1 / 43

The Strong Exponential Time Hypothesis

The Strong Exponential Time Hypothesis. Daniel Lokshtnaov. Tight lower bounds. Have seen that ETH can give tight lower bounds How tight ? ETH « ignores » constants in exponent How to distinguish 1.85 n from 1.0001 n ?. SAT.

jerica
Download Presentation

The Strong Exponential Time Hypothesis

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. The StrongExponential Time Hypothesis Daniel Lokshtnaov

  2. Tightlowerbounds Have seenthatETHcangivetightlowerbounds How tight? ETH «ignores» constants in exponent How to distinguish1.85n from 1.0001n?

  3. SAT Input:Formulawithmclauses over nboolean variables. Question:Doesthereexist an assignment to the variables thatsatisfiesall clauses? Note: Input can have sizesuperpolynomial in n! Fastest algorithm for SAT: 2npoly(m)

  4. d-SAT Here all clauses have size d Input size nd Fastest algorithm for 2-SAT: n+mFastest algorithm for 3-SAT: 1.31nFastest algorithm for 4-SAT: 1.47n … Fastest algorithm for d-SAT: Fastest algorithm for SAT:

  5. Strong ETH Let d-SAT has a algorithm Let Know:0 Sd 1 SETH: 1 ETH: s3 0

  6. ShowingLowerBounds under SETH Your Problem d-SAT Too fast algorithm? The numberof9’s MUSTbe independentofd

  7. Dominating Set Input:nvertices, integerk Question: Is there a setSof at most kverticessuchthatN[S] = V(G)? Naive: nk+1Smarter: nk+o(1) Assuming ETH: nof(k)no(k) nk/10? nk-1?

  8. SAT  k-Dominating Set Variables SAT-formula k groups, each on n/k variables. One vertex for each of the 2n/kassignments to the variables in the group.

  9. Variables SAT-formula k groups, each on n/k variables. x x x x x y y y y y Selecting one vertex from each cloud corrsponds to selecting an assignment to the variables. Cliques

  10. Variables SAT-formula k groups, each on n/k variables. x x x x x y y y y y Edge if the partial assignmentsatisfies the clause One vertex per clause in the formula

  11. SAT  k-Dominating Setanalysis Too fast algorithmfor k-Dominating Set: nk-0.01 For anyfixedk (like k=3) If m 2n/kthen2n is at most mk, which is polynomial! So m The output graph has k2n/k + m 2k 2n/kvertices

  12. Dominating Set, wrapping up A O(n2.99)algorithm for 3-Dominating Set, or a O(n3.99)algorithm for 4-Dominating Set, or a a O(n4.99)algorithm for 5-Dominating Set, or a … … wouldviolateSETH.

  13. Independent Set / Treewidth Input: Graph G, integerk, tree-decompositionofGofwidth t. Question:DoesG have an independentsetofsize at leastk? DP: O(2tn) time algorithm Canwe do it in 1.99t poly(n) time? Next: If yes, then SETH fails!

  14. Independent Set / Treewidth Will reducen-variable d-SAT to Independent Setin graphsoftreewidtht, wheret n+d. So a 1.99tpoly(N)algorithm for Independent Setgives a 1.99n+dpoly(n) O(1.999n) time algorithm for d-SAT.

  15. Independent Sets on an Even Path t f t f t f t f t f True False then False first True In independentset: Not in solution:

  16. d-SAT Independent Setproof by example = (a b c) (a c d) (b c d) b c c a c a d b d a t f t f t f b t f t f t f c t f t f t f d t f t f t f

  17. Independent Sets Assignments = (a b c) (a c d) (b c d) b c c Butwhataboutthe firsttruethenfalseindependentsets? a c a d b d True a t f t f t f False b t f t f t f True c t f t f t f False d t f t f t f

  18. Dealingwithtruefalse Clause gadgets 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 a b c d Every variable flipstruefalseat most once!

