For many problems, the important instances from practice possess certain structure that one should reflect in the design of specific algorithms. As data reduction is an important and inextricable part of today's computation, we employ one of the most successful models of such precomputation -- the kernelization. Within this framework, we focus on Traveling Salesperson Problem (TSP) and some of its generalizations. We provide a kernel for TSP with size polynomial in either the feedback edge set number or the size of a modulator to constant-sized components. For its generalizations, we also consider other structural parameters such as the vertex cover number and the size of a modulator to constant-sized paths. We complement our results from the negative side by showing that the existence of a polynomial-sized kernel with respect to the fractioning number, the combined parameter maximum degree and treewidth, and, in the case of Subset-TSP, modulator to disjoint cycles (i.e., the treewidth two graphs) is unlikely.

Abstract Consider a vertex-weighted graphGwith a sourcesand a targett.Tracking Pathsrequires finding a minimum weight set of vertices (trackers) such that the sequence of trackers in each path fromstotis unique. In this work, we derive a factor 66-approximation algorithm forTracking Pathsin weighted graphs and a factor 4-approximation algorithm if the input is unweighted. This is the first constant factor approximation for this problem. While doing so, we also study approximation of the closely relatedr-Fault Tolerant Feedback Vertex Setproblem. There, for a fixed integer rand a given vertex-weighted graphG, the task is to find a minimum weight set of vertices intersecting every cycle of Gin at least $$r+1$$ r+1 vertices. We give a factor $$\mathcal {O}(r^2)$$ O(r2) approximation algorithm forr-Fault Tolerant Feedback Vertex Setifris a constant.

A feedback vertex set in a hypergraph H is a set of vertices S such that deleting S from H results in an acyclic hypergraph. Here, deleting a vertex means removing the vertex and all incident hyperedges, and a hypergraph is acyclic if its vertex-edge incidence graph is acyclic. We study the (parameterized complexity of) the Hypergraph Feedback Vertex Set (HFVS) problem: given as input a hypergraph H and an integer k, determine whether H has a feedback vertex set of size at most k. It is easy to see that this problem generalizes the classic Feedback Vertex Set (FVS) problem on graphs. Remarkably, despite the central role of FVS in parameterized algorithms and complexity, the parameterized complexity of a generalization of FVS to hypergraphs has not been studied previously. In this paper, we fill this void. Our main results are as follows - HFVS is W[2]-hard (as opposed to FVS, which is fixed parameter tractable). - If the input hypergraph is restricted to a linear hypergraph (no two hyperedges intersect in more than one vertex), HFVS admits a randomized algorithm with running time 2^{𝒪(k³log k)}n^{𝒪(1)}. - If the input hypergraph is restricted to a d-hypergraph (hyperedges have cardinality at most d), then HFVS admits a deterministic algorithm with running time d^{𝒪(k)}n^{𝒪(1)}. The algorithm for linear hypergraphs combines ideas from the randomized algorithm for FVS by Becker et al. [J. Artif. Intell. Res., 2000] with the branching algorithm for Point Line Cover by Langerman and Morin [Discrete & Computational Geometry, 2005].

Abstract Given a graphG, and terminal verticessandt, theTracking Pathsproblem asks to compute a set of minimum number of vertices to be marked as trackers, such that the sequence of trackers encountered in each $$s$$ s - $$t$$ t path is unique.Tracking PathsisNP-hard in both directed and undirected graphs in general. In this paper we give a collection of polynomial time algorithms for some restricted versions ofTracking Paths. We prove thatTracking Pathsis polynomial time solvable for undirected chordal graphs and tournament graphs. We also show thatTracking PathsisNP-hard in graphs with bounded maximum degree $$\Delta \ge 6$$ Δ≥6 , and give a $$2(\Delta +1)$$ 2(Δ+1) -approximate algorithm for this case. Further, we give a polynomial time algorithm which, given an undirected graphG, a tracking set $$T\subseteq V(G)$$ T⊆V(G) , and a sequence of trackers $$\pi $$ π , returns the unique $$s$$ s - $$t$$ t path inGthat corresponds to $$\pi $$ π , if one exists. Finally we analyze the version of tracking $$s$$ s - $$t$$ t paths where paths are tracked using edges instead of vertices, and we give a polynomial time algorithm for the same.