We introduce a natural knapsack intersection hierarchy for strengthening linear programming relaxations of packing integer programs, i.e., $\max\{w^Tx:x\in P\cap\{0,1\}^n\}$ where $P=\{x\in[0,1]^n:Ax \leq b\}$ and $A,b,w\ge0$. The $t^{th}$ level $P^{t}$ corresponds to adding cuts associated with the integer hull of the intersection of any $t$ knapsack constraints (rows of the constraint matrix). This model captures the maximum possible strength of "$t$-row cuts", an approach often used by solvers for small $t$. If $A$ is $m \times n$, then $P^m$ is the integer hull of $P$ and $P^1$ corresponds to adding cuts for each associated single-row knapsack problem. Thus, even separating over $P^1$ is NP-hard. However, for fixed $t$ and any $\epsilon>0$, results of Pritchard imply there is a polytime $(1+\epsilon)$-approximation for $P^{t}$. We then investigate the hierarchy's strength in the context of the well-studied all-or-nothing flow problem in trees (also called unsplittable flow on trees). For this problem, we show that the integrality gap of $P^t$ is $O(n/t)$ and give examples where the gap is $\Omega(n/t)$. We then examine the stronger formulation $P_{\text{rank}}$ where all rank constraints are added. For $P_{\text{rank}}^t$, our best lower bound drops to $\Omega(1/c)$ at level $t=n^c$ for any $c>0$. Moreover, on a well-known class of "bad instances" due to Friggstad and Gao, we show that we can achieve this gap; hence a constant integrality gap for these instances is obtained at level $n^c$.
翻译:我们引入了一个自然的 knapsack 交叉层次, 用于加强包装整数程序的线性编程松绑, 即 $\ max$( Px:x\ in P\ cap @ 0, 1\\ n$( 美元) $( 美元) ax\leq b ⁇ 美元 和 $A, b, w\\ ge0 美元。 $t} 水平相当于增加任何美元( 美元) knapsack 限制( 约束矩阵的行距) 的整形框的切削。 这个模型可以捕捉到“ 美元( 美元) 美元( 美元) 削减 。 这个模型可以捕捉到“ 美元( 美元) 美元( 美元), 美元( 美元) 美元( 美元) 的整数( 美元) 。 美元( 美元( 美元) 美元( 美元) 的整数( 美元( 美元) 美元( 美元) 美元( 美元) 的整数( =( 美元) 美元( 美元) 美元( 美元) 问题( 问题) 。