arXiv subjects

Saket Saurabh

Publications and source records attributed to Saket Saurabh.

At least 37 records · Page 2Linked to original sources

arXiv2025

When Distances Lie: Euclidean Embeddings in the Presence of Outliers and Distance Violations

Distance geometry explores the properties of distance spaces that can be exactly represented as the pairwise Euclidean distances between points in $\mathbb{R}^d$ ($d \geq 1$), or equivalently, distance spaces that can be isometrically embedded in $\mathbb{R}^d$. In this work, we investigate whether a distance space can be isometrically embedded in $\mathbb{R}^d$ after applying a limited number of modifications. Specifically, we focus on two types of modifications: outlier deletion (removing points) and distance modification (adjusting distances between points). The central problem, Euclidean Embedding Editing (EEE), asks whether an input distance space on $n$ points can be transformed, using at most $k$ modifications, into a space that is isometrically embeddable in $\mathbb{R}^d$. We present several fixed-parameter tractable (FPT) and approximation algorithms for this problem. Our first result is an algorithm that solves EEE in time $(dk)^{\mathcal{O}(d+k)} + n^{\mathcal{O}(1)}$. The core subroutine of this algorithm, which is of independent interest, is a polynomial-time method for compressing the input distance space into an equivalent instance of EEE with $\mathcal{O}((dk)^2)$ points. For the special but important case of EEE where only outlier deletions are allowed, we improve the parameter dependence of the FPT algorithm and obtain a running time of $\min\{(d+3)^k, 2^{d+k}\} \cdot n^{\mathcal{O}(1)}$. Additionally, we provide an FPT-approximation algorithm for this problem, which outputs a set of at most $2 \cdot {\rm OPT}$ outliers in time $2^d \cdot n^{\mathcal{O}(1)}$. This 2-approximation algorithm improves upon the previous $(3+\varepsilon)$-approximation algorithm by Sidiropoulos, Wang, and Wang [SODA '17]. Furthermore, we complement our algorithms with hardness results motivating our choice of parameterizations.

cs.CG↗

arXiv2025

A Quadratic Vertex Kernel and a Subexponential Algorithm for Subset-FAST

In the Subset Feedback Arc Set in Tournaments, Subset-FAST problem we are given as input a tournament $T$ with a vertex set $V(T)$ and an arc set $A(T)$, along with a terminal set $S \subseteq V(T)$, and an integer $ k$. The objective is to determine whether there exists a set $ F \subseteq A(T) $ with $|F| \leq k$ such that the resulting graph $T-F $ contains no cycle that includes any vertex of $S$. When $S=V(T)$ this is the classic Feedback Arc Set in Tournaments (FAST) problem. We obtain the first polynomial kernel for this problem parameterized by the solution size. More precisely, we obtain an algorithm that, given an input instance $(T, S, k)$, produces an equivalent instance $(T',S',k')$ with $k'\leq k$ and $V(T')=O(k^2)$. It was known that FAST admits a simple quadratic vertex kernel and a non-trivial linear vertex kernel. However, no such kernel was previously known for Subset-FAST. Our kernel employs variants of the most well-known reduction rules for FAST and introduces two new reduction rules to identify irrelevant vertices. As a result of our kernelization, we also obtain the first sub-exponential time FPT algorithm for Subset-FAST.

cs.DM↗

arXiv2025

Subexponential Parameterized Algorithms for Hitting Subgraphs

For a finite set $\mathcal{F}$ of graphs, the $\mathcal{F}$-Hitting problem aims to compute, for a given graph $G$ (taken from some graph class $\mathcal{G}$) of $n$ vertices (and $m$ edges) and a parameter $k\in\mathbb{N}$, a set $S$ of vertices in $G$ such that $|S|\leq k$ and $G-S$ does not contain any subgraph isomorphic to a graph in $\mathcal{F}$. As a generic problem, $\mathcal{F}$-Hitting subsumes many fundamental vertex-deletion problems that are well-studied in the literature. The $\mathcal{F}$-Hitting problem admits a simple branching algorithm with running time $2^{O(k)}\cdot n^{O(1)}$, while it cannot be solved in $2^{o(k)}\cdot n^{O(1)}$ time on general graphs assuming the ETH. In this paper, we establish a general framework to design subexponential parameterized algorithms for the $\mathcal{F}$-Hitting problem on a broad family of graph classes. Specifically, our framework yields algorithms that solve $\mathcal{F}$-Hitting with running time $2^{O(k^c)}\cdot n+O(m)$ for a constant $c<1$ on any graph class $\mathcal{G}$ that admits balanced separators whose size is (strongly) sublinear in the number of vertices and polynomial in the size of a maximum clique. Examples include all graph classes of polynomial expansion and many important classes of geometric intersection graphs. Our algorithms also apply to the \textit{weighted} version of $\mathcal{F}$-Hitting, where each vertex of $G$ has a weight and the goal is to compute the set $S$ with a minimum weight that satisfies the desired conditions. The core of our framework is an intricate subexponential branching algorithm that reduces an instance of $\mathcal{F}$-Hitting (on the aforementioned graph classes) to $2^{O(k^c)}$ general hitting-set instances, where the Gaifman graph of each instance has treewidth $O(k^c)$, for some constant $c<1$ depending on $\mathcal{F}$ and the graph class.

