The maximum/minimum bisection problems are, given an edge-weighted graph, to find a bipartition of the vertex set into two sets whose sizes differ by at most one, such that the total weight of edges between the two sets is maximized/minimized. Although these two problems are known to be NP-hard, there is an efficient algorithm for bounded-treewidth graphs. In particular, Jansen et al. (SIAM J. Comput. 2005) gave an $O(2^t{n}^3)$-time algorithm when given a tree decomposition of width t of the input graph, where n is the number of vertices of the input graph. Eiben et al. (ESA 2019, JCSS 2021) improved the dependency of n in the running time by giving an $O(8^t{t}^5{n}^2\log n)$-time algorithm. Moreover, they showed that there is no $O(n^{2-\epsilon })$-time algorithm for trees under some reasonable complexity assumption.

In this paper, we show an $O(2^t{(tn)}^2)$-time algorithm for both problems, which is asymptotically tight to their conditional lower bound. We also show that the exponential dependency of the treewidth is asymptotically optimal under the Strong Exponential Time Hypothesis. Finally, we discuss the (in)tractability of both problems with respect to special graph classes.

在给定边缘加权图的情况下，最大/最小二等分问题是将顶点集分成两部分，其大小相差最大为一个，从而最大程度地/最小化了两组之间的边的总权重。尽管已知这两个问题都是NP难的，但是有一种有效的有界树图算法。特别是，Jansen等。（SIAM J.Comput.2005）给出了$Ø（{\mathrm{2个}}^{Ť}{ñ}^3）$给定输入图宽度t的树分解的时间算法，其中n是输入图的顶点数。埃本（Eiben）等人。（ESA 2019，2021 JCSS）改善的依赖性Ñ通过给出在运行时间$Ø（8^{Ť}{Ť}^5{ñ}^{\mathrm{2个}}\mathrm{日志}ñ）$时间算法。而且，他们表明没有$Ø（{ñ}^{\mathrm{2个}-\epsilon }）$合理复杂度假设下的树木实时算法。

在本文中，我们展示了 $Ø（{\mathrm{2个}}^{Ť}{（Ťñ）}^{\mathrm{2个}}）$这两个问题的时间算法，渐近地紧贴其条件下界。我们还表明，在强指数时间假设下，树宽的指数依赖性是渐近最优的。最后，我们讨论关于特殊图类的两个问题的（难）难点。