Simple and Improved Parameterized Algorithms for Multiterminal Cuts

An O *(1.84 k ) Parameterized Algorithm for the Multiterminal Cut Problem

“…A matching algorithm for NODE MULTIWAY CUT was later reported by Cygan et al [8], who used a very interesting branching scheme based on a novel usage of the LP for multicommodity flow. With the benefit of hindsight, we are able to point out that techniques and results of Xiao [22] and Cygan et al [8] are essentially the same: the former algorithm can also be re-interpreted with the same branching scheme using the same flow-based LP.…”

“…On each neighbor v of t i , we have two options: including it and merging {v,t i } into t i , or excluding it and then edge (t i , v) is a crossing edge. This branching rule was first presented by Chen et al [5] and later used by Xiao [22]. The original analysis in [5] uses 2k − d(t i ) as the measure, which delivers a 4 k bound on the number of leaves the algorithm traverses in the search tree.…”

“…The original analysis in [5] uses 2k − d(t i ) as the measure, which delivers a 4 k bound on the number of leaves the algorithm traverses in the search tree. Inspired by the observation on isolating cuts, Xiao [22] used the new measure 2k − ∑d(ti). As it always holds that ∑ d(t i ) > k in a nontrivial instance, the bound can be tightened to 2 k .…”

Multicut Algorithms via Tree Decompositions

“…a parameter k, if a problem is solvable in time f (k) · n O (1) , where n denotes the size of the input instance. The function f is usually exponential but only depends on k. In case of multicut problems, various such parameters have been studied like solution size [7,8], cardinality |H| plus solution size [9,10], |H| plus the treewidth of the graph G [11,12], or the treewidth of the structure representing both G and H [13]. The result in [13] was proved by showing that the solutions to any of the above multicut problems can be described by a monadic second-order (MSO) formula over the structure representing G and H. The FPT follows by an extension of Courcelle's Theorem [14] proved in [15]: optimization problems expressible by an MSO formula over the input structures are fixed-parameter tractable w.r.t.…”

An FPT algorithm for edge subset feedback edge set

“…Note that for any two items X, X ′ ∈ π of a tight crucial partition π, there is an edge e ∈ S with one endpoint in X and the other in X ′ , otherwise X and X ′ can be merged together in π to get another crucial partition. There are only |S| edges in S, and then for any tight crucial partition π holds |π| ≤ T (n, m))-time algorithm presented in [8] to solve it. Then we get the running time bound claimed in the lemma.…”