cs.DS↗

arXiv2025

Exponential-Time Approximation (Schemes) for Vertex-Ordering Problems

In this paper, we begin the exploration of vertex-ordering problems through the lens of exponential-time approximation algorithms. In particular, we ask the following question: Can we simultaneously beat the running times of the fastest known (exponential-time) exact algorithms and the best known approximation factors that can be achieved in polynomial time? Following the recent research initiated by Esmer et al. (ESA 2022, IPEC 2023, SODA 2024) on vertex-subset problems, and by Inamdar et al. (ITCS 2024) on graph-partitioning problems, we focus on vertex-ordering problems. In particular, we give positive results for Feedback Arc Set, Optimal Linear Arrangement, Cutwidth, and Pathwidth. Most of our algorithms build upon a novel ``balanced-cut'' approach, which is our main conceptual contribution. This allows us to solve various problems in very general settings allowing for directed and arc-weighted input graphs. Our main technical contribution is a (1+ε)-approximation for any ε > 0 for (weighted) Feedback Arc Set in O*((2-δ)^n) time, where δ > 0 is a constant only depending on ε.

cs.DS↗

arXiv2025

MaxMin Separation Problems: FPT Algorithms for $st$-Separator and Odd Cycle Transversal

In this paper, we study the parameterized complexity of the MaxMin versions of two fundamental separation problems: Maximum Minimal $st$-Separator and Maximum Minimal Odd Cycle Transversal (OCT), both parameterized by the solution size. In the Maximum Minimal $st$-Separator problem, given a graph $G$, two distinct vertices $s$ and $t$ and a positive integer $k$, the goal is to determine whether there exists a minimal $st$-separator in $G$ of size at least $k$. Similarly, the Maximum Minimal OCT problem seeks to determine if there exists a minimal set of vertices whose deletion results in a bipartite graph, and whose size is at least $k$. We demonstrate that both problems are fixed-parameter tractable parameterized by $k$. Our FPT algorithm for Maximum Minimal $st$-Separator answers the open question by Hanaka, Bodlaender, van der Zanden and Ono (TCS 2019). One unique insight from this work is the following. We use the meta-result of Lokshtanov, Ramanujan, Saurabh and Zehavi (ICALP 2018) that enables us to reduce our problems to highly unbreakable graphs. This is interesting, as an explicit use of the recursive understanding and randomized contractions framework of Chitnis, Cygan, Hajiaghayi, Pilipczuk and Pilipczuk (SICOMP 2016) to reduce to the highly unbreakable graphs setting (which is the result that Lokshtanov et al. tries to abstract out in their meta-theorem) does not seem obvious because certain ``extension'' variants of our problems are W[1]-hard.

cs.CC↗

arXiv2025

Multivariate Exploration of Metric Dilation

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 \cdot 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 $t\leq 2$ and the problem is W[1]-hard for $t\geq 3$ 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 $Γ$.

