In this paper we study a resource allocation problem that encodes correlation between items in terms of \conflict and maximizes the minimum utility of the agents under a conflict free allocation. Admittedly, the problem is computationally hard even under stringent restrictions because it encodes a variant of the {\sc Maximum Weight Independent Set} problem which is one of the canonical hard problems in both classical and parameterized complexity. Recently, this subject was explored by Chiarelli et al.~[Algorithmica'22] from the classical complexity perspective to draw the boundary between {\sf NP}-hardness and tractability for a constant number of agents. The problem was shown to be hard even for small constant number of agents and various other restrictions on the underlying graph. Notwithstanding this computational barrier, we notice that there are several parameters that are worth studying: number of agents, number of items, combinatorial structure that defines the conflict among the items, all of which could well be small under specific circumstancs. Our search rules out several parameters (even when taken together) and takes us towards a characterization of families of input instances that are amenable to polynomial time algorithms when the parameters are constant. In addition to this we give a superior $2^{m}|I|^{\Co{O}(1)}$ algorithm for our problem where $m$ denotes the number of items that significantly beats the exhaustive $\Oh(m^{m})$ algorithm by cleverly using ideas from FFT based fast polynomial multiplication; and we identify simple graph classes relevant to our problem's motivation that admit efficient algorithms.

翻译：本文研究了一个资源分配问题，该问题通过冲突关系编码项目之间的关联，并在无冲突分配下最大化代理人的最小效用。诚然，即使在严格的约束条件下，该问题在计算上也是困难的，因为它编码了最大权重独立集问题的一个变体，而该问题是经典复杂性和参数化复杂性中的典型困难问题之一。最近，Chiarelli等人[Algorithmica'22]从经典复杂性的角度探索了这一主题，以划定在常数数量代理人情况下NP难与可解性之间的边界。研究表明，即使代理人数量为小的常数且底层图受到各种其他限制，该问题仍然是困难的。尽管存在这一计算障碍，我们注意到仍有几个值得研究的参数：代理人数量、项目数量、定义项目间冲突的组合结构，这些参数在特定情况下可能很小。我们的探索排除了几个参数（即使组合在一起），并使我们能够描述当参数为常数时，适用于多项式时间算法的输入实例族特征。此外，我们为问题提出了一种更优的$2^{m}|I|^{\Co{O}(1)}$算法（其中$m$表示项目数量），通过巧妙利用基于FFT的快速多项式乘法思想，显著超越了穷举的$\Oh(m^{m})$算法；同时我们识别了与问题动机相关的简单图类，这些图类支持高效算法。