arXiv subjects

Tanmay Inamdar

Publications and source records attributed to Tanmay Inamdar.

At least 19 recordsLinked to original sources

arXiv2026

Reaching Fairness by Reallocating Goods

Fair allocation of indivisible goods has largely been studied under the assumption that no prior allocation exists. Motivated by practical settings with pre-existing (and possibly unfair) allocations, we study how to achieve fairness through limited reallocations. Building on recent work on reformability/reallocations, we consider three fairness notions---envy-freeness (EF), envy-freeness up to one good (EF1), and envy-freeness up to any good (EFX)---and optimize the number of goods reallocated. We analyze both the classical and parameterized complexity of these problems, providing a comprehensive analysis across multiple fairness notions.

cs.GT↗

arXiv2026

Minimum Envy Graphical House Allocation Beyond Identical Valuations

House allocation is an extremely well-studied problem in the field of fair allocation, where the goal is to assign $n$ houses to $n$ agents while satisfying certain fairness criterion, e.g., envy-freeness. To model social interactions, the Graphical House Allocation framework introduces a social graph $G$, in which each vertex corresponds to an agent, and an edge $(u, v)$ corresponds to the potential of agent $u$ to envy the agent $v$, based on their allocations and valuations. In undirected social graphs, the potential for envy is in both the directions. In the Minimum Envy Graphical House Allocation (ME-GHA) problem, given a set of $n$ agents, $n$ houses, a social graph, and agent's valuation functions, the goal is to find an allocation that minimizes the total envy summed up over all the edges of $G$. Recent work, [Hosseini et al., AAMAS 2023, AAMAS 2024] studied ME-GHA in the regime of polynomial-time algorithms, and designed exact and approximation algorithms, for certain graph classes under identical agent valuations. We initiate the study of \gha with non-identical valuations, a setting that has so far remained unexplored. We investigate the multivariate (parameterized) complexity of \gha by identifying structural restrictions on the social graph and valuation functions that yield tractability. We also design moderately exponential-time algorithms for several graph classes, and a polynomial-time algorithm for {binary valuations that returns an allocation with envy at most one when the social graph has maximum degree at most one.

cs.GT↗

arXiv2026

FPT Approximations for Connected Maximum Coverage

We revisit connectivity-constrained coverage through a unifying model, Partial Connected Red-Blue Dominating Set. Given a red-blue bipartite graph $G$ and an auxiliary connectivity graph $G_{conn}$ on red vertices, and integers $k, t$, the task is to find a $k$-sized subset of red vertices that dominates at least $t$ blue vertices, and that induces a connected subgraph in $G_{conn}$. This formulation captures connected variants of Max Coverage, Partial Dominating Set, and Partial Vertex Cover studied in prior literature. After identifying (parameterized) inapproximability results inherited from known problems, we first show that the problem is fixed-parameter tractable by $t$. Furthermore, when the bipartite graph excludes $K_{d,d}$ as a subgraph, we design (resp. efficient) parameterized approximation schemes for approximating $t$ (resp. $k$). Notably, these FPT approximations do not impose any restrictions on $G_{conn}$. Together, these results chart the boundary between hardness and FPT-approximability for connectivity-constrained coverage.

cs.DS↗

arXiv2025

Tight Parameterized (In)tractability of Layered Crossing Minimization: Subexponential Algorithms and Kernelization

