An optimal FPT algorithm parametrized by treewidth for Weighted-Max-Bisection given a tree decomposition as advice assuming SETH and the hardness of MinConv

The weighted maximal bisection problem is, given an edge weighted graph, to find a bipartition of the vertex set into two sets such that their cardinality differs by at most one and the sum of the weight of the edges between vertices that are not in the same set is maximized. This problem is known to be NP-hard, even when a tree decomposition of width $t$ and $\mathcal O(n)$ nodes is given as an advice as part of the input, where $n$ is the number of vertices of the input graph. But, given such an advice, the problem is decidable in FPT time in $n$ parametrized by $t$. In particular Jansen et al. presented an algorithm with running time $\mathcal O(2^tn^3)$. Hanaka, Kobayashi, and Sone enhanced the analysis of the complexity to $\mathcal O(2^t(nt)^2)$. By slightly modifying the approach, we improve the running time to $\mathcal O(2^tn^2)$ in the RAM model, which is asymptotically optimal in $n$ under the hardness of MinConv. We proof that this is also asymptotically optimal in its dependence on $t$ assuming SETH by showing for a slightly easier problem (maximal cut) that there is no $\mathcal O(2^{\epsilon t} \operatorname{poly} n)$ algorithm for any $\varepsilon < 1$ under SETH. This was already claimed by Hanaka, Kobayashi, and Sone but without a correct proof. We also present a hardness result (no $\mathcal O(2^t n^{2-\varepsilon})$ algorithm for any $\varepsilon > 0$) for a broad family of subclasses of the weighted maximal bisection problem that are characterized only by the dependence of $t$ from $n$, more precisely, all instances with $t = f(n)$ for an arbitrary but fixed $f(n) \in \operatorname o(\log n)$. This holds even when only considering planar graphs. Moreover we present a detailed description of the implementation details and assumptions that are necessary to achieve the optimal running time.