Venkatesan Guruswami Yuan Zhou (Carnegie Mellon University) Approximating bounded occurrence ordering CSPs 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 Random assignment Max-Cut >= 0.878... [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

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) 5 ordering constraints, OPT = 4 1 2 3 4 5

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] Step 2. Extend t-ordering to full ordering by random (within each bin) 1 2 3 4 ... ... 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

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. 1 when σt(i) < σt(j) w(σt(i), σt(j)) = 0 when σt(i) > σt(j) 1/2 when σt(i) = σt(j) Answer. To maximize regular CSP with domain size t !

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))

|{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 1 2 3 4 ... ... n A B=[n]\A < 2 bins: |{i: iэA, i+1эB}|+ |{i: i,i+1эA}|/2+|{i: i,i+1эB}|/2 A

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

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!