Quasi-polynomial-sized vertex multicut-mimicking networks and kernelization for Vertex Multiway Cut
Eduard Eiben, Roohani Sharma, Magnus Wahlström
原文摘要
We give deterministic polynomial-time constructions of vertex multicut-mimicking networks of quasi-polynomial size. Given a graph G, a terminal set T, and a positive integer c, our algorithm constructs a graph with |T|^7 c^{O( log c ) } vertices that preserves every minimum vertex multiway cut value of at most c, for every partition of the terminals. Setting c=|T| gives an exact vertex multicut-mimicking network with |T|^{O( log |T| ) } vertices, resolving an open question of Wahlström [ACM Transactions on Algorithms 2022], who showed a bound of κ^{O( log^3 κ)} for the edge multicut-mimicking variant, where κ:= \sum_{t \in T} |N_G(t)| is the capacity of the terminal set, which was later improved to κ^{O( log^2 κ)} by Anand, Lee, Li and Saranurak [STOC 2024]. As applications of our vertex multicut-mimicking network, we obtain deterministic kernels of size k^{O( log k )} for Vertex Multiway Cut, p^{O(1)} k^{O(log k)} for Vertex Multicut with p requests and |Γ|^{O(1)} k^{O( log k)} for Group Feedback Vertex Set, where k is the deletion budget and Γis a finite group. Our first kernelization result, in particular, resolves an open question by Wahlström [ACM Transactions on Algorithms 2022], which was reiterated in a recent survey on kernelization by Jansen [Computer Science Reviews 2026]. Also, to the best of our knowledge, our kernel for Vertex Multicut is the first one whose size is completely polynomial in p and (quasi-polynomial) super-polynomial only in k. Finally, again using our vertex multicut-mimicking network for graphs, we obtain a hyperedge multicut-mimicking network for hypergraphs, which preserves hypergraph edge-multicuts of size up to c, of size |T|^{O(1)} c^{O(log c)}. Our result removes the dependence on the rank of the hypergraph from the exponent of the size bound, that existed in prior work.
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