6 research outputs found
On the Parameterized Complexity of Multiway Near-Separator
Full text linkWe study a new graph separation problem called Multiway Near-Separator. Given an undirected graph G, integer k, and terminal set T ⊆ V(G), it asks whether there is a vertex set S ⊆ V(G) ⧵ T of size at most k such that in graph G-S, no pair of distinct terminals can be connected by two pairwise internally vertex-disjoint paths. Hence each terminal pair can be separated in G-S by removing at most one vertex. The problem is therefore a generalization of (Node) Multiway Cut, which asks for a vertex set for which each terminal is in a different component of G-S. We develop a fixed-parameter tractable algorithm for Multiway Near-Separator running in time 2^{(k log k)} ⋅ n^{(1)}. Our algorithm is based on a new pushing lemma for solutions with respect to important separators, along with two problem-specific ingredients. The first is a polynomial-time subroutine to reduce the number of terminals in the instance to a polynomial in the solution size k plus the size of a given suboptimal solution. The second is a polynomial-time algorithm that, given a graph G and terminal set T ⊆ V(G) along with a single vertex x ∈ V(G) that forms a multiway near-separator, computes a 14-approximation for the problem of finding a multiway near-separator not containing x
On the Hardness of Compressing Weights
Full text linkWe investigate computational problems involving large weights through the lens of kernelization, which is a framework of polynomial-time preprocessing aimed at compressing the instance size. Our main focus is the weighted Clique problem, where we are given an edge-weighted graph and the goal is to detect a clique of total weight equal to a prescribed value. We show that the weighted variant, parameterized by the number of vertices n, is significantly harder than the unweighted problem by presenting an (n^{3 - ε}) lower bound on the size of the kernel, under the assumption that NP ̸ ⊆ coNP/poly. This lower bound is essentially tight: we show that we can reduce the problem to the case with weights bounded by 2^(n), which yields a randomized kernel of (n³) bits.
We generalize these results to the weighted d-Uniform Hyperclique problem, Subset Sum, and weighted variants of Boolean Constraint Satisfaction Problems (CSPs). We also study weighted minimization problems and show that weight compression is easier when we only want to {preserve the collection of} optimal solutions. Namely, we show that for node-weighted Vertex Cover on bipartite graphs it is possible to maintain the set of optimal solutions using integer weights from the range [1, n], but if we want to maintain the ordering of the weights of all inclusion-minimal solutions, then weights as large as 2^Ω(n) are necessary
Circumventing Connectivity for Kernelization
No full textClassical vertex subset problems demanding connectivity are of the following form: given an input graph G on n vertices and an integer k, find a set S of at most k vertices that satisfies a property and G[S] is connected. In this paper, we initiate a systematic study of such problems under a specific connectivity constraint, from the viewpoint of Kernelization (Parameterized) Complexity. The specific form that we study does not demand that G[S] is connected but rather G[S] has a closed walk containing all the vertices in S. In particular, we study Closed Walk-Subgraph Vertex Cover (CW-SVC, in short), where given a graph G, a set X⊆ E(G), and an integer k; the goal is to find a set of vertices S that hits all the edges in X and can be traversed by a closed walk of length at most k in G. When X is E(G), this corresponds to Closed Walk-Vertex Cover (CW-VC, in short). One can similarly define these variants for Feedback Vertex Set, namely Closed Walk-Subgraph Feedback Vertex Set (CW-SFVS, in short) and Closed Walk-Feedback Vertex Set (CW-FVS, in short). Our results are as follows: CW-VC and CW-SVC both admit a polynomial kernel, in contrast to Connected Vertex Cover that does not admit a polynomial kernel unless NP⊆ coNP/ poly.CW-FVS admits a polynomial kernel. On the other hand CW-SFVS does not admit even a polynomial Turing kernel unless the polynomial-time hierarchy collapses. We complement our kernelization algorithms by designing single-exponential FPT algorithms – 2 O ( k )n O ( 1 ) – for all the problems mentioned above. </p
