Lijie Chen MIT Todays Topic Background What is FineGrained Complexity The Methodology of FineGrained Complexity Frontier FineGrained Hardness for Approximation Problems The Connection ID: 1002810
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1. Recent Developments in Fine-Grained Complexity via Communication ComplexityLijie ChenMIT
2. Today’s TopicBackground What is Fine-Grained Complexity?The Methodology of Fine-Grained ComplexityFrontier: Fine-Grained Hardness for Approximation ProblemsThe Connection[ARW’17]: Connection between Fine-Grained Complexity and Communication Protocols. ([Rub’18, CLM’18]: Further developments.)Our Results[Chen’18]: Hardness for Furthest Pair[CW’19]: A New Equivalence Class in Fine-Grained Complexity[CGLRR’19]: Fine-Grained Complexity Meets IP = PSPACE
3. What is Fine-Grained Complexity Theory? The goal of algorithm design and complexity theory What problems are efficiently solvable? If yes, find a fast algorithm!(algorithm designer’s job) If no, prove there are no fast algorithms! (complexity theorist’s job)What is “efficiently solvable”?Answer from Classical Complexity Theory: polynomial time! (e.g., , or… )
4. Classical Complexity TheoryPoly-Time vs. Super-Poly-TimeEfficient algorithmsPolynomial-timeInefficient algorithmsSuper-polynomial-timeShortest PathEdit DistanceRecognizing Map Graphs SATHamiltonian PathApproximate Nash Equilibrium
5. Why Poly-Time is not Always “Efficient”Case Study: Edit DistanceEdit Distance on DNA sequences : Measure how “close” two DNA sequences are Textbook algorithm: time given DNA sequences of length .Classical complexity theorists: This is efficient! GOOOOD Biologists: But I have data of 100GBs, is too slow…Is the best we can do?Classical complexity theorists: I don’t care, it is already efficientBiologists: @#$@$%#$^#%$#$%$#$#@!@#$#Fine-Grained complexity theorists: I care!
6. Fine-Grained ComplexityMotivationThe difference between and is HUGE in practice.But classical complexity theory says nothing about it except “I don’t care”. Goal of Fine-Grained Complexity TheoryFigure out the “exact exponent” for a problem! (Is it linear-time or quadratic time?)For example, is the best we can do for Edit Distance?Is the best we can do for All-Pair-Shortest-Path?Is the best we can do for Knapsack problem? Time limit exceeded on test 27Acceptedvs.
7. Methodology of Fine-Grained Complexity TheoryHow does Fine-Grained Complexity Theory work?
8. How does Classical Complexity Work?Ideally, want to unconditionally prove there is no polynomial-time algorithm for certain problems (like Hamiltonian Path).This appears to be too hard…(Require to show ).But still, there are two weapons: “assumptions” and “reductions”
9. Two Weapons of Complexity TheoristAssumptionsWe assume something without proving it (for example, or ).Under , the NP-complete problem has no poly-time problem. ReductionsThe surprising part is how much we get from a single assumption . Problem Problem reduction is harder than
10. Hardness via ReductionSATGiven a formula ,Is it satisfiable? Hamiltonian PathGiven a graph , is there a path visiting all nodes exactly once? has a Hamiltonian path is satisfiable s is not satisfiable Formula A graph Therefore, Hamiltonian Path is harder than SAT. Since SAT doesn’t have poly-time algorithms under . Neither does Hamiltonian Path.
