We continue and extend previous work on the parameterized complexity analysis of the NP-hard Stable Roommates with Ties and Incomplete Lists problem, thereby strengthening earlier results both on the side of parameterized hardness as well as on the side of fixed-parameter tractability. Other than for its famous sister problem Stable Marriage which focuses on a bipartite scenario, Stable Roommates with Incomplete Lists allows for arbitrary acceptability graphs whose edges specify the possible matchings of each two agents (agents are represented by graph vertices). Herein, incomplete lists and ties reflect the fact that in realistic application scenarios the agents cannot bring all other agents into a linear order. Among our main contributions is to show that it is W[1]-hard to compute a maximum-cardinality stable matching for acceptability graphs of bounded treedepth, bounded tree-cut width, and bounded disjoint paths modulator number (these are each time the respective parameters). However, if we `only' ask for perfect stable matchings or the mere existence of a stable matching, then we obtain fixed-parameter tractability with respect to tree-cut width but not with respect to treedepth. On the positive side, we also provide fixed-parameter tractability results for the parameter feedback edge set number.
翻译:我们继续并延长了以前对NP硬稳定室友的参数复杂性分析工作,该室友有铁链和不完整列表问题,从而在参数硬度方面以及在固定参数可移动性方面加强了早期结果。除了其著名的姊妹问题之外,稳定的婚姻以双片情景为重点,具有不完整列表的室友允许使用任意的可接受性图表,其边缘指定了两种物剂的可能匹配(试剂由图形顶点代表),因此,不完整的名单和联系反映了这样一个事实,即在现实应用假设中,代理人不能将所有其他物剂都置于线性顺序。我们的主要贡献之一是表明,用最硬的心性稳定地匹配可接受的深层树木图、捆绑的树切宽度和接合不接的路径调制式数字(这些是各自参数的每个时间)。如果我们“只”要求完全稳定匹配或仅仅存在稳定的匹配,那么我们获得固定的距离比分度,那么我们获得的距离比差值是W[1]硬度稳定,以正比度稳定度稳定比值来测量我们树底平面的比值。