OKlibrary  0.2.1.6
general.hpp File Reference

General plans regarding investigations on Hindman problems. More...

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## Detailed Description

General plans regarding investigations on Hindman problems.

The hyperedges of "Hindman problems" are of the form {x_1, ..., x_k, s_1, ..., s_m, p_1, ..., p_m}, where k >= 1, m = 2^k-1-k, and where the s_i resp. p_i correspond to the non-empty sums resp. products of the x_i. The standard versions don't require the x_i to be distinct, while the injective version do require it. A further parameter a >= 1 allows ranges {a, ..., n} of vertices to be considered (default is a = 1).

The (normal) numbers are called hindman_r(k_1, ..., k_r) resp. hindmani_r(k_1, ..., k_r). The generalised versions are hindman_r^a(k_1, ..., k_r) and hindmani_r^a(k_1, ..., k_r).

Todo:
Connections
Todo:
Generation
• hindman_k2_ohg allows x=y (as with Schur numbers).
• x <> y is given by hindmani_k2_ohg.
• Both above cases concern k=2, while the general cases are treated by hindman_ohg resp. hindmani_ohg.
Todo:
Only products
• Also the problems similar to Folkman problems, but using the product instead of the sum, should be considered.
• How to call the corresponding hypergraphs and numbers? "allproducts" resp. "allproductsi"?
Todo:
The case k=1
• The case k=1 is trivial for Hindman-problems, since summation happens only over subsets.
• However one could consider hyperedges {x,2*x,x^2}.
• One could consider a generalised form, where one considers a k-subset, and adds all sums and products of tuples of length from 1 to p over this subset. Then the above would be the case k=1, p=2.
• For p=3 we would obtain {x, 2*x, x^2, 3*x, x^3}.
Todo:
hindman_2(2) = 39
• Generated by output_hindman_a1k2_stdname(2,n).
• OKsolver_2002:
```kullmann-0:OKplatform> OKsolver_2002-O3-DNDEBUG Hindman_a1k2_2_38.cnf
s SATISFIABLE
c sat_status=1 initial_maximal_clause_length=4 initial_number_of_variables=76 initial_number_of_clauses=226 initial_number_of_literal_occurrences=664 running_time(s)=0.0 number_of_nodes=1 number_of_single_nodes=0 number_of_quasi_single_nodes=0 number_of_2-reductions=0 number_of_pure_literals=0 number_of_autarkies=3 number_of_missed_single_nodes=0 max_tree_depth=0 number_of_table_enlargements=0 reduced_maximal_clause_length=0 reduced_number_of_variables=0 reduced_number_of_clauses=0 reduced_number_of_literal_occurrences=0 number_of_1-autarkies=0 number_of_initial_unit-eliminations=0 number_of_new_2-clauses=0 maximal_number_of_added_2-clauses=0 initial_number_of_2-clauses=80 file_name=Hindman_a1k2_2_38.cnf
kullmann-0:OKplatform> OKsolver_2002-O3-DNDEBUG Hindman_a1k2_2_39.cnf
s UNSATISFIABLE
c sat_status=0 initial_maximal_clause_length=4 initial_number_of_variables=78 initial_number_of_clauses=232 initial_number_of_literal_occurrences=682 running_time(s)=0.0 number_of_nodes=1 number_of_single_nodes=0 number_of_quasi_single_nodes=0 number_of_2-reductions=1 number_of_pure_literals=0 number_of_autarkies=0 number_of_missed_single_nodes=0 max_tree_depth=0 number_of_table_enlargements=0 reduced_maximal_clause_length=0 reduced_number_of_variables=0 reduced_number_of_clauses=0 reduced_number_of_literal_occurrences=0 number_of_1-autarkies=0 number_of_initial_unit-eliminations=0 number_of_new_2-clauses=0 maximal_number_of_added_2-clauses=0 initial_number_of_2-clauses=82 file_name=Hindman_a1k2_2_39.cnf
```
Todo:
hindman_3(2) > 80000
• Local search seems ineffective:
```> ubcsat-okl -alg adaptnovelty+ -runs 10 -cutoff 10000000 -i Hindmand_3_7000.cnf
sat  min     osteps     msteps       seed
1 0     1     128712   10000000 1974571835
2 0     1     164697   10000000  141907909
3 0     1     137570   10000000 2844493766
4 0     1     156528   10000000 2907523445
5 0     1     152484   10000000 1281678395
6 0     1     121241   10000000 1148261914
7 0     1     110357   10000000  361512568
8 0     1     116423   10000000 3049483154
9 0     1     153049   10000000 2648332806
10 0     1     166842   10000000 1270017279
Clauses = 122728
Variables = 21000
TotalLiterals = 420663
FlipsPerSecond = 10437
