2
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ I can't find one either, so ... $\:$ $\endgroup$ – user6973 May 15 '13 at 5:34
3
$\begingroup$

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 $\left[\hspace{.03 in}j_0,j_1,j_2,...,j_{(3\cdot k)-1}\hspace{-0.01 in}\right]$

.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.