Fixed Parameter Approximation Scheme for Min-Max k-Cut

We consider the graph k-partitioning problem under min-max objective, termed as \(\textsc {Minmax}\;k\text {-}\textsc {cut}\). The input here is a \(G=(V,E)\) with non-negative edge weights \(w:E\rightarrow \mathbb {R}_+\) and an integer \(k\ge 2\) goal to partition vertices into k non-empty parts \(V_1, \ldots , V_k\) so minimize \(\max _{i=1}^k w(\delta (V_i))\). Although minimizing sum objective \(\sum (V_i))\), {Minsum}\;k\text {cut}\), has been studied extensively in literature, very little known about max objective. initiate study of {cut}\) by showing that it NP-hard W[1]-hard when parameterized k, design approximation scheme k. main ingredient our exact algorithm for runs time \((\lambda k)^{O(k^2)}n^{O(1)}\), where \(\lambda \) value optimum n number vertices. Our algorithmic technique builds on Lokshtanov, Saurabh, Surianarayanan (FOCS, 2020) who showed similar result techniques are more general can be used obtain schemes \(\ell _p\)-norm measures every \(p\ge 1\).