cs.DM↗

arXiv2024

Robust Contraction Decomposition for Minor-Free Graphs and its Applications

We prove a robust contraction decomposition theorem for $H$-minor-free graphs, which states that given an $H$-minor-free graph $G$ and an integer $p$, one can partition in polynomial time the vertices of $G$ into $p$ sets $Z_1,\dots,Z_p$ such that $\operatorname{tw}(G/(Z_i \setminus Z')) = O(p + |Z'|)$ for all $i \in [p]$ and $Z' \subseteq Z_i$. Here, $\operatorname{tw}(\cdot)$ denotes the treewidth of a graph and $G/(Z_i \setminus Z')$ denotes the graph obtained from $G$ by contracting all edges with both endpoints in $Z_i \setminus Z'$. Our result generalizes earlier results by Klein [SICOMP 2008] and Demaine et al. [STOC 2011] based on partitioning $E(G)$, and some recent theorems for planar graphs by Marx et al. [SODA 2022], for bounded-genus graphs (more generally, almost-embeddable graphs) by Bandyapadhyay et al. [SODA 2022], and for unit-disk graphs by Bandyapadhyay et al. [SoCG 2022]. The robust contraction decomposition theorem directly results in parameterized algorithms with running time $2^{\widetilde{O}(\sqrt{k})} \cdot n^{O(1)}$ or $n^{O(\sqrt{k})}$ for every vertex/edge deletion problems on $H$-minor-free graphs that can be formulated as Permutation CSP Deletion or 2-Conn Permutation CSP Deletion. Consequently, we obtain the first subexponential-time parameterized algorithms for Subset Feedback Vertex Set, Subset Odd Cycle Transversal, Subset Group Feedback Vertex Set, 2-Conn Component Order Connectivity on $H$-minor-free graphs. For other problems which already have subexponential-time parameterized algorithms on $H$-minor-free graphs (e.g., Odd Cycle Transversal, Vertex Multiway Cut, Vertex Multicut, etc.), our theorem gives much simpler algorithms of the same running time.

cs.DS↗

arXiv2024

Parameterized Geometric Graph Modification with Disk Scaling

The parameterized analysis of graph modification problems represents the most extensively studied area within Parameterized Complexity. Given a graph $G$ and an integer $k\in\mathbb{N}$ as input, the goal is to determine whether we can perform at most $k$ operations on $G$ to transform it into a graph belonging to a specified graph class $\mathcal{F}$. Typical operations are combinatorial and include vertex deletions and edge deletions, insertions, and contractions. However, in many real-world scenarios, when the input graph is constrained to be a geometric intersection graph, the modification of the graph is influenced by changes in the geometric properties of the underlying objects themselves, rather than by combinatorial modifications. It raises the question of whether vertex deletions or adjacency modifications are necessarily the most appropriate modification operations for studying modifications of geometric graphs. We propose the study of the disk intersection graph modification through the scaling of disks. This operation is typical in the realm of topology control but has not yet been explored in the context of Parameterized Complexity. We design parameterized algorithms and kernels for modifying to the most basic graph classes: edgeless, connected, and acyclic. Our technical contributions encompass a novel combination of linear programming, branching, and kernelization techniques, along with a fresh application of bidimensionality theory to analyze the area covered by disks, which may have broader applicability.

cs.CG↗

arXiv2024

Parameterized Approximation for Capacitated $d$-Hitting Set with Hard Capacities

The \textsc{Capacitated $d$-Hitting Set} problem involves a universe $U$ with a capacity function $\mathsf{cap}: U \rightarrow \mathbb{N}$ and a collection $\mathcal{A}$ of subsets of $U$, each of size at most $d$. The goal is to find a minimum subset $S \subseteq U$ and an assignment $ϕ: \mathcal{A} \rightarrow S$ such that for every $A \in \mathcal{A}$, $ϕ(A) \in A$, and for each $x \in U$, $|ϕ^{-1}(x)| \leq \mathsf{cap}(x)$. For $d=2$, this is known as \textsc{Capacitated Vertex Cover}. In the weighted variant, each element of $U$ has a positive integer weight, with the objective of finding a minimum-weight capacitated hitting set. Chuzhoy and Naor [SICOMP 2006] provided a factor-3 approximation for \textsc{Capacitated Vertex Cover} and showed that the weighted case lacks an $o(\log n)$-approximation unless $P=NP$. Kao and Wong [SODA 2017] later independently achieved a $d$-approximation for \textsc{Capacitated $d$-Hitting Set}, with no $d - ε$ improvements possible under the Unique Games Conjecture. Our main result is a parameterized approximation algorithm with runtime $\left(\frac{k}ε\right)^k 2^{k^{O(kd)}}(|U|+|\mathcal{A}|)^{O(1)}$ that either concludes no solution of size $\leq k$ exists or finds $S$ of size $\leq 4/3 \cdot k$ and weight at most $2+ε$ times the minimum weight for solutions of size $\leq k$. We further show that no FPT-approximation with factor $c > 1$ exists for unweighted \textsc{Capacitated $d$-Hitting Set} with $d \geq 3$, nor with factor $2 - ε$ for the weighted version, assuming the Exponential Time Hypothesis. These results extend to \textsc{Capacitated Vertex Cover} in multigraphs. Additionally, a variant of multi-dimensional \textsc{Knapsack} is shown hard to FPT-approximate within $2 - ε$.