The starting point of our work is a decade-old open question concerning the subexponential parameterized complexity of \textsc{2-Layer Crossing Minimization}. In this problem, the input is an $n$-vertex graph $G$ whose vertices are partitioned into two independent sets $V_1$ and $V_2$, and a non-negative integer $k$. The question is whether $G$ admits a 2-layered drawing with at most $k$ crossings, where each $V_i$ lies on a distinct line parallel to the $x$-axis, and all edges are straight lines. We resolve this open question by giving the first subexponential fixed-parameter algorithm for this problem, running in time $2^{O(\sqrt{k}\log k)} + n \cdot k^{O(1)}$. We then ask whether the subexponential phenomenon extends beyond two layers. In the general $h$-Layer Crossing Minimization problem, the vertex set is partitioned into $h$ independent sets $V_1, \ldots, V_h$, and the goal is to decide whether an $h$-layered drawing with at most $k$ crossings exists. We present a subexponential FPT algorithm for three layers with running time $2^{O(k^{2/3}\log k)} + n \cdot k^{O(1)}$ for $h = 3$ layers. In contrast, we show that for all $h \ge 5$, no algorithm with running time $2^{o(k/\log k)} \cdot n^{O(1)}$ exists unless the Exponential-Time Hypothesis fails. Finally, we address polynomial kernelization. While a polynomial kernel was already known for $h=2$, we design a new polynomial kernel for $h=3$. These kernels are essential ingredients in our subexponential algorithms. Finally, we rule out polynomial kernels for all $h \ge 4$ unless the polynomial hierarchy collapses.

cs.DS↗

arXiv2025

Clustering under Constraints: Efficient Parameterized Approximation Schemes

We present a unified framework that yields EPASes for constrained $(k,z)$-clustering in metric spaces of bounded (algorithmic) scatter dimension, a notion introduced by Abbasi et al. (FOCS 2023). They showed that several well known metric families, including continuous Euclidean spaces, bounded doubling spaces, planar metrics, and bounded treewidth metrics, have bounded scatter dimension. Subsequently, Bourneuf and Pilipczuk (SODA 2025) proved that this also holds for metrics induced by graphs from any fixed proper minor closed class. Our result, in particular, addresses a major open question of Abbasi et al., whose approach to $k$-clustering in such metrics was inherently limited to \emph{Voronoi-based} objectives, where each point is connected only to its nearest chosen center. As a consequence, we obtain EPASes for several constrained clustering problems, including capacitated and matroid $(k,z)$-clustering, fault tolerant and fair $(k,z)$-clustering, as well as for metrics of bounded highway dimension. In particular, our results on capacitated and fair $k$-Median and $k$-Means provide the first EPASes for these problems across broad families of structured metrics. Previously, such results were known only in continuous Euclidean spaces, due to the works of Cohen-Addad and Li (ICALP 2019) and Bandyapadhyay, Fomin, and Simonov (ICALP 2021; JCSS 2024), respectively. Along the way, we also obtain faster EPASes for uncapacitated $k$-Median and $k$-Means, improving upon the running time of the algorithm by Abbasi et al. (FOCS 2023).

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+{\epsilon})-approximation for any {\epsilon} > 0 for (weighted) Feedback Arc Set in O*((2-{\delta})^n) time, where {\delta} > 0 is a constant only depending on {\epsilon}.

cs.DS↗

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 $\Gamma$. Even in this specific case, Dilation $t$-Augmentation remains computationally challenging. In particular, the problem is W[2]-hard parameterized by $k$ when $\Gamma$ 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 $k+t+\Delta$, where $\Delta$ is the maximum degree of the graph $G$ or $\Gamma$.

