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Approximating Bounded Occurrence Ordering CSPs

This paper discusses the approximability of bounded occurrence ordering CSPs and provides a proof for its non-approximation resistance. It also presents a generalization of Berger-Shor's theorem and explores the concept of t-ordering.

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Approximating Bounded Occurrence Ordering CSPs

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  1. 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 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 1 2 3 4 5

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

  13. 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 !

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

  15. Negative news for t = 2 • Take MAS for example opt(I) = n-1 opt(I2) = max <= n/2 1 2 3 4 ... ... n B=[n]\A 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 opt(I3) >= (n-1) * 2/3 1 2 3 4 ... ... n < < 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!

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