# partition to min the max number of intersections

Given $n$ items and $m$ customers, each of whom is interested in some subset of the items, partition the set of items among $k$ different stores so that the maximum number of customers visiting any store is minimized.

Does anyone recognize this problem? I have the slight feeling that it is some standard set or cut problem in disguise, but I just cannot find the right one.

• I can't find one either, so ... $\:$ – user6973 May 15 '13 at 5:34

This is NP-hard, by reduction from 3-partition.

Given $\left[\hspace{.03 in}j_0,j_1,j_2,...,j_{(3\cdot k)-1}\hspace{-0.01 in}\right]$, $\:$ where $\; J = \operatorname{sum}\left(\left[\hspace{.03 in}j_0,j_1,j_2,...,j_{(3\cdot k)-1}\hspace{-0.01 in}\right]\right) \;$, $\;$ if there are

$k$ different stores, $\:J$ items, and $\:(3\cdot k)+(4\cdot J)\:$ customers,
with customer $i$ for $\:i\in [0,1,2,...,(3\cdot k)-1]\:$ interested in the obvious $j_i$ items
and four other customers interested in each one of the items,

then

the maximum can't be less than $\:\frac{4\cdot J}k+3\:$ and the partitions whose maximum is
$\frac{4\cdot J}k+3\:$ are exactly those corresponding to $3$-partitions of $\left[\hspace{.03 in}j_0,j_1,j_2,...,j_{(3\cdot k)-1}\hspace{-0.01 in}\right]$

.