1. Venkatesan Guruswami Yuan Zhou (Carnegie Mellon University)
Approximating bounded
occurrence ordering CSPs
Venkatesan Guruswami
Yuan Zhou
(Carnegie Mellon University)

2. (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

3. (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

4. Approximability of some Boolean CSPs
Random assignment
Max-Cut
>=
[GW95]
1/2
Max-3LIN
< 1/2 + ε
[Hastad01]
Max-3SAT
< 7/8 + ε
7/8
approx. resistant
approx. resistant
Approximation resistant: when random assignment is the best approximation algorithm

5. 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

6. 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

7. Ordering CSPs (cont'd) Example Maximum Acyclic Subgraph (MAS)
Constraints: for each (i, j) э E, σ(i) < σ(j)
5 ordering constraints, OPT = 4

8. 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)

9. 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

10. 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

11. Technical Part : Proof of Theorems

12. Proof sketch < < < <
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) n
n variables:
σt:
<
<
t bins:
1,3,7
2,5
4,6,8
random assignment
full ordering:
3 < 7 < 1
<
2 < 5
<
4 < 8 < 6

13. regular CSP with domain size t !
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.
when σt(i) < σt(j)
w(σt(i), σt(j)) = when σt(i) > σt(j)
1/ when σt(i) = σt(j)
Answer. To maximize
regular CSP with domain size t !

14. Proof sketch (cont'd) Ordering CSP I final ordering
random
(variant of) Hastad's alg.
t-ordering CSP It (regular CSP)
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))

15. |{i: i,i+1эA}|/2+|{i: i,i+1эB}|/2
Negative news for t = 2
Take MAS for example
opt(I) = n-1
opt(I2) = max
<= n/2 n A
B=[n]\A
<
2 bins:
|{i: iэA, i+1эB}|+
|{i: i,i+1эA}|/2+|{i: i,i+1эB}|/2 A

16. What about t = 3 ? < < Take MAS for example opt(I) = n-1
3 bins:
1,4,7,...
<
2,5,8,...
<
3,6,9,...

17. 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.

18. Proof sketch of the lemma
Write objective value of I4 as the maximum value of a function over Boolean cube
f : {-1, 1} -> R
(encode each of the n values with 2 Boolean bits)
Fourier expansion.
Observation.
Definition.
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 ≥0 2n

19. 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...

20. 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?

21. Thanks!