binary numerics precedence invariant_knapsack mixed_binary

Submitter Variables Constraints Density Status Group Objective MPS File
Kelly Eurek 348921 661133 1.66674e-05 open -833555203.4089005* mining.mps.gz

Unspecified mining application. Mark Zuckerberg suggests the following tightening of one class of constraints: The variables in this problem (except for the last one, x348921, which is fixed to 1 and does not appear in any constraint) are arranged in groups of 20, and there are three classes of precedence constraints (or more accurately, implication constraints): 1) For each variable index i=…,348920, if i mod 20 > 0 then there is a constraint x_i 2) If i mod 20 = and i>20 then there is a constraint x_i = x21 >= x41 >=…) 3) and for all other i there is a constraint x_i Graphically, we can represent the variables as a table with height twenty and width 348920 / 20, with indices 1,…,20 going down the left-most column, then 21,…,40 going down the next column to the right, etc. There is then a precedence constraint (1) from each variable in a column to the one below it, a precedence constraint (2) from the topmost entry in each column to its neighbor on the left, and an “OR” precedence constraint (3) from each entry that isn’t on top of a column, requiring that if it has value 1 then either the one above it or the one to its left must have value 1 as well. But in fact this OR precedence (3) can be replaced with a simple precedence constraint from each variable to the one at its left. To see this note that in any column, the first type of precedence constraint will require that a feasible solution must be a contiguous sequence of 1’s from some point in the column until the bottom. Consider column c > 1. The topmost 1 in this column, by precedence constraints (3) (or if the topmost is at the top of the column, then by precedence constraints (2)) implies that the variable to its left (in column c-1) is also 1, which implies that the contiguous sequence of 1’s in column c-1 must be at least as high as the sequence in column c. Thus for every 1 in column c, the entry to its left in column c-1 must also be 1. Imported from MIPLIB2010.

Instance Statistics

Detailed explanation of the following tables can be found here.

Size Related Properties
Original Presolved
Variables 348921 348920
Constraints 661133 661132
Binaries 348920 348920
Integers 0 0
Continuous 1 0
Implicit Integers 0 0
Fixed Variables 0 0
Nonzero Density 1.66674e-05 1.66674e-05
Nonzeroes 3844880 3844880
Constraint Classification Properties
Original Presolved
Total 661133 661132
Empty 0 0
Free 0 0
Singleton 1 0
Aggregations 0 0
Precedence 347953 347953
Variable Bound 0 0
Set Partitioning 0 0
Set Packing 0 0
Set Covering 0 0
Cardinality 0 0
Invariant Knapsack 313101 313101
Equation Knapsack 0 0
Bin Packing 0 0
Knapsack 0 0
Integer Knapsack 0 0
Mixed Binary 78 78
General Linear 0 0
Indicator 0 0


Available nonzero structure and decomposition information. Further information can be found here.

value min median mean max
Constraint %
Variable %

Best Known Solution(s)

Find solutions below. Download the archive containing all solutions from the Download page.

## Warning in lapply(df["exactobjval"], as.numeric): NAs introduced by coercion
ID Objective Exact Int. Viol Cons. Viol Obj. Viol Submitter Date Description
3 -833555203 -833555203 0 0 4.0e-07 Edward Rothberg 2020-07-17 Obtained with the Gurobi 9.0 solution improvement heuristic
2 -797686257 0 0 5.0e-07 Robert Ashford and Alkis Vazacopoulus 2019-12-18 Found using ODH|CPlex
1 531134255 531134255 0 0 2.7e-06 - 2018-10-12 Solution found during MIPLIB2017 problem selection.

Similar instances in collection

The following instances are most similar to mining in the collection. This similarity analysis is based on 100 scaled instance features describing properties of the variables, objective function, bounds, constraints, and right hand sides.

Instance Status Variables Binaries Integers Continuous Constraints Nonz. Submitter Group Objective Tags
rmine21 open 162547 162547 0 0 1441651 3514884 Daniel Espinoza rmine -10618.75083837232* binary precedence mixed_binary
rmine25 open 326599 326599 0 0 2953849 7182744 Daniel Espinoza rmine -15541.66928749976* binary precedence mixed_binary
rmine15 open 42438 42438 0 0 358395 879732 Daniel Espinoza rmine -5018.819990999996* binary precedence mixed_binary
rmine13 open 23980 23980 0 0 197155 485784 Daniel Espinoza rmine -3495.3706624382535* binary precedence mixed_binary
opm2-z12-s8 hard 10800 10800 0 0 319508 725377 Daniel Espinoza opm2 -58540 binary precedence knapsack


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