The Minimum Consistent Subset (MCS) problem arises naturally in the context of supervised clustering and instance selection. In supervised clustering, one aims to infer a meaningful partitioning of data using a small labeled subset. However, the sheer volume of training data in modern applications poses a significant computational challenge. The MCS problem formalizes this goal: given a labeled dataset X in a metric space, the task is to compute a smallest subset S of X such that every point in X shares its label with at least one of its nearest neighbors in S. Recently, the MCS problem has been extended to graph metrics, where distances are defined by shortest paths. Prior work has shown that MCS remains NP-hard even on simple graph classes like trees, and presented an fixed-parameter tractable (FPT) algorithm parameterized by the number of colors for MCS on trees. This raises the challenge of identifying graph classes that admit algorithms efficient in both input size (n) and the number of colors (c). In this work, we study the Minimum Consistent Subset problem on graphs, focusing on two well-established measures: the vertex cover number (vc) and the neighborhood diversity (nd). Specifically, we design efficient algorithms for graphs exhibiting small vc or small nd, which frequently arise in real-world domains characterized by local sparsity or repetitive structure. These parameters are particularly relevant because they capture structural properties that often correlate with the tractability of otherwise hard problems. Graphs with small vertex cover sizes are "almost independent sets", representing sparse interactions, while graphs with small neighborhood diversity exhibit a high degree of symmetry and regularity. Importantly, small neighborhood diversity can occur even in dense graphs, a property frequently observed in domains such as social networks with modular communities or knowledge graphs with repeated relational patterns. Thus, algorithms designed to work efficiently for graphs with small neighborhood diversity are capable of efficiently solving MCS in complex settings where small vertex covers may not exist. We show that MCS is FPT when parameterized by the vertex cover number and by neighborhood diversity. In each case, we present an algorithm whose running time is polynomial in n and c, and the non-polynomial part depends solely on the chosen parameter. Notably, our algorithms remain efficient for arbitrarily many colors, as their complexity is polynomially dependent on the number of colors.

Packing and Covering problems with geometric regions in the plane have been extensively studied and several notions of "complexity" of the regions involved have been developed and exploited to obtain good approximation algorithms. Examples of such complexity measures are VC-dimension, union complexity, shallow-cell complexity, fatness, etc. While these restrictions lead to constant-factor approximation algorithms in many cases, they typically do not lead to PTASs. In fact, several geometric Set Cover and Discrete Independent Set variants remain APX-hard even when these parameters are small, as demonstrated in earlier work by Chan and Grant (Exact algorithms and APX-hardness results for geometric packing and covering problems. Comput. Geom., 2014), and by Har-Peled and Quanrud (Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs. SIAM J. Comput., 2017). A key feature of these hardness constructions is that many pairs of regions in the input pierce one another. Motivated by this observation, we initiate a systematic study of geometric families parameterized by their piercing complexity. A connected region A is said to pierce a connected region B if B ⧵ A has more than one connected component; we consider instances in which every region is pierced by at most a constant number of others. This framework smoothly interpolates between the classical non-piercing case-where local-search PTASs are known due to Raman and Ray (Constructing Planar Support for Non-Piercing Regions, Discret. Comput. Geom., 2020), and the fully general case, where APX-hardness persists. Our main contribution is to show that bounded-piercing families admit efficient approximation schemes for fundamental geometric optimization problems. For regions in the plane with a constant piercing bound, we obtain PTASs for the (unweighted) Discrete Independent Set and Set Cover problems, and constant-factor approximation algorithms for their weighted variants. These results strictly generalize the known PTASs for non-piercing families and yield improved guarantees for several long-standing special cases, including Independent Set and Set Cover with axis-parallel rectangles under bounded piercing. Overall, our work identifies piercing complexity as a robust and expressive topological parameter-distinct from geometric notions such as density or fatness-and demonstrates that bounding this parameter yields a broad family of geometric instances for which PTASs become achievable.