11. Two Weapons of Fine-Grained Complexity Theorist(Stronger) AssumptionWe assume something without proving it, for example SETH (Strong Exponential Time Hypothesis).SETH: (Informally) SAT requires -time.SETH implies Orthogonal Vectors (OV) requires -time. OVFind an orthogonal pair, among vectors in (). Fine-Grained ReductionsProblem Problem -timereduction has no algos
12. SummaryClassical ComplexityWhich problems require super-poly time?Fine-Grained ComplexityWhich problems require (say) time? Basic QuestionsAssumptions SAT requires time. (for instance)OV requires time. ReductionsKarp-reductionFine-Grained ReductionIn short, Fine-Grained Complexity studied “more fine-grained” questions, with “more fine-grained” assumptions and reductions
13. The Success of Fine-Grained Complexityfor Exact Problemsdynamic data structures [Pat10, AV14, AW14, HKNS15, KPP16, AD16, HLNW17, GKLP17]computational geometry [Bri14,Wil18, DKL16] pattern matching [AVW14, BI15, BI16, BGL16,BK18]graph algorithms [RV13, GIKW17, AVY15, KT17]SETHA lot of success for exact problems (e.g. computing the edit distance exactly requires )
14. Dialogue ContinuedEdit Distance on DNA sequences : Measure how “close” two DNA sequences areTextbook algorithm: time given DNA sequences of length .Classical Complexity Theorists (Not here, trying to prove circuit lower bounds but no progress)Fine-Grained complexity theorists: I care! I can show very likely that is the best we can do for Edit Distance.Biologists: …OK, a (say) -approximation is also good enough! Any better algorithms for that?Fine-Grained complexity theorists: Probably not... Emmm…
15. Frontier: Fine-Grained Complexity for Approximation HardnessFor many natural problems, a good enough approximation is as good as an exact solution.Can we figure out the best exact exponent on those approximation algorithms?ExampleWhat is the best algorithm for 1.1-approximation to Edit Distance?
16. Challenge: How to Show Approximate Hardness?Exact CaseSETHEdit DistanceOVSETH1.1-approx. to Edit DistanceOVApproximation Case Yes No ?OVFind an orthogonal pair, among vectors in ().
17. Classical Solution: The PCP Theorem 0.88-approx. to 3-SATSAT is satisfiable Yes<0.88 fractions of clausesin is satisfiable No PCPs0.88-approx. to is as hard as determining whether is satisfiable
18. Major Challenge: How to Show Approximation Hardness in Fine-Grained Setting?The PCP theorem is too “coarse” to be applied in the fine-grained setting.SETHSAT of vars requires time PCP TheoremSAT of vars approx. to SAT of vars Approx. to SAT of vars.Requires time OVDrops by more than a polynomial comparing to !
19. Some Earlier Works[Roditty-Vassilevska’13]Distinguishing Diameter or requires time. (Approximation to Graph Diameter better than is HARD.)[Abboud-Backurs’17]Deterministic time algorithm for constant factor approximation to Longest Common Subsequence implies circuit lower bound(Approximate LCS may be hard to get.)
20. SummaryClassical complexity theory only cares about polynomial or not. This is very “coarse” for real world applications.Even vs can make a HUGE difference in the practice.Fine-Grained Complexity theory cares about the exact exponent on the running time.This program is very successful for exact problems, the complexity of many fundamental problems are characterized.It was less successful for approximation problems, due to the lack of techniques.PCP Theorem doesn’t work because of the blowup.
21. Today’s TopicBackground What is Fine-Grained Complexity?The Methodology of Fine-Grained ComplexityFrontier: Fine-Grained Hardness for Approximation ProblemsThe Connection[ARW’17]: Connection between Fine-Grained Complexity and Communication Protocols.[Rub’18, CLM’18]: Further developments.Our Results[Chen’18]: Hardness for Furthest Pair[CW’19]: A New Equivalence Class in Fine-Grained Complexity[CGLRR’19]: Fine-Grained Complexity Meets IP = PSPACE
22. [ARW’17]: Hardness of Approximation in P Via Communication Protocols!Key Contribution of [ARW’17]There is a framework to show fine-grained approximation result!The key: Communication Protocols!Max-IP: sets of vectors from .Compute . [ARW’17] approximation to Max-IP with dimensions requires Hardness for many other problems [ARW’17] Bichromatic LCS Closest Pair Over Permutations,Approximate Regular Expression Matching, and Diameter in Product Metrics
23. Merlin-Arthur(MA) ProtocolsMA Communication ProtocolF(x,y) = 1 exists a proof, .F(x,y) = 0 for all proofs, .Complexity = (Proof Length, Communication) Alice holds , Bob holds , want to compute
24. Set-DisjointnessDefinitionAlice holds , Bob holds Want to determine whether The NameLet and are disjoint
25. Lemma (Informal)An efficient MA protocol for Set-Disjointness A Fine-Grained Reduction from OV to Approx. Max-IP Merlin-Arthur Protocols Implies Reduction to Approx. Max-IP[AW’09]There is a good MA protocol for Set-Disjointness [ARW’17] approximation to Max-IP with dimensions requires OVOV requires time under SETH.
