We study the balanced $k$-way hypergraph partitioning problem, with a special focus on its practical applications to manycore scheduling. Given a hypergraph on $n$ nodes, our goal is to partition the node set into $k$ parts of size at most $(1+\epsilon)\cdot \frac{n}{k}$ each, while minimizing the cost of the partitioning, defined as the number of cut hyperedges, possibly also weighted by the number of partitions they intersect. We show that this problem cannot be approximated to within a $n^{1/\text{poly} \log\log n}$ factor of the optimal solution in polynomial time if the Exponential Time Hypothesis holds, even for hypergraphs of maximal degree 2. We also study the hardness of the partitioning problem from a parameterized complexity perspective, and in the more general case when we have multiple balance constraints. Furthermore, we consider two extensions of the partitioning problem that are motivated from practical considerations. Firstly, we introduce the concept of hyperDAGs to model precedence-constrained computations as hypergraphs, and we analyze the adaptation of the balanced partitioning problem to this case. Secondly, we study the hierarchical partitioning problem to model hierarchical NUMA (non-uniform memory access) effects in modern computer architectures, and we show that ignoring this hierarchical aspect of the communication cost can yield significantly weaker solutions.

翻译：我们研究平衡$k$路超图划分问题，特别关注其在多核调度中的实际应用。给定一个包含$n$个节点的超图，目标是将节点集划分为$k$个部分，每个部分的大小至多为$(1+\epsilon)\cdot \frac{n}{k}$，同时最小化划分代价（定义为被切割的超边数量，可能还按其所跨分区数量加权）。我们证明：若指数时间假说成立，则即使在最大度为2的超图中，该问题在多项式时间内也无法近似达到最优解的$n^{1/\text{poly} \log\log n}$因子以内。我们还从参数化复杂性角度研究了划分问题的难度，以及在具有多重平衡约束的更一般情形下的难度。此外，我们考虑了由实际需求驱动的两个扩展问题：首先，引入超有向无环图概念将带优先级约束的计算建模为超图，并分析平衡划分问题在该情形下的适应性；其次，研究层次化划分问题以模拟现代计算机架构中的层次化非均匀内存访问效应，并证明忽略通信代价的层次化特性可能产生显著较差的解。