Let G be a weighted graph embedded in a metric space (M, d_M). The vertices of G correspond to the points in M, with the weight of each edge uv being the distance d_M(u,v) between their respective points in M. The dilation (or stretch) of G is defined as the minimum factor t such that, for any pair of vertices u,v, the distance between u and v - represented by the weight of a shortest u,v-path - is at most t⋅ d_M(u,v). We study Dilation t-Augmentation, where the objective is, given a metric M, a graph G, and numerical values k and t, to determine whether G can be transformed into a graph with dilation t by adding at most k edges. Our primary focus is on the scenario where the metric M is the shortest path metric of an unweighted graph Γ. Even in this specific case, Dilation t-Augmentation remains computationally challenging. In particular, the problem is W[2]-hard parameterized by k when Γ is a complete graph, already for t = 2. Our main contribution lies in providing new insights into the impact of combinations of various parameters on the computational complexity of the problem. We establish the following. - The parameterized dichotomy of the problem with respect to dilation t, when the graph G is sparse: Parameterized by k, the problem is FPT for graphs excluding a biclique K_{d,d} as a subgraph for t ≤ 2 and the problem is W[1]-hard for t ≥ 3 even if G is a forest consisting of disjoint stars. - The problem is FPT parameterized by the combined parameter k+t+Δ, where Δ is the maximum degree of the graph G or Γ.

Vertex (s, t)-Cut and Vertex Multiway Cut are two fundamental graph separation problems in algorithmic graph theory. We study matroidal generalizations of these problems, where in addition to the usual input, we are given a representation R ∈ 𝔽^{r × n} of a linear matroid ℳ = (V(G), ℐ) of rank r in the input, and the goal is to determine whether there exists a vertex subset S ⊆ V(G) that has the required cut properties, as well as is independent in the matroid ℳ. We refer to these problems as Independent Vertex (s, t){-cut}, and Independent Multiway Cut, respectively. We show that these problems are fixed-parameter tractable (FPT) when parameterized by the solution size (which can be assumed to be equal to the rank of the matroid ℳ). These results are obtained by exploiting the recent technique of flow augmentation [Kim et al. STOC '22], combined with a dynamic programming algorithm on flow-paths á la [Feige and Mahdian, STOC '06] that maintains a representative family of solutions w.r.t. the given matroid [Marx, TCS '06; Fomin et al., JACM]. As a corollary, we also obtain FPT algorithms for the independent version of Odd Cycle Transversal. Further, our results can be generalized to other variants of the problems, e.g., weighted versions, or edge-deletion versions.

In the Minimum Consistent Subset (MCS) problem, we are presented with a connected simple undirected graph G, consisting of a vertex set V(G) of size n and an edge set E(G). Each vertex in V(G) is assigned a color from the set {1,2,…, c}. The objective is to determine a subset V' ⊆ V(G) with minimum possible cardinality, such that for every vertex v ∈ V(G), at least one of its nearest neighbors in V' (measured in terms of the hop distance) shares the same color as v. The decision problem, indicating whether there exists a subset V' of cardinality at most l for some positive integer l, is known to be NP-complete even for planar graphs. In this paper, we establish that the MCS problem is NP-complete on trees. We also provide a fixed-parameter tractable (FPT) algorithm for MCS on trees parameterized by the number of colors (c) running in O(2^{6c} n^6) time, significantly improving the currently best-known algorithm whose running time is O(2^{4c} n^{2c+3}). In an effort to comprehensively understand the computational complexity of the MCS problem across different graph classes, we extend our investigation to interval graphs. We show that it remains NP-complete for interval graphs, thus enriching graph classes where MCS remains intractable.

