Integrality gap analysis for bin packing games

A cooperative bin packing game is an $N$-person game, where the player set $N$ consists of $k$ bins of capacity 1 each and $n$ items of sizes $a_1,\dots,a_n$. The value $v(S)$ of a coalition $S$ of players is defined to be the maximum total size of items in $S$ that can be packed into the bins of $S$. We analyze the integrality gap of the corresponding 0–1 integer program of the value $v(N)$, thereby presenting an alternative proof for the non-emptiness of the 1/3-core for all bin packing games. Further, we show how to improve this bound $\epsilon\leq1/3$ (slightly) and point out that the conclusion in Matsui (2000) [9] is wrong (claiming that the bound 1/3 was tight). We conjecture that the true best possible value is $\epsilon=1/7$. The results are obtained using a new “rounding technique‿ that we develop to derive good (integral) packings from given fractional ones.