cs.DS↗

arXiv2024

Parameterized Saga of First-Fit and Last-Fit Coloring

The classic greedy coloring (first-fit) algorithm considers the vertices of an input graph $G$ in a given order and assigns the first available color to each vertex $v$ in $G$. In the {\sc Grundy Coloring} problem, the task is to find an ordering of the vertices that will force the greedy algorithm to use as many colors as possible. In the {\sc Partial Grundy Coloring}, the task is also to color the graph using as many colors as possible. This time, however, we may select both the ordering in which the vertices are considered and which color to assign the vertex. The only constraint is that the color assigned to a vertex $v$ is a color previously used for another vertex if such a color is available. Whether {\sc Grundy Coloring} and {\sc Partial Grundy Coloring} admit fixed-parameter tractable (FPT) algorithms, algorithms with running time $f(k)n^{\OO(1)}$, where $k$ is the number of colors, was posed as an open problem by Zaker and by Effantin et al., respectively. Recently, Aboulker et al. (STACS 2020 and Algorithmica 2022) resolved the question for \Grundycol\ in the negative by showing that the problem is W[1]-hard. For {\sc Partial Grundy Coloring}, they obtain an FPT algorithm on graphs that do not contain $K_{i,j}$ as a subgraph (a.k.a. $K_{i,j}$-free graphs). Aboulker et al.~re-iterate the question of whether there exists an FPT algorithm for {\sc Partial Grundy Coloring} on general graphs and also asks whether {\sc Grundy Coloring} admits an FPT algorithm on $K_{i,j}$-free graphs. We give FPT algorithms for {\sc Partial Grundy Coloring} on general graphs and for {\sc Grundy Coloring} on $K_{i,j}$-free graphs, resolving both the questions in the affirmative. We believe that our new structural theorems for partial Grundy coloring and ``representative-family'' like sets for $K_{i,j}$-free graphs that we use in obtaining our results may have wider algorithmic applications.

cs.DM↗

arXiv2024

Packing Short Cycles

Cycle packing is a fundamental problem in optimization, graph theory, and algorithms. Motivated by recent advancements in finding vertex-disjoint paths between a specified set of vertices that either minimize the total length of the paths [Björklund, Husfeldt, ICALP 2014; Mari, Mukherjee, Pilipczuk, and Sankowski, SODA 2024] or request the paths to be shortest [Lochet, SODA 2021], we consider the following cycle packing problems: Min-Sum Cycle Packing and Shortest Cycle Packing. In Min-Sum Cycle Packing, we try to find, in a weighted undirected graph, $k$ vertex-disjoint cycles of minimum total weight. Our first main result is an algorithm that, for any fixed $k$, solves the problem in polynomial time. We complement this result by establishing the W[1]-hardness of Min-Sum Cycle Packing parameterized by $k$. The same results hold for the version of the problem where the task is to find $k$ edge-disjoint cycles. Our second main result concerns Shortest Cycle Packing, which is a special case of Min-Sum Cycle Packing that asks to find a packing of $k$ shortest cycles in a graph. We prove this problem to be fixed-parameter tractable (FPT) when parameterized by $k$ on weighted planar graphs. We also obtain a polynomial kernel for the edge-disjoint variant of the problem on planar graphs. Deciding whether Min-Sum Cycle Packing is FPT on planar graphs and whether Shortest Cycle Packing is FPT on general graphs remain challenging open questions.