For a connected graph G = (V, E) and s, t ∈ V, a non-separating s-t path is a path P between s and t such that the set of vertices of P does not separate G, that is, G - V(P) is connected. An s-t path P is non-disconnecting if G - E(P) is connected. The problems of finding shortest non-separating and non-disconnecting paths are both known to be NP-hard. In this paper, we consider the problems from the viewpoint of parameterized complexity. We show that the problem of finding a non-separating s-t path of length at most k is W[1]-hard parameterized by k, while the non-disconnecting counterpart is fixed-parameter tractable (FPT) parameterized by k. We also consider the shortest non-separating path problem on several classes of graphs and show that this problem is NP-hard even on bipartite graphs, split graphs, and planar graphs. As for positive results, the shortest non-separating path problem is FPT parameterized by k on planar graphs and on unit disk graphs (where no s, t is given). Further, we give a polynomial-time algorithm on chordal graphs if k is the distance of the shortest path between s and t.

We study the priority set cover problem for simple geometric set systems in the plane. For pseudo-halfspaces in the plane we obtain a PTAS via local search by showing that the corresponding set system admits a planar support. We show that the problem is APX-hard even for unit disks in the plane and argue that in this case the standard local search algorithm can output a solution that is arbitrarily bad compared to the optimal solution. We then present an LP-relative constant factor approximation algorithm (which also works in the weighted setting) for unit disks via quasi-uniform sampling. As a consequence we obtain a constant factor approximation for the capacitated set cover problem with unit disks. For arbitrary size disks, we show that the problem is at least as hard as the vertex cover problem in general graphs even when the disks have nearly equal sizes. We also present a few simple results for unit squares and orthants in the plane.

Given $n$ pairs of points, $\mathcal{S} = \{\{p_1, q_1\}, \{p_2, q_2\}, \dots, \{p_n, q_n\}\}$, in some metric space, we study the problem of two-coloring the points within each pair, red and blue, to optimize the cost of a pair of node-disjoint networks, one over the red points and one over the blue points. In this paper we consider our network structures to be spanning trees, traveling salesman tours or matchings. We consider several different weight functions computed over the network structures induced, as well as several different objective functions. We show that some of these problems are NP-hard, and provide constant factor approximation algorithms in all cases.

In this paper, we study the problem of data gathering in ad-hoc sensor networks using a mobile entity called mule. The mule traverses the children of failed sensors, to prevent loss of data. Our objective is to define the optimal communication tree and the mule's placement such that the mule's overall traveling distance is minimized. We explore this problem in several network topologies including: unit disc graph on a line (UDL), general unit disc graph (UDG), and a complete graph with failing probabilities on the nodes (CGFP). We provide an optimal solution for the UDL problem and two approximation algorithms for the UDG problem. For the CGFP problem we outline the two possible structures of an optimal solution and provide near optimal approximation algorithms.

Frechet distance is an important geometric measure that captures the distance between two curves or more generally point sets. In this paper, we consider a natural variant of Frechet distance problem with multiple choice, provide an approximation algorithm and address its parameterized and kernelization complexity. A multiple choice problem consists of a set of color classes Q={Q_1,Q_2,...,Q_n}, where each class Q_i consists of a pair of points Q_i = {q_i, bar{q_i}}. We call a subset A subset {q_i , bar{q_i}:1 <= i <= n} conflict free if A contains at most one point from each color class. The standard objective in multiple choice problem is to select a conflict free subset that optimizes a given function. Given a line segment l and set Q of a pair of points in R^2, our objective is to find a conflict free subset that minimizes the Frechet distance between l and the point set, where the minimum is taken over all possible conflict free subsets. We first show that this problem is NP-hard, and provide a 3-approximation algorithm. Then we develop a simple randomized FPT algorithm which is later derandomized using universal family of sets. We believe that this technique can be of independent interest, and can be used to solve other parameterized multiple choice problems. The randomized algorithm runs in O(2^k * n * log^2(n)) time, and the derandomized deterministic algorithm runs in O(2^k * k^{O(log(k))} * n * log^2(n)) time, where k, the parameter, is the number of elements in the conflict free subset solution. Finally we present a simple branching algorithm for the problem running in O(2^k * n^{2} *log(n)) time. We also show that the problem is unlikely to have a polynomial sized kernel under standard complexity theoretic assumption.