26. The High-Level ideaLet be an MA protocol for Set-Disjointness. OVGiven of vectors from , is there such that ? -Satisfying-PairGiven of vectors from , is there such that accepts? Approximate Max-IPEmbeddingsuch that is the acceptance probability of Approximation to Max-IP on solves OV on
27. SummaryHardness of Approximation in is the natural next step of the Fine-Grained Complexity program.[Abboud-Rubinstein-Williams’17]: Established the connection between fine-grained complexity and MA communication protocols. Proved many inapproximability results.Some Further Developments[Rubinstein’18]: Improved the MA protocols. Proved hardness of Approximate Nearest Neighbor Search.[C. S.-Laekhanukit-Manurangsi]: Generalize this to the -player setting. Proved hardness of Approximate -Dominating Set.
28. Motivation of Our WorksExplore More on connection between Fine-Grained Complexity and Communication ProtocolsCommunication protocols other than Merlin-Arthur protocols?
29. Today’s TopicBackground What is Fine-Grained Complexity?The Methodology of Fine-Grained ComplexityFrontier: Fine-Grained Hardness for Approximation ProblemsThe Connection[ARW’17]: Connection between Fine-Grained Complexity and Communication Protocols.[Rub’18, CLM’18]: Further developments.Our Results[Chen’18]: Hardness for Furthest Pair[CW’19]: A New Equivalence Class in Fine-Grained Complexity[CGLRR’19]: Fine-Grained Complexity Meets IP = PSPACE
30. [Chen’18] Protocols and Hardness of Furthest Pair
31. Closest Pair vs. Furthest PairClosest PairFurthest PairFind the pair with minimum distanceGiven points in Find the pair with maximum distance
32. Closest Pair vs. Furthest PairClosest PairFurthest Pair Best AlgorithmsWhen Always Goes to Is Furthest Pair “Far Harder” Than Closest Pair?HARDEASY
33. Closest Pair vs. Furthest PairTheorem Under SETH, Furthest Pair in dimensions requires time grows extremely slowly! is effectively a constant 17
34. Comparing to [Wil’18][Wil’18] Under SETH, Furthest Pair in dimensions (ours)requires time An “infinite” improvement
35. Closest Pair vs. Furthest Pair: UpdatedClosest PairFurthest Pair Best Algorithms Goes to Furthest Pair is “Far Harder” Than Closest Pair! Requires
36. Technique: Protocols MA Communication ProtocolF(x,y) = 1 exists a proof, .F(x,y) = 0 for all proofs, .Complexity = (Proof Length, Communication) Alice holds , Bob holds , want to compute Communication Protocol F(x,y) = 1 exists a proof, .F(x,y) = 0 for all proofs, .Complexity = (Proof Length, Communication)
37. Technique: Protocols Implies SETH-Hardness LemmaAn protocol for Set-Disjointness with proof length communication complexity under SETH, Furthest Pair in dimensions requires time
38. Technique: Protocols Via Recursive Chinese Remainder Theorem Theorem There is an protocol for Set-Disjointness with proof length communication complexity Proved by an involved recursive application of Chinese Remainder Theorem(See the paper )
39. Open QuestionCan we show that Furthest Pair in dimensions for any requires time?
40. SummaryFurthest Pair/ Closest Pair look similar. But we show that Furthest Pair is “far harder than” Closest Pair.In dimensions, closest pair is in time, furthest pair requires time under SETH protocols are natural relaxation of MA protocols.Fast protocols for Set-Disjointness Hardness for Furthest Pair.We construct an protocols with sub-linear proof complexity and communication complexity.