  19. TreewidthBoundby picture b c c d a c a d b d … a t f t f t f … b t f t f t f n … c t f t f t f … d t f t f t f Formal proof - exercise

  20. Independent Set / Treewidthwrap up Reducedn-variable d-SAT to Independent Setin graphsoftreewidtht, wheret n+d. A 1.99tpoly(N)algorithm for Independent Setgives a 1.99n+dpoly(n) O(1.999n) time algorithm for d-SAT. Thus, no1.99talgorithmfor Independent SetassumingSETH

  21. AssumingSETH, the following algorithms are optimal: • 2t poly(n) for Independent Set • 3tpoly(n) for Dominating Set • ctpoly(n) for c-Coloring • 3tpoly(n) for Odd Cycle Transversal • 2tpoly(n) for Partition Into Triangles • 2tpoly(n) for Max Cut • …

  22. 3tlowerbound for Dominating Set? Need to reducek-SATformulasonn-variables to Dominating Set in graphsoftreewidtht, where So

  23. Hitting Set / n Input: Family F = {S1,…,Sm}ofsets over universeU = {v1, …, vn}, integerk. Question:Doesthereexist a setX Uofsize at most ksuchthat for everySi F, SiX? Naive algorithm runs in time. Next:impliesthatSETHfails

  24. d-SAT Hitting Set = (a b c) (a c d) (b c d) Budget = 4

  25. d-SAT vsHitting Set A cnalgorithm for Hitting Set makes a c2nalgorithm for d-SAT. Since1.412n < 1.9999n, a 1.41nalgorithm for Hitting Set violatestheSETH. Have a 2nalgorithmand a 1.41nlowerbound. Next:2nlowerbound

  26. Hitting Set For anyfixed, willreducek-SATwithn variables to Hitting Setwithuniversewith at most (1+)n elements. So a 1.99nalgorithm for Hitting Set gives a 1.99n(1+) 1.999n time algorithm for k-SAT

  27. Somedeepmath For everythereexists a naturalnumbergsuchthat, for t = we have: Why is this relevant?

  28. d-SAT Hitting Set Group the variables intogroupsofsizeg, and sett =. Solution budget from eachgroup Will force from eachgroup  Exactly from eachgroup g g g Variables: g g t t t t t Elements: Elements (1 + ) variables

  29. Analyzing a group Group ofg variables 2gassignments to variables Injection subsetsof elements ofsize exactly. Group oft elements

  30. Forcingsolutionverticesin a group? Add all subsetsofthegroupofsize to thefamilyF. Lets callthesesetsguards Any setthatpicks less than elements thegroup misses a guard. Anysetthatpicks at least elements from eachgrouphits all theguards

  31. Analyzing a group Group ofg variables assignments to variables Injection subsetsof elements ofsizeexactly. Group oft elements Whataboutthe element subsetsofsizethat do not correspond to assignments?

  32. Sets ofsize Adding a setofsize to thefamilyFensuresthatthe «groupcomplement» set is not picked. All othersetsofsize in thegroupmay still be picked in solution. Forbidsetsofsizethat do not correspond to assignments.

  33. d-SAT Hitting Set g g g Variables: g g assignments potentialsolutions t t t t t Elements: Want: SolutionsSatisfyingassignments

  34. Forbiddingpartialassignments Pick anydgroupsof variables, and considersomeassignment to these variables. If thisassignmentfalsifieswewant to forbidthecorrespondingset in theHitting Set instance from beingselected.

  35. Forbiddingpartialassignments Bad assignment … Variables … Set added to F to forbidthe bad assignment

  36. Forbiddingpartialassignments For each bad assignment to at most dgroups, forbid it by adding a «bad assignmentguard» This addsO(nd2gd) = O(nd) sets to F.

  37. SatisfyingAssignmentsHitting Sets A satisfyingassignment has no bad sub-assignments corresponds to a hittingset. A hittingsetcorresponds to an assignment. If thisassignmentfalsified a clauseC, theassignmentwould be bad for the dgroupsC lives in, and miss a bad assignmentguard.

  38. Hitting Set wrap up Can reducen variable d-SAT to n(1+) element Hitting Set. So a cnalgorithm for Hitting Set yields a (c+)nalgorithm for d-SAT. A 1.99nalgorithm for Hitting Set wouldviolate SETH.

  39. Conclusions SETHcan be used to giveverytightrunning time bounds. SETHrecently has been used to givelowerbounds for polynomial time solvable problems, and for running timeofapproximationalgorithms.

  40. Important Open Problems Canwe show a 2nlowerbound for Set Cover assumingSETH? Canwe show a 1.00001lowerbound for 3-SATassumingSETH?

  41. Exercises Show thatSETHimpliesETH (Hint: usethesparsification lemma) Exercise 4.16, 4.17, 4.18, 4.19.

  42. Postdoc in Bergen? Muchbetterthan Budapest. Really.

  43. Thank You

More Related