1. Fast Hamiltonicity Checking via Bases of Perfect Matchings
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Marek Cygan, Stefan Kratsch, Jesper Nederlof
STOC 2013

2. Hamiltonicity (aka Hamiltonian cycle)
Held&Karp (‘61), Bellman (‘62):                time and space (Dynamic Programming).
Gurevich&Shelah (‘71):                time, poly space (Divide & Conquer).
Kohn et al. (‘77), Karp:                time, poly space (Inclusion-Exclusion).
Björklund (‘10):                     time, poly space (Determinant approach).

3. Hamiltonicity (aka Hamiltonian cycle) Traveling Salesman
Held&Karp (‘61), Bellman (‘62):                time and space (Dynamic Programming).
Gurevich&Shelah (‘71):                time, poly space (Divide & Conquer).
Kohn et al. (‘77), Karp:                time, poly space (Inclusion-Exclusion).
Björklund (‘10):                     time, poly space (Determinant approach).

4. Hamiltonicity (aka Hamiltonian cycle)
Decompose HC into subpaths, keep track of visited vertices (     ) to make sure subpaths combine.
Held&Karp (‘61), Bellman (‘62):                time and space (Dynamic Programming).
Gurevich&Shelah (‘71):                time, poly space (Divide & Conquer).
Kohn et al. (‘77), Karp:                time, poly space (Inclusion-Exclusion).
Björklund (‘10):                     time, poly space (Determinant approach).

5. Hamiltonicity (aka Hamiltonian cycle)
Decompose HC into subpaths, keep track of visited vertices (     ) to make sure subpaths combine.
Held&Karp (‘61), Bellman (‘62):                time and space (Dynamic Programming).
Gurevich&Shelah (‘71):                time, poly space (Divide & Conquer).
Kohn et al. (‘77), Karp:                time, poly space (Inclusion-Exclusion).
Björklund (‘10):                     time, poly space (Determinant approach).
Split input graph into two subgraphs. Decompose HC into intersection with subgraphs. Keep track of.. what exactly?

6. Directed Bipartite graphs

7. Directed Bipartite graphs

8. Graph with separator S
     
     

9. Graph with separator 2
     
      S

10. Graph with separator 2 2 2
     
      S

11. Graph with separator 2 2 2
      S
     
Seems to require storing information for every matching. ->                        

12. Our contribution
We determine the required amount of stored information via studying the rank of the ‘Matchings Connectivity matrix’:
For even integer           , and perfect matchings                 of the complete graph on                   ,
                                       

13. Our contribution We show that and use it to
over GF(2),        has rank at most              by giving
an explicit basis       and corresponding factorization.
changing representation between can be done quick.
      induces a permutation matrix.
and use it to
Solve Hamiltonicity on Directed Bipartite graphs in
                       time and space.
Solve Hamiltonicity in                                 time and space
if a path decomposition of width        is given.
Exclude                                        time under SETH.

14. k-CNF-SAT on n vars cannot be solved in time when tends to infinity.
Our contribution
We show that
over GF(2),        has rank at most              by giving
an explicit basis       and corresponding factorization.
changing representation between can be done quick.
      induces a permutation matrix.
and use it to
Solve Hamiltonicity on Directed Bipartite graphs in
                       time and space.
Solve Hamiltonicity in                                 time and space
if a path decomposition of width        is given.
Exclude                                        time under SETH.
k-CNF-SAT on n vars cannot be solved in                          time when     tends to infinity.

15. Base cases t=4 1

16. Base cases t=6

17. Basis of H_t
We define a set of pm’s         as follows:
Divide base set                    into groups:
1 | | | . . | . t-1 | t

18. Basis of H_t
We define a set of pm’s         as follows:
Divide base set                    into groups:
      are all matchings using only edges between
consecutive groups.
1-1 correspondence with               -length bitstrings
1 | | | . . | . t-1 | t

19. Basis of H_t
We define a set of pm’s         as follows:
Divide base set                    into groups:
      are all matchings using only edges between
consecutive groups.
1-1 correspondence with               -length bitstrings
1 | | | . . | . t-1 | t 1 1

20. Basis of H_t
We define a set of pm’s         as follows:
Divide base set                    into groups:
      are all matchings using only edges between
consecutive groups.
1-1 correspondence with               -length bitstrings
1 | | | . . | . t-1 | t 1

21. Basis of H_t
We define a set of pm’s         as follows:
Divide base set                    into groups:
      are all matchings using only edges between
consecutive groups.
1-1 correspondence with               -length bitstrings
                          HC. iff complementary bitstrings.
Let         be obtained by using order     rather then               
1 | | | . . | . t-1 | t 1

22. Base cases t=6

23. Proof idea of basis Observation: Every matching belongs to for some .
Let      be obtained from     by swapping to consecutive elements.
We show that for                   , the row                  is the sum of at most 3 rows from        .
This relies on the base cases: All rows share all but at most three edges, which can therefore be contracted. We are left with pm’s from       .

24. HC in directed bipartite graphs
Focus on computing parity of number of Hamiltonian cycles.
Adding isolation lemma gives monte carlo algorithm.

25. HC in directed bipartite graphs

26. HC in directed bipartite graphs

27. HC in directed bipartite graphs1 1 1 1 1 1 1 1 1 1 1

28. HC in directed bipartite graphs1 1 1 1 1 1 1 1 1 1 1

29. HC in directed bipartite graphs 1 1 1 1 1 1 1 1 1 1 1

30. HC in directed bipartite graphs 1 1 1 1 1 1 1 1 1 1 1

31. HC in directed bipartite graphs 1 1 1 1 1 1 1 1 1 1 1 1

32. HC in directed bipartite graphs
Compute the representation amounts to, for every                   compute #given pm’s that are HC with M.
We compute this representation using DP over all subsets of matchings in        .
Since         is a permutation matrix, we can easily compute the number of 1’s from the two vectors in               time.

33. Graphs with small pathwidth2 2 2
     
      S

34. Graphs with small pathwidth
Again, counted #solutions modulo 2
Use dynamic programming, for every partition of the bag into degree 0,1,2 vertices store a representation of the matchings that can be established by partial solutions.
Non-trivial when a new edge e is inserted: change representation to         for some     that puts e at the end.

35. Further results
Lower bound under Strong ETH is rather technical reduction using some generic gadgets and the permutation matrix property from the basis
Recently, a superset of the authors showed that rank upper bounds such as the one presented can also be used to give deterministic algorithms, and extend to weighted problems.

36. Conclusions
Rank upper abounds on the partial solution versus partial solutions matrix are useful for designing algorithms.
Can Hamiltonicity be solved in                   time?
Can our tools be used to obtain faster deterministic algorithms and algorithms for TSP?

37. Conclusions
Rank upper abounds on the partial solution versus partial solutions matrix are useful for designing algorithms.
Can Hamiltonicity be solved in                   time?
Can our tools be used to obtain faster deterministic algorithms and algorithms for TSP?
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