41. [CW’19] Communication Protocols and An Equivalence Class for OV
42. Fine-Grained Complexity:“Modern” NP-completenessMany Conceptual SimilaritiesNP-CompletenessWhich problems require super-poly time?Fine-Grained ComplexityWhich problems require (say) time? Basic QuestionsBasic Assumptions SAT requires time. (for instance)OV requires time. Weapons (Reductions)Karp-reductionFine-Grained ReductionPreserve being in PPreserve less-than-
43. The Key Conceptual DifferenceNP-completenessFine-Grained ComplexityHamiltonian PathVertex CoverMax-CliqueOrthogonal VectorsEdit DistanceSparse-Graph-DiameterApprox. Bichrom.Closest PairRubinstein 2018Backurs and Indyk 2015Except for the APSP equivalence classThousands of NP-complete problemsform an equivalence classFew Problems are knownTo be Equivalent to OVRoditty and V.Williams 2013
44. Why we want an Equivalence Class? IWhat does an equivalence class mean?All problems are essentially the same problem!Hamiltonian PathVertex CoverMax-CliqueThese NP-complete problemsare just SAT “in disguise”!A super strong understanding of the nature of computation!We cannot say “Edit Distance is just OV in disguise”
45. Why we want an Equivalence Class? IIConsequence of an equivalence class If “just one” NP-complete problem requires super-poly time, then all of them doHamiltonian CycleVertex CoverMax-CliqueIf “just one” NP-complete problem is in , then all problems are as well. OV in time doesn’t necessarily imply anything for OV-hard problems. Orthogonal VectorsEdit DistanceSparse-Graph-DiameterApprox. Bichrom.Closest Pair
46. This WorkAn Equivalence Class for Orthogonal Vectors in dims.In particular, OV is equivalent to approx. bichromatic closest pair. Two Frameworks for Reductions to OVwith communication protocols (this talk)with Locality Sensitive Hashing Families(see the paper)
47. OVFind an orthogonal pair, among vectors in (). A New Equivalence Class for OVMax-IP/Min-IPFind a red-blue pair of vectors with minimum (resp. maximum) inner product, among vectors in . Approx. Bichrom.-Closest-Pair: Compute a -approx. to the distance between the closest red-blue pair among points.Approx. Furthest-Pair:Compute a -approx. to the distance between the furthest pair among points Apx-Min-IP/-Max-IP Compute a approximation to Max-IP/Min-IP. Theorem (Informal)Either all of these problems are in sub-quadratic time ( for ), or none of them are.
48. Technique: Two Reduction FrameworksKnown Directions [R. Williams 05, Rubinstein 18]: OV Other Problems This work: Other Problems OV via two reduction frameworks Framework I (this talk)Based on Communication Protocols A Fast protocols A reduction to OV Framework II (see the paper)Based on Locality-Sensitive Hashing (LSH)An efficient LSH family A reduction to OV
49. Framework : communication protocols Communication Protocol for from Merlins.t. from Megan,Alice accepts after communicating with Bob.
50. Framework : communication protocols -Satisfying Pair ProblemGiven , s.t. Theorem (Informal)Efficient protocols for -Satisfying Pair can be reduced to OV. (Decisional) Max-IPGiven and a target , is there s.t. ? Define Max-IP is just -Satisfying PairThere is an efficient protocol for , so Max-IP can be reduced to OV. Application
51. Open ProblemsFind More Problems Equivalent to OVUnequivalence Results?
52. SummaryFine-Grained Complexity Mimics The Theory of NP-completeness and is very successful. One Important difference is that Fine-Grained Complexity lacks equivalence class for OV. protocols are analogy of in communication complexity.Efficient protocols implies fine-grained reductions to Orthogonal Vectors (OV).We construct efficient protocols and show an equivalence class for OV.In particular, OV is equivalent to Approximate Bichromatic Closest Pair.
53. [CGLRR’19] Fine-Grained Complexity Meets IP = PSPACE
54. IP ProtocolsMA Communication ProtocolF(x,y) = 1 exists a proof, .F(x,y) = 0 for all proofs, .Complexity = (Proof Length, Communication) Alice holds , Bob holds , want to compute IP Communication ProtocolAlice and Merlin now interact in several rounds.Complexity = (Total Proof Length, Communication)
55. IP = PSPACE and Communication Complexity[AW’09] (Informal) space algorithm for efficient IP protocols for . Closest-LCS-PairInput: Two sets of strings with max length Output: Theorem (Informal)efficient IP protocols for . Closest--Pair can be reduced to approx. Closest-LCS-Pair Closest-LCS-Pair can be reduced to approx. Closest-LCS-Pair. (That is, it is equivalent to its approximation version.)
56. SummaryIP protocols are generalization of Merlin-Arthur protocols where Merlin and Arthur interact for more than one round.Utilizing IP protocols, we show an equivalence between exact closest-LCS-pair and approximate closest-LCS-pair.There are many other results in the paper.
57. ConclusionFine-Grained Complexity want to understand the exact running time for problems in P.Still old weapons: assumptions and reductionsThe frontier: hardness for approximation algorithms[ARW’17]: Connect fine-grained complexity to communication complexity to show approximation hardness.Our work: Further explore this direction.[Chen’18]: Hardness for Furthest Pair with protocols[CW’19]: Equivalence Class for OV with protocols[CGLRR’19]: Applying IP = PSPACE to Fine-Grained Complexity