Venkatesan Guruswami
Yuan Zhou
(Carnegie Mellon University)
(Boolean) Constraint Satisfaction Problems
• Given:
– a set of variables: V = {1, 2, 3, ..., n}
– a set of values: Ω = {0, 1}
– a set of "local constraints": E
• Goal: find an assignment σ : V -> Ω to maximize
#satisfied constraints in E
• Examples (prob. name and typical local constraint)
– Max-Cut: σ(i) ≠ σ(j)
– Max-3LIN: σ(i)+σ(j)+σ(k) = 0/1 (mod 2)
– Max-3SAT: σ(i) + σ(j) + σ(k) >= 1
(Boolean) Constraint Satisfaction Problems (cont'd)
• Given:
– a set of variables: V = {1, 2, 3, ..., n}
– a set of values: Ω = {0, 1}
– a set of "local constraints": E
• Goal: find an assignment σ : V -> Ω to maximize
#satisfied constraints in E
• α-approximation algorithm: always outputs a
solution of value at least α*OPT
Approximability of some Boolean CSPs
Max-Cut
Approximability
Random
assignment
>= 0.878...
1/2
[GW95]
Max-3LIN
< 1/2 + ε
1/2
approx.
resistant
7/8
approx.
resistant
[Hastad01]
Max-3SAT
< 7/8 + ε
[Hastad01]
• Approximation resistant: when random
assignment is the best approximation algorithm
Bounded occurrence CSPs
• B-bounded occurrence: each variable appears in
at most B constraints
• Theorem. [Hastad00] B-bounded occurrence
Boolean CSPs admit (random + Ω(1/B))approximation algorithm
==> Not approximation resistant
Ordering CSPs
• Given:
– a set of variables: V = {1, 2, 3, ..., n}
– "local constraints" E, on the order of related
variables
• Goal: find an ordering σ : V -> [n] to maximize
#satisfied constraints in E
Ordering CSPs (cont'd)
• Example
– Maximum Acyclic Subgraph (MAS)
• Constraints: for each (i, j) э E, σ(i) < σ(j)
1
2
3
4
5
5 ordering constraints, OPT = 4
Ordering CSPs (cont'd)
• More Examples
– Maximum Acyclic Subgraph (MAS)
• Constraints: for each (i, j) э E, σ(i) < σ(j)
– k-ary monotone constraint
• (i1, i2, ..., ik) э E, σ(i1) < σ(i2) < ... < σ(ik)
– Betweenness
• (i, j, k) э E, σ(i) < σ(j) < σ(k) or σ(k) < σ(j) < σ(i)
Approximability of ordering CSPs
• Theorem. [GMR08, CGM09, CGHMR11] Assuming the
Unique Games Conjecture, every ordering CSP is
approximation resistant.
• Bounded occurrence ordering CSPs?
• Theorem. [Berger-Shor97] The B-bounded
occurrence maximum acyclic subgraph problem
admits a (1/2+Ω(1/√B))-approximation algorithm
Our results
• Goal. Every bounded occurrence ordering CSP is
not approximation resistant (generalization of
Hastad's theorem for CSPs)
• Theorem. Every B-bounded occurrence
monotone ordering CSP can be approximated by
(1/(k!) + Ω(1/B))
– A generalization of Berger-Shor
• Theorem. Every 3-ary bounded occurrence CSP
is not approximation resistant
Technical Part : Proof of Theorems
Proof sketch
• Step 1. Find t-ordering instead of full ordering
– t-ordering: a mapping σt : V -> [t]
n variables:
1 2 3 4
... ...
n
σt:
t bins:
1,3,7
random
assignment
full ordering: 3 < 7 < 1
<
2,5
<
2<5
<
4,6,8
< 4<8<6
• Step 2. Extend t-ordering to full ordering by
random (within each bin)
Proof sketch (cont'd)
• Step 1. Find t-ordering instead of full ordering
– t-ordering: a mapping σt : V -> [t]
• Step 2. Extend t-ordering to full ordering by
random (within each bin)
• Problem. What kind of t-ordering do we want?
(Take MAS as example,) in Step 2,
constraint σ(i) < σ(j) is satisfied w.p.
1
0
1/2
when σt(i) < σt(j)
w(σt(i), σt(j)) =
when σt(i) > σt(j)
when σt(i) = σt(j) regular CSP
w( t (i ),  t ( j )) with domain
• Answer. To maximize
( i , j )E
size t !

Proof sketch (cont'd)
Ordering CSP I
final ordering
random
t-ordering CSP
It (regular CSP)
(variant of)
Hastad's alg.
t-ordering for
It
• Theorem. [Hastad00] Given an B-bounded
occurrence CSP instance It, there is an
algorithm finding a solution of value at least
rand(It) + Ω(opt(It) - rand(It))/B
• Goal. Suffices to show that for some constant t,
opt(It) - rand(I) = Ω(opt(I) - rand(I))
Negative news for t = 2
• Take MAS for example
1
2
opt(I) = n-1
2 bins:
3
4
... ...
n
B=[n]\A
A
<
|{i: iэA, i+1эB}|+
opt(I2) = max
A |{i: i,i+1эA}|/2+|{i: i,i+1эB}|/2
<= n/2
What about t = 3 ?
• Take MAS for example
1
2
3
4
... ...
n
opt(I) = n-1
3 bins:
1,4,7,...
< 2,5,8,...
opt(I3) >= (n-1) * 2/3
<
3,6,9,...
In general...
• Lemma. t = 4 works for
– monotone bounded occurrence ordering CSPs
– every 3-ary bounded occurrence ordering CSP
I.e., for any instance I from the two cases above,
opt(I4) - rand(I) = Ω(opt(I) - rand(I))
• Remark. t = 3 might also work -- but we do not
have a proof.
Proof sketch of the lemma
• Write objective value of I4 as the maximum
value of a function over Boolean cube
f : {-1, 1}2n -> R≥0
(encode each of the n values with 2 Boolean bits)
f ( x)   fˆ ( S )  S ( x)
• Fourier expansion.
S [ 2 n ]
• Observation. rand ( I 4 )  E[ f ]  fˆ ( )
• Definition. adv(f)   fˆ ( S )
S 
• Technical Lemma. [Hastad00] If f has constant
degree and is "B-occurrence bounded", there is
an algorithm finding x such that
f(x) = E[f] + Ω(adv(f))/B
Proof sketch of the lemma (cont'd)
• Technical Lemma. [Hastad00] If f has constant
degree and is "B-occurrence bounded", there is
an algorithm finding x such that
f(x) = E[f] + Ω(adv(f))/B
• Lemma. For monotone/3-ary ordering CSPs,
adv(f) = Ω(opt(I) - rand(I))
• Proof. Fourier analysis, and...
stare at the Fourier spectrum of the pay-off
functions in the 4-ordering instances...
Conclusion & open questions
• It is easy to beat random assignments for many
bounded occurrence ordering CSPs
• Hard instances for ordering CSPs cannot be
bounded occurrence
• Question 1. Algorithm for all bounded
occurrence ordering CSPs?
• Question 2. Improve the Ω(1/B) bound? -Where Berger-Shor gets Ω(1/√B).
– Maybe monotone constraint is the first step?
Thanks!