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Tanmay Inamdar

Publications and source records attributed to Tanmay Inamdar.

At least 37 records · Page 2Linked to original sources

arXiv2024

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 $Δ$-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 $Δ$-space, and asked whether there is a ZFC example of a ladder system $L$ over some cardinal $κ$ for which $X_L$ is not countably metacompact, in particular, not a $Δ$-space. We prove that an affirmative answer holds for the cardinal $κ=cf(\beth_{ω+1})$. Assuming $\beth_ω=\aleph_ω$, we get an example at a much lower cardinal, namely $κ=2^{2^{2^{\aleph_0}}}$, and our ladder system $L$ is moreover $ω$-bounded.

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arXiv2023

Was Ulam right? II: Small width and general ideals

We continue our study of Sierpinski-type colourings. In contrast to the prequel paper, we focus here on colourings for ideals stratified by their completeness degree. In particular, improving upon Ulam's theorem and its extension by Hajnal, it is proved that if $κ$ is a regular uncountable cardinal that is not weakly compact in L, then there is a universal witness for non-weak-saturation of $κ$-complete ideals. Specifically, there are $κ$-many decompositions of $κ$ such that, for every $κ$-complete ideal $J$ over $κ$, and every $B\in J^+$, one of the decompositions shatters $B$ into $κ$-many $J^+$-sets. A second focus here is the feature of narrowness of colourings, one already present in the theorem of Sierpinski. This feature ensures that a colouring suitable for an ideal is also suitable for all superideals possessing the requisite completeness degree. It is proved that unlike successors of regulars, every successor of a singular cardinal admits such a narrow colouring.

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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., ($1\pm ε$)-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.

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arXiv2023

FPT Approximation and Subexponential Algorithms for Covering Few or Many Edges

We study the \textsc{$α$-Fixed Cardinality Graph Partitioning ($α$-FCGP)} problem, the generic local graph partitioning problem introduced by Bonnet et al. [Algorithmica 2015]. In this problem, we are given a graph $G$, two numbers $k,p$ and $0\leqα\leq 1$, the question is whether there is a set $S\subseteq V$ of size $k$ with a specified coverage function $cov_α(S)$ at least $p$ (or at most $p$ for the minimization version). The coverage function $cov_α(\cdot)$ counts edges with exactly one endpoint in $S$ with weight $α$ and edges with both endpoints in $S$ with weight $1 - α$. $α$-FCGP generalizes a number of fundamental graph problems such as \textsc{Densest $k$-Subgraph}, \textsc{Max $k$-Vertex Cover}, and \textsc{Max $(k,n-k)$-Cut}. A natural question in the study of $α$-FCGP is whether the algorithmic results known for its special cases, like \textsc{Max $k$-Vertex Cover}, could be extended to more general settings. One of the simple but powerful methods for obtaining parameterized approximation [Manurangsi, SOSA 2019] and subexponential algorithms [Fomin et al. IPL 2011] for \textsc{Max $k$-Vertex Cover} is based on the greedy vertex degree orderings. The main insight of our work is that the idea of greed vertex degree ordering could be used to design fixed-parameter approximation schemes (FPT-AS) for $α> 0$ and the subexponential-time algorithms for the problem on apex-minor free graphs for maximization with $α> 1/3$ and minimization with $α< 1/3$.

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arXiv2023

Parameterized and Approximation Algorithms for the Maximum Bimodal Subgraph Problem

A vertex of a plane digraph is bimodal if all its incoming edges (and hence all its outgoing edges) are consecutive in the cyclic order around it. A plane digraph is bimodal if all its vertices are bimodal. Bimodality is at the heart of many types of graph layouts, such as upward drawings, level-planar drawings, and L-drawings. If the graph is not bimodal, the Maximum Bimodal Subgraph (MBS) problem asks for an embedding-preserving bimodal subgraph with the maximum number of edges. We initiate the study of the MBS problem from the parameterized complexity perspective with two main results: (i) we describe an FPT algorithm parameterized by the branchwidth (and hence by the treewidth) of the graph; (ii) we establish that MBS parameterized by the number of non-bimodal vertices admits a polynomial kernel. As the byproduct of these results, we obtain a subexponential FPT algorithm and an efficient polynomial-time approximation scheme for MBS.

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arXiv2023

Parameterized Complexity of Fair Bisection: FPT-Approximation meets Unbreakability

In the Minimum Bisection problem, input is a graph $G$ and the goal is to partition the vertex set into two parts $A$ and $B$, such that $||A|-|B|| \le 1$ and the number $k$ of edges between $A$ and $B$ is minimized. This problem can be viewed as a clustering problem where edges represent similarity, and the task is to partition the vertices into two equally sized clusters, while minimizing the number of pairs of similar objects that end up in different clusters. In this paper, we initiate the study of a fair version of Minimum Bisection. In this problem, the vertices of the graph are colored using one of $c \ge 1$ colors. The goal is to find a bisection $(A, B)$ with at most $k$ edges between the parts, such that for each color $i\in [c]$, $A$ has exactly $r_i$ vertices of color $i$. We first show that Fair Bisection is $W$[1]-hard parameterized by $c$ even when $k = 0$. On the other hand, our main technical contribution shows that is that this hardness result is simply a consequence of the very strict requirement that each color class $i$ has {\em exactly} $r_i$ vertices in $A$. In particular, we give an $f(k,c,ε)n^{O(1)}$ time algorithm that finds a balanced partition $(A, B)$ with at most $k$ edges between them, such that for each color $i\in [c]$, there are at most $(1\pm ε)r_i$ vertices of color $i$ in $A$. Our approximation algorithm is best viewed as a proof of concept that the technique introduced by [Lampis, ICALP '18] for obtaining FPT-approximation algorithms for problems of bounded tree-width or clique-width can be efficiently exploited even on graphs of unbounded width. The key insight is that the technique of Lampis is applicable on tree decompositions with unbreakable bags (as introduced in [Cygan et al., SIAM Journal on Computing '14]). Along the way, we also derive a combinatorial result regarding tree decompositions of graphs.

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arXiv2023

Kernelization for Spreading Points

We consider the following problem about dispersing points. Given a set of points in the plane, the task is to identify whether by moving a small number of points by small distance, we can obtain an arrangement of points such that no pair of points is ``close" to each other. More precisely, for a family of $n$ points, an integer $k$, and a real number $d > 0$, we ask whether at most $k$ points could be relocated, each point at distance at most $d$ from its original location, such that the distance between each pair of points is at least a fixed constant, say $1$. A number of approximation algorithms for variants of this problem, under different names like distant representatives, disk dispersing, or point spreading, are known in the literature. However, to the best of our knowledge, the parameterized complexity of this problem remains widely unexplored. We make the first step in this direction by providing a kernelization algorithm that, in polynomial time, produces an equivalent instance with $O(d^2k^3)$ points. As a byproduct of this result, we also design a non-trivial fixed-parameter tractable (FPT) algorithm for the problem, parