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Combinatorial algorithms

Explore combinatorial algorithms tackling the Hamilton cycle problem. Learn about its history, conjectures, and recent breakthroughs like the Middlelevels conjecture. Dive into proof ideas and results surrounding this challenging computational puzzle.

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Combinatorial algorithms

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  1. Combinatorial algorithms Torsten Mütze

  2. The Hamilton cycleproblem • Problem:Given a graph, doesithave a Hamilton cycle? • fundamentalproblemwithmanyapplications(specialcase of travellingsalesmanproblem) • computational point of view: • no efficientalgorithmknown (NP-complete[Karp 72]), i.e. brute-forceapproachessentially best possible

  3. The Hamilton cycleproblem • named after Sir William Rowan Hamilton(1805-1865) and his `Icosian Game’ • Most advanced version of this puzzle: Conjecture[Lovász 70]: Everyconnectedvertex-transitivegraph has a Hamilton cycle (apart fromfiveexceptions). • vertex-transitive = graph `looks the same’ from every vertex • various related/opposing conjectures, e.g. [Babai 95] • This thesis: solutions of several long-standing special cases

  4. Partial results • Known cases: • and prime • [Turner 67], [Alspach 79], [Marušič 86, 87], [Chen 98], [Kutnar, Marušič 08] • Every connected and vertex-transitive graph has a cycle of length at least [Babai 79] • Many results known for Cayley graphs: • abelian groups [Marušič 83] • finite p-groups [Witte 86] • every finite group has gen. set of size[Pak, Radoičič 09]

  5. Combinatorialalgorithms • Goal:generate all objects of a combinatorialclassefficiently • Examples: • binarytrees • triangulations • permutations • words • partitions • linear extensions • spanningtrees • matchings • … aabc 1234 ccba 1243 ccca 1423

  6. Flip graphs • Goal:generate all objects of a combinatorialclassefficiently • fundamental taskwithmanyapplications • ultimately: eachnewobject in constanttime • consecutiveobjectsmaydifferonly`a littlebit‘ • defines a highlysymmetricgraph, oftenevenvertex-transitive 111 321 101 011 110 231 312 213 132 001 100 010 123 • associahedron • permutahedron • hypercube 000 • Questions: Existence of a Hamilton cycle? Efficientalgorithm?

  7. Applications Song Duration Problem: Partition songsintosubsets A and B of sizeorwithminimum play lengthdifference. • 2:432 3:203 2:18…7 1:58 Partition is hard! 1234567 1234567 1000011 1000011 0110110 1000111 Question: Isthere a cyclicsingle-fliplisting of all bitstrings of lengthwithormany 1-bits forevery ?

  8. The middlelevelsconjecture Define a graph : 10110 10101 01101 01011 00111 11100 11010 11001 10011 01110 10001 01100 00110 00101 00011 11000 10100 10010 01010 01001 • is a subgraph of the -dimensional hypercube • isvertex-transitive Question: Isthere a cyclicsingle-fliplisting of all bitstrings of lengthwithormany 1-bits forevery ? Middlelevelsconjecture: has a Hamilton cyclefor all .

  9. The middlelevelsconjecture • notoriouslydifficultproblemraised in the 1980s • hardestopenproblem in [Knuth 11] withdifficulty 49/50 • mentioned in popularbooks[Diaconis, Graham 12], [Winkler 04], and in survey[Gowers 17] • ≈25 previouspaperswith partial results ([Kierstead, Trotter 88], [Savage, Winkler 95], [Felsner, Trotter 95], [Johnson 04], …) • has severalfar-rangingimplications • specialcase of Lovász‘ conjecture Middlelevelsconjecture: has a Hamilton cyclefor all .

  10. Ourresults Theorem 1 [M. 16; Proc. LMS]: The middlelevelsgraph has a Hamilton cycleforevery . Theorem 2 [M. 16; Proc. LMS]: The middlelevelsgraph has distinct Hamilton cycles. Remarks: number of automorphismsisonly ,so Theorem 2 isnot an immediate consequence of Theorem 1

  11. Ourresults Theorem 1 [M. 16; Proc. LMS]: The middlelevelsgraph has a Hamilton cycleforevery . Theorem 2 [M. 16; Proc. LMS]: The middlelevelsgraph has distinct Hamilton cycles. Remarks: number of Hamilton cyclesis at most ,so Theorem 2 is best possible

  12. Ourresults Theorem 1 [M. 16; Proc. LMS]: The middlelevelsgraph has a Hamilton cycleforevery . Theorem 2 [M. 16; Proc. LMS]: The middlelevelsgraph has distinct Hamilton cycles. • Remarks: • Werecentlyfound a short and moreaccessibleproofforthosetheorems[Gregor, M., Nummenpalo 18; Discrete Analysis] • 40 pages 9 pages • 27 lemmas 2 lemmas • 88 formulas 9 formulas

  13. Proofideas Step 1:Build a 2-factor in thegraph 2-factor

  14. Proofideas Step 1:Build a 2-factor in thegraph Step 2: Connect the cycles in the 2-factor to a singlecycle 2-factor flippable pair

  15. Proofideas Step 1:Build a 2-factor in thegraph Step 2: Connect the cycles in the 2-factor to a singlecycle 2-factor flippable pair

  16. Proofideas Step 1:Build a 2-factor in thegraph Step 2: Connect the cycles in the 2-factor to a singlecycle The middlelevelsgraphdoes not have 4-cycles, so weuse 6-cycles 2-factor flippable pair

  17. Proofideas Step 1:Build a 2-factor in thegraph Step 2: Connect the cycles in the 2-factor to a singlecycle The middlelevelsgraphdoes not have 4-cycles, so weuse6-cycles 2-factor flippable pair

  18. Proofideas Step 1:Build a 2-factor in thegraph Step 2: Connect the cycles in the 2-factor to a singlecycle 2-factor Auxiliarygraph 1 1 4 4 3 3 2 2 8 5 8 6 5 6 flippable pairs (disjoint) 7 7

  19. Proofideas Lemma 1:Ifisconnected, then has a Hamilton cycle. Lemma 2:If has distinctspanningtrees, then has distinct Hamilton cycles. 2-factor Auxiliarygraph 1 1 4 4 3 3 2 2 8 5 8 6 5 6 flippable pairs (disjoint) 7 7

  20. The crucialreduction Provethat auxiliarygraph isconnected (has manyspanningtrees) Provethat middlelevelsgraph has a Hamilton cycle (many Hamilton cycles)

  21. Algorithmicresults Theorem 3 [M., Nummenpalo 17; SODA]: Thereis an algorithmwhichfor a givenvertex of the middlelevelsgraphcomputes the nextone on a Hamilton cycle in time . • initialization time is and requiredspaceis • Remarks • C++ codeavailable on ourwebsite

  22. BipartiteKnesergraphs • integer parameters and • vertices = all -element and -element subsets of • edges = • iff is the middle • Examples levelsgraph {2,3,4} {1,3,4} {1,2,4} {1,2,3} {1,2} {1,3} {2,3} • 1 {1} {2} {3} {4} {1} {2} {3}

  23. IsHamiltonian? • Conjecture: • For all and the graph has Hamilton cycle. • raisedby[Simpson 91], and Roth (see[Gould 91], [Hurlbert 94]) • anotherinstance of Lovász‘ conjecture • middlelevelsconjecture

  24. IsHamiltonian? • Knownresults: • has a Hamilton cycleif • [Shields, Savage 94] • [Chen 03] (followingearlierworkby[Simpson 94], [Hurlbert 94], [Chen 00]) • ???

  25. Ourresults • Theorem 4 [M., Su 17; Combinatorica]: • For all and the graph has Hamilton cycle. Remark: simple inductionproof, assuming the validity of the middlelevelsconjecture

  26. Generalized MLC 11...1 Conjecture[Savage 93], [Gregor, Škrekovski 10]:For any and ,the middlelevels ofhave a Hamilton cycle. levels • Known results: 00...0 [Gray 53] [El-Hashash, Hassan 01], [Locke, Stong 03] [Gregor, Škrekovski 10] ? [Gregor, Jäger, M.; ICALP 18] [M. 16] Middlelevelsconjecture

  27. Generalized MLC 11...1 Conjecture[Savage 93], [Gregor, Škrekovski 10]:For any and ,the middlelevels ofhave a Hamilton cycle. levels 00...0 • Theorem 5 [Gregor, M. 17; STACS+TCS]: • For any and anyinterval not part of thisconjecture,the subgraph of withlevels in thisinterval has an `almost‘ Hamilton cycle, and wehavecorrespondingconstant-time generation algorithms.

  28. Knesergraphs Knesergraphs • integer parameters and • vertices = all -element subsets of • edges = • iff Petersen graph Completegraph {1,2} {1} {2} {3,5} {3,4} {4,5} {2,3} {1,5} {4} {3} {2,4} {1,4} {2,5} {1,3}

  29. Knesergraphs Knesergraphs • introducedby[Lovász 78] to proveKneser‘sconjecture • have long been conjectured to have Hamilton cycle, with one notable exception, the Petersen graph • sparsest and therefore hardest case is when • anotherinstance of Lovász‘ conjecture • odd graphs • degree , which is logarithmic in number of vertices Observation:Hamiltonicity of impliesHamiltonicity of .

  30. Knesergraphs Knesergraphs • conjecture that is Hamiltonian for raised in the 70s[Meredith, Lloyd 72], [Biggs 79] • , [Balaban 72] • , [Meredith, Lloyd 72+73] • [Mather 76] • [Shields, Savage 04] has a Hamilton cycle if • [Heinrich, Wallis 78] • [B. Chen, Lih 87] • [Y. Chen 00], [Y. Chen, Füredi 02] • [Y. Chen 03] several other partial results [Johnson 04], [Johnson, Kierstead 04], etc.

  31. Ourresults • Theorem 6 [M., Nummenpalo, Walczak 18; STOC]: • For all , the oddgraph has aHamilton cycle. • Theorem 7 [M., Nummenpalo, Walczak 18; STOC]: • For all and , the graph has aHamilton cycle. • Theorem 8 [M., Nummenpalo, Walczak 18; STOC]: • For all , the oddgraph has at least distinctHamilton cycles.

  32. Proofideas Step 1:Build a 2-factor in thegraph 2-factor • based on Dyck words of length[M., Standke, Wiechert 17; EurJC] • all cycles have the same length • number of cycles = th Catalan number

  33. Proofideas Step 1:Build a 2-factor in thegraph Step 2: Connect the cycles in the 2-factor to a singlecycle 2-factor flippable triple flipping 6-cycle flippable triples

  34. Proofideas Step 1:Build a 2-factor in thegraph Step 2: Connect the cycles in the 2-factor to a singlecycle 2-factor auxiliaryhypergraph 2 2 1 1 5 5 4 4 3 3 7 7 6 6 flippable triples

  35. Proofideas • translates problem into proving that has a loosespanning tree • connectivity is not enough; construct spanning tree directly 2-factor auxiliaryhypergraph 2 2 1 1 5 5 4 4 3 3 7 7 6 6 flippable triples

  36. Resultssummary • Middle levels conjecture • [M., Weber 2012; Journal of CombinatorialTheorySeries A] • [M. 2016; Proceedings of the London Mathematical Society] • [Gregor, M., Nummenpalo 2018; Discrete Analysis] • [M., Nummenpalo 2015; ESA] • [M., Nummenpalo 2017; SODA] • [Gregor, M. 2017; STACS], [Gregor, M. 2017; Theoretical Computer Science] • (Bipartite) Kneser graphs • [M., Su 2017; Combinatorica] • [M., Standke, Wiechert 2018; European Journal of Combinatorics] • [M., Nummenpalo, Walczak 2018; STOC]

  37. Thankyou!

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