The skiving stock problem, also called bin covering, asks for the largest number of disjoint groups of items whose total size reaches a fixed threshold 
. Items may remain unused. It reverses the aim of the
bin-packing problem, which minimizes the number
of groups subject to an upper capacity bound (Zak 2003). For example, items of sizes
0.6, 0.6, 0.4, and 0.4 can cover two bins with threshold one.
For item types of sizes 
and available multiplicities 
, a pattern is a nonnegative integer vector 
satisfying 
. Assigning a nonnegative integer
multiplicity 
to each pattern gives the optimum
 |
Relaxing 
to nonnegative real numbers gives a linear
programming bound 
. The proper relaxation additionally restricts patterns
by 
.
The modified integer round-down conjecture asserts 
, where 
is the floor function,
equivalently 
(Martinovic and Scheithauer 2019).
