).

• One of the hottest conjectures in Theoretical CS

• If true, optimal approx algorithms for many classical

problems, incl. all Max CSP(Γ) [Raghavendra’08].

• If false, there is a new powerful approx technique

On Constant-Factor Approximable Valued Constraint ... · On Constant-Factor Approximable Valued Constraint Satisfaction Problems Andrei Krokhin Durham University, UK Joint work with

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