cs.DM↗

arXiv2025

Dimension-Free Parameterized Approximation Schemes for Hybrid Clustering

Hybrid $k$-Clustering is a model of clustering that generalizes two of the most widely studied clustering objectives: $k$-Center and $k$-Median. In this model, given a set of $n$ points $P$, the goal is to find $k$ centers such that the sum of the $r$-distances of each point to its nearest center is minimized. The $r$-distance between two points $p$ and $q$ is defined as $\max\{d(p, q)-r, 0\}$ -- this represents the distance of $p$ to the boundary of the $r$-radius ball around $q$ if $p$ is outside the ball, and $0$ otherwise. This problem was recently introduced by Fomin et al. [APPROX 2024], who designed a $(1+\varepsilon, 1+\varepsilon)$-bicrtieria approximation that runs in time $2^{(kd/\varepsilon)^{O(1)}} \cdot n^{O(1)}$ for inputs in $\mathbb{R}^d$; such a bicriteria solution uses balls of radius $(1+\varepsilon)r$ instead of $r$, and has a cost at most $1+\varepsilon$ times the cost of an optimal solution using balls of radius $r$. In this paper we significantly improve upon this result by designing an approximation algorithm with the same bicriteria guarantee, but with running time that is FPT only in $k$ and $\varepsilon$ -- crucially, removing the exponential dependence on the dimension $d$. This resolves an open question posed in their paper. Our results extend further in several directions. First, our approximation scheme works in a broader class of metric spaces, including doubling spaces, minor-free, and bounded treewidth metrics. Secondly, our techniques yield a similar bicriteria FPT-approximation schemes for other variants of Hybrid $k$-Clustering, e.g., when the objective features the sum of $z$-th power of the $r$-distances. Finally, we also design a coreset for Hybrid $k$-Clustering in doubling spaces, answering another open question from the work of Fomin et al.

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

Walks on uncountable ordinals and non-structure theorems for higher Aronszajn lines

It is proved that if there is an $\aleph_2$-Aronszajn line, then there is one that does not contain an $\aleph_2$-Countryman line. This solves a problem of Moore and stands in a sharp contrast with his Basis Theorem for linear orders of size $\aleph_1$. The proof combines walks on ordinals, club guessing, strong colourings of three different types, and a bit of finite combinatorics. This and further non-structure theorems for Aronszajn lines and trees are established for successors of regulars, successors of singulars, as well as inaccessibles.

math.LO↗

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 \epsilon>0, produces a hybrid k-clustering with balls of radius (1+\epsilon)r. This algorithm achieves a cost at most 1+\epsilon of the optimum, and it operates in time 2^{(kd/\epsilon)^{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 \'a 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

A Ramsey theorem for the reals

We prove that for every colouring of pairs of reals with finitely-many colours, there is a set homeomorphic to the rationals which takes no more than two colours. This was conjectured by Galvin in 1970, and a colouring of Sierpi{\'n}ski from 1933 witnesses that the number of colours cannot be reduced to one. Previously in 1985 Shelah had shown that a stronger statement is consistent with a forcing construction assuming the existence of large cardinals. Then in 2018 Raghavan and Todor\v{c}evi\'c had proved it assuming the existence of large cardinals. We prove it in $ZFC$. In fact Raghavan and Todor\v{c}evi\'c proved, assuming more large cardinals, a similar result for a large class of topological spaces. We prove this also, again in $ZFC$.

math.LO↗

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 $1 \le \lambda \le k$, the committee size k, a voter derives satisfaction from only the $\lambda$-th favorite committee member; the goal is to maximize the satisfaction of the least satisfied voter. For the special case of $\lambda = 1$, 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 $\lambda$ is $1$ and $k$, the problem is polynomial-time solvable in $\mathbb{R}^2$ and general metric space, respectively. However, for $\lambda = k-1$, it is NP-hard even in $\mathbb{R}^2$. Thus, we have ``double-dichotomy'' in $\mathbb{R}^2$ with respect to the value of {\lambda}, 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 $\lambda=1$. 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↗

arXiv2024

Satisfiability to Coverage in Presence of Fairness, Matroid, and Global Constraints

In MaxSAT with Cardinality Constraint problem (CC-MaxSAT), we are given a CNF-formula $\Phi$, and $k \ge 0$, and the goal is to find an assignment $\beta$ with at most $k$ variables set to true (also called a weight $k$-assignment) such that the number of clauses satisfied by $\beta$ is maximized. MaxCov can be seen as a special case of CC-MaxSAT, where the formula $\Phi$ is monotone, i.e., does not contain any negative literals. CC-MaxSAT and MaxCov are extremely well-studied problems in the approximation algorithms as well as parameterized complexity literature. Our first contribution is that the two problems are equivalent to each other in the context of FPT-Approximation parameterized by $k$ (approximation is in terms of number of clauses satisfied/elements covered). We give a randomized reduction from CC-MaxSAT to MaxCov in time $O(1/\epsilon)^{k} \cdot (m+n)^{O(1)}$ that preserves the approximation guarantee up to a factor of $1-\epsilon$. Furthermore, this reduction also works in the presence of fairness and matroid constraints. Armed with this reduction, we focus on designing FPT-Approximation schemes (FPT-ASes) for MaxCov and its generalizations. Our algorithms are based on a novel combination of a variety of ideas, including a carefully designed probability distribution that exploits sparse coverage functions. These algorithms substantially generalize the results in Jain et al. [SODA 2023] for CC-MaxSAT and MaxCov for $K_{d,d}$-free set systems (i.e., no $d$ sets share $d$ elements), as well as a recent FPT-AS for Matroid-Constrained MaxCov by Sellier [ESA 2023] for frequency-$d$ set systems.

