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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)
Andrei Krokhin - On constant-factor approximable Min CSPs 2
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
Andrei Krokhin - On constant-factor approximable Min CSPs 3
(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
Andrei Krokhin - On constant-factor approximable Min CSPs 4
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
Andrei Krokhin - On constant-factor approximable Min CSPs 5
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.
Andrei Krokhin - On constant-factor approximable Min CSPs 6
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.
Andrei Krokhin - On constant-factor approximable Min CSPs 7
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]
Andrei Krokhin - On constant-factor approximable Min CSPs 8
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?
Andrei Krokhin - On constant-factor approximable Min CSPs 9
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
Andrei Krokhin - On constant-factor approximable Min CSPs 10
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
Andrei Krokhin - On constant-factor approximable Min CSPs 11
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
Andrei Krokhin - On constant-factor approximable Min CSPs 12
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).
Andrei Krokhin - On constant-factor approximable Min CSPs 13
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
Andrei Krokhin - On constant-factor approximable Min CSPs 14
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})
Andrei Krokhin - On constant-factor approximable Min CSPs 15
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)
Andrei Krokhin - On constant-factor approximable Min CSPs 16
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.
Andrei Krokhin - On constant-factor approximable Min CSPs 17
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)
Andrei Krokhin - On constant-factor approximable Min CSPs 18
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.
Andrei Krokhin - On constant-factor approximable Min CSPs 19
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
Andrei Krokhin - On constant-factor approximable Min CSPs 20
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).
Andrei Krokhin - On constant-factor approximable Min CSPs 21
Fractional Symmetric Operation Example
1/6 1/6 1/8
1/8 1/6 1/4
Andrei Krokhin - On constant-factor approximable Min CSPs 22
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.
Andrei Krokhin - On constant-factor approximable Min CSPs 23
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)).
Andrei Krokhin - On constant-factor approximable Min CSPs 24
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.
Andrei Krokhin - On constant-factor approximable Min CSPs 25
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).
Andrei Krokhin - On constant-factor approximable Min CSPs 26
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.
Andrei Krokhin - On constant-factor approximable Min CSPs 27
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.
Andrei Krokhin - On constant-factor approximable Min CSPs 28
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