BestStep_Mean = 140790.300000
Steps_Mean = 10000000.000000
Steps_Max = 10000000.000000
```
• minisat2 seems to perform best (also on satisfiable instances):
```kullmann-0:OKplatform> minisat2 Hindmand_3_40000.cnf
This is MiniSat 2.0 beta
WARNING: for repeatability, setting FPU to use double precision
============================[ Problem Statistics ]=============================
|                                                                             |
|  Number of variables:  120000                                               |
|  Number of clauses:    805390                                               |
|  Parsing time:         3.06         s                                       |
============================[ Search Statistics ]==============================
| Conflicts |          ORIGINAL         |          LEARNT          | Progress |
|           |    Vars  Clauses Literals |    Limit  Clauses Lit/Cl |          |
===============================================================================
|         0 |   62629   613294  3095858 |   204431        0    nan |  0.000 % |
|       100 |   62629   613294  3095858 |   224874      100     75 |  0.000 % |
|       250 |   62629   613294  3095858 |   247361      250     92 |  0.000 % |
|       475 |   62629   613294  3095858 |   272098      475    103 |  0.000 % |
===============================================================================
restarts              : 4
conflicts             : 545            (2 /sec)
decisions             : 334862         (1.70 % random) (1256 /sec)
propagations          : 502545         (1885 /sec)
conflict literals     : 49960          (2.41 % deleted)
Memory used           : 153.79 MB
CPU time              : 266.641 s
SATISFIABLE
```
• n=80000 still found easily satisfiable (4 restarts) by minisat2.
• For larger problems we need a C++ program.
• Neil Hindman remarked that for the infinite versions (partitioning the set of natural numbers, and asking for arbitrarily long sequences), which are implied by the finite versions, the degenerated form (which just asks for some sequence) and the non-degenerated form (asking for an injective sequence) are equivalent.
Todo:
hindmani_2(2) = 252
• Generated by output_hindmani_a1k2_stdname(2,n).
• n=502, c=1908, c/n ~ 3.80
• Note that these instances use the translation from non-boolean to boolean variables.
• OKsolver_2002
```kullmann-0:OKplatform> OKsolver_2002-O3-DNDEBUG Hindmani_a1k2_2_251.cnf
s SATISFIABLE
c sat_status=1 initial_maximal_clause_length=4 initial_number_of_variables=502 initial_number_of_clauses=1908 initial_number_of_literal_occurrences=6130 running_time(s)=0.0 number_of_nodes=6 number_of_single_nodes=0 number_of_quasi_single_nodes=0 number_of_2-reductions=27 number_of_pure_literals=0 number_of_autarkies=64 number_of_missed_single_nodes=0 max_tree_depth=3 number_of_table_enlargements=0 reduced_maximal_clause_length=0 reduced_number_of_variables=0 reduced_number_of_clauses=0 reduced_number_of_literal_occurrences=0 number_of_1-autarkies=2328 number_of_initial_unit-eliminations=0 number_of_new_2-clauses=0 maximal_number_of_added_2-clauses=0 initial_number_of_2-clauses=502 file_name=Hindmani_a1k2_2_251.cnf
kullmann-0:OKplatform> OKsolver_2002-O3-DNDEBUG Hindmani_a1k2_2_252.cnf
s UNSATISFIABLE
c sat_status=0 initial_maximal_clause_length=4 initial_number_of_variables=504 initial_number_of_clauses=1928 initial_number_of_literal_occurrences=6204 running_time(s)=0.1 number_of_nodes=11 number_of_single_nodes=0 number_of_quasi_single_nodes=0 number_of_2-reductions=36 number_of_pure_literals=0 number_of_autarkies=98 number_of_missed_single_nodes=0 max_tree_depth=3 number_of_table_enlargements=0 reduced_maximal_clause_length=0 reduced_number_of_variables=0 reduced_number_of_clauses=0 reduced_number_of_literal_occurrences=0 number_of_1-autarkies=4624 number_of_initial_unit-eliminations=0 number_of_new_2-clauses=0 maximal_number_of_added_2-clauses=0 initial_number_of_2-clauses=504 file_name=Hindmani_a1k2_2_252.cnf
```
Interesting that the problems are so easy.

Definition in file general.hpp.