cs.DS↗

arXiv2024

Fixed-Parameter Tractability of Hedge Cut

In the Hedge Cut problem, the edges of a graph are partitioned into groups called hedges, and the question is what is the minimum number of hedges to delete to disconnect the graph. Ghaffari, Karger, and Panigrahi [SODA 2017] showed that Hedge Cut can be solved in quasipolynomial-time, raising the hope for a polynomial time algorithm. Jaffke, Lima, Masarík, Pilipczuk, and Souza [SODA 2023] complemented this result by showing that assuming the Exponential Time Hypothesis (ETH), no polynomial-time algorithm exists. In this paper, we show that Hedge Cut is fixed-parameter tractable parameterized by the solution size $\ell$ by providing an algorithm with running time $\binom{O(\log n) + \ell}{\ell} \cdot m^{O(1)}$, which can be upper bounded by $c^{\ell} \cdot (n+m)^{O(1)}$ for any constant $c>1$. This running time captures at the same time the fact that the problem is quasipolynomial-time solvable, and that it is fixed-parameter tractable parameterized by $\ell$. We further generalize this algorithm to an algorithm with running time $\binom{O(k \log n) + \ell}{\ell} \cdot n^{O(k)} \cdot m^{O(1)}$ for Hedge $k$-Cut.

cs.DS↗

arXiv2024

Efficient Approximation of Fractional Hypertree Width

We give two new approximation algorithms to compute the fractional hypertree width of an input hypergraph. The first algorithm takes as input $n$-vertex $m$-edge hypergraph $H$ of fractional hypertree width at most $ω$, runs in polynomial time and produces a tree decomposition of $H$ of fractional hypertree width $O(ω\log n \log ω)$. As an immediate corollary this yields polynomial time $O(\log^2 n \log ω)$-approximation algorithms for (generalized) hypertree width as well. To the best of our knowledge our algorithm is the first non-trivial polynomial-time approximation algorithm for fractional hypertree width and (generalized) hypertree width, as opposed to algorithms that run in polynomial time only when $ω$ is considered a constant. For hypergraphs with the bounded intersection property we get better bounds, comparable with that recent algorithm of Lanzinger and Razgon [STACS 2024]. The second algorithm runs in time $n^ωm^{O(1)}$ and produces a tree decomposition of $H$ of fractional hypertree width $O(ω\log^2 ω)$. This significantly improves over the $(n+m)^{O(ω^3)}$ time algorithm of Marx [ACM TALG 2010], which produces a tree decomposition of fractional hypertree width $O(ω^3)$, both in terms of running time and the approximation ratio. Our main technical contribution, and the key insight behind both algorithms, is a variant of the classic Menger's Theorem for clique separators in graphs: For every graph $G$, vertex sets $A$ and $B$, family ${\cal F}$ of cliques in $G$, and positive rational $f$, either there exists a sub-family of $O(f \cdot \log^2 n)$ cliques in ${\cal F}$ whose union separates $A$ from $B$, or there exist $f \cdot \log |{\cal F}|$ paths from $A$ to $B$ such that no clique in ${\cal F}$ intersects more than $\log |{\cal F}|$ paths.

cs.DS↗

arXiv2024

The Parameterized Complexity Landscape of Two-Sets Cut-Uncut

In Two-Sets Cut-Uncut, we are given an undirected graph $G=(V,E)$ and two terminal sets $S$ and $T$. The task is to find a minimum cut $C$ in $G$ (if there is any) separating $S$ from $T$ under the following ``uncut'' condition. In the graph $(V,E \setminus C)$, the terminals in each terminal set remain in the same connected component. In spite of the superficial similarity to the classic problem Minimum $s$-$t$-Cut, Two-Sets Cut-Uncut is computationally challenging. In particular, even deciding whether such a cut of any size exists, is already NP-complete. We initiate a systematic study of Two-Sets Cut-Uncut within the context of parameterized complexity. By leveraging known relations between many well-studied graph parameters, we characterize the structural properties of input graphs that allow for polynomial kernels, fixed-parameter tractability (FPT), and slicewise polynomial algorithms (XP). Our main contribution is the near-complete establishment of the complexity of these algorithmic properties within the described hierarchy of graph parameters. On a technical level, our main results are fixed-parameter tractability for the (vertex-deletion) distance to cographs and an OR-cross composition excluding polynomial kernels for the vertex cover number of the input graph (under the standard complexity assumption NP is not contained in coNP/poly).

