1 / 21

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

dunne
Download Presentation

Venkatesan Guruswami Yuan Zhou (Carnegie Mellon University)

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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!

More Related