cs.DS↗

arXiv2024

Controlling Delegations in Liquid Democracy

In liquid democracy, agents can either vote directly or delegate their vote to a different agent of their choice. This results in a power structure in which certain agents possess more voting weight than others. As a result, it opens up certain possibilities of vote manipulation, including control and bribery, that do not exist in standard voting scenarios of direct democracy. Here we formalize a certain kind of election control -- in which an external agent may change certain delegation arcs -- and study the computational complexity of the corresponding combinatorial problem.

cs.GT↗

arXiv2023

FPT Approximations for Packing and Covering Problems Parameterized by Elimination Distance and Even Less

For numerous graph problems in the realm of parameterized algorithms, using the size of a smallest deletion set (called a modulator) into well-understood graph families as parameterization has led to a long and successful line of research. Recently, however, there has been an extensive study of structural parameters that are potentially much smaller than the modulator size. In particular, recent papers [Jansen et al. STOC 2021; Agrawal et al. SODA 2022] have studied parameterization by the size of the modulator to a graph family $\mathcal{H}$ ($\textbf{mod}_{\mathcal{H}}$), elimination distance to $\mathcal{H}$ ($\textbf{ed}_{\mathcal{H}}$), and $\mathcal{H}$-treewidth ($\textbf{tw}_{\mathcal{H}}$). While these new parameters have been successfully exploited to design fast exact algorithms their utility (especially that of latter two) in the context of approximation algorithms is mostly unexplored. The conceptual contribution of this paper is to present novel algorithmic meta-theorems that expand the impact of these structural parameters to the area of FPT Approximation, mirroring their utility in the design of exact FPT algorithms. Precisely, we show that if a covering or packing problem is definable in Monadic Second Order Logic and has a property called Finite Integer Index, then the existence of an FPT Approximation Scheme (FPT-AS, i.e., ($1\pm \epsilon$)-approximation) parameterized these three parameters is in fact equivalent. As concrete exemplifications of our meta-theorems, we obtain FPT-ASes for well-studied graph problems such as Vertex Cover, Feedback Vertex Set, Cycle Packing and Dominating Set, parameterized by these three parameters.

cs.DS↗

arXiv2023

Diamond on ladder systems and countably metacompact topological spaces

The property of countable metacompactness of a topological space gets its importance from Dowker's 1951 theorem that the product of a normal space X with the unit interval is again normal iff X is countably metacompact. In a recent paper, Leiderman and Szeptycki studied $\Delta$-spaces, which are a subclass of the class of countably metacompact spaces. They proved that a single Cohen real introduces a ladder system $L$ over the first uncountable cardinal for which the corresponding space $X_L$ is not a $\Delta$-space, and asked whether there is a ZFC example of a ladder system $L$ over some cardinal $\kappa$ for which $X_L$ is not countably metacompact, in particular, not a $\Delta$-space. We prove that an affirmative answer holds for the cardinal $\kappa=cf(\beth_{\omega+1})$. Assuming $\beth_\omega=\aleph_\omega$, we get an example at a much lower cardinal, namely $\kappa=2^{2^{2^{\aleph_0}}}$, and our ladder system $L$ is moreover $\omega$-bounded.

math.LO↗