cs.DS↗

arXiv2024

Bipartizing (Pseudo-)Disk Graphs: Approximation with a Ratio Better than 3

In a disk graph, every vertex corresponds to a disk in $\mathbb{R}^2$ and two vertices are connected by an edge whenever the two corresponding disks intersect. Disk graphs form an important class of geometric intersection graphs, which generalizes both planar graphs and unit-disk graphs. We study a fundamental optimization problem in algorithmic graph theory, Bipartization (also known as Odd Cycle Transversal), on the class of disk graphs. The goal of Bipartization is to delete a minimum number of vertices from the input graph such that the resulting graph is bipartite. A folklore (polynomial-time) $3$-approximation algorithm for Bipartization on disk graphs follows from the classical framework of Goemans and Williamson [Combinatorica'98] for cycle-hitting problems. For over two decades, this result has remained the best known approximation for the problem (in fact, even for Bipartization on unit-disk graphs). In this paper, we achieve the first improvement upon this result, by giving a $(3-α)$-approximation algorithm for Bipartization on disk graphs, for some constant $α>0$. Our algorithm directly generalizes to the broader class of pseudo-disk graphs. Furthermore, our algorithm is robust in the sense that it does not require a geometric realization of the input graph to be given.

cs.DS↗

arXiv2024

Hybrid k-Clustering: Blending k-Median and k-Center

We propose a novel clustering model encompassing two well-known clustering models: k-center clustering and k-median clustering. In the Hybrid k-Clusetring problem, given a set P of points in R^d, an integer k, and a non-negative real r, our objective is to position k closed balls of radius r to minimize the sum of distances from points not covered by the balls to their closest balls. Equivalently, we seek an optimal L_1-fitting of a union of k balls of radius r to a set of points in the Euclidean space. When r=0, this corresponds to k-median; when the minimum sum is zero, indicating complete coverage of all points, it is k-center. Our primary result is a bicriteria approximation algorithm that, for a given ε>0, produces a hybrid k-clustering with balls of radius (1+ε)r. This algorithm achieves a cost at most 1+εof the optimum, and it operates in time 2^{(kd/ε)^{O(1)}} n^{O(1)}. Notably, considering the established lower bounds on k-center and k-median, our bicriteria approximation stands as the best possible result for Hybrid k-Clusetring.

cs.DS↗

arXiv2024

Cuts in Graphs with Matroid Constraints

{\sc Vertex $(s, t)$-Cut} and {\sc 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 $R \in \mathbb{F}^{r \times n}$ of a linear matroid $\mathcal{M} = (V(G), \mathcal{I})$ of rank $r$ in the input, and the goal is to determine whether there exists a vertex subset $S \subseteq V(G)$ that has the required cut properties, as well as is independent in the matroid $\mathcal{M}$. We refer to these problems as {\sc Independent Vertex $(s, t)$-cut}, and {\sc Independent Multiway Cut}, respectively. We show that these problems are fixed-parameter tractable ({\sf FPT}) when parameterized by the solution size (which can be assumed to be equal to the rank of the matroid $\mathcal{M}$). 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 {\sf FPT} algorithms for the independent version of {\sc Odd Cycle Transversal}. Further, our results can be generalized to other variants of the problems, e.g., weighted versions, or edge-deletion versions.

cs.DM↗

arXiv2024

When far is better: The Chamberlin-Courant approach to obnoxious committee selection

Classical work on metric space based committee selection problem interprets distance as ``near is better''. In this work, motivated by real-life situations, we interpret distance as ``far is better''. Formally stated, we initiate the study of ``obnoxious'' committee scoring rules when the voters' preferences are expressed via a metric space. To this end, we propose a model where large distances imply high satisfaction and study the egalitarian avatar of the well-known Chamberlin-Courant voting rule and some of its generalizations. For a given integer value $1 \le λ\le k$, the committee size k, a voter derives satisfaction from only the $λ$-th favorite committee member; the goal is to maximize the satisfaction of the least satisfied voter. For the special case of $λ= 1$, this yields the egalitarian Chamberlin-Courant rule. In this paper, we consider general metric space and the special case of a $d$-dimensional Euclidean space. We show that when $λ$ is $1$ and $k$, the problem is polynomial-time solvable in $\mathbb{R}^2$ and general metric space, respectively. However, for $λ= k-1$, it is NP-hard even in $\mathbb{R}^2$. Thus, we have ``double-dichotomy'' in $\mathbb{R}^2$ with respect to the value of λ, where the extreme cases are solvable in polynomial time but an intermediate case is NP-hard. Furthermore, this phenomenon appears to be ``tight'' for $\mathbb{R}^2$ because the problem is NP-hard for general metric space, even for $λ=1$. Consequently, we are motivated to explore the problem in the realm of (parameterized) approximation algorithms and obtain positive results. Interestingly, we note that this generalization of Chamberlin-Courant rules encodes practical constraints that are relevant to solutions for certain facility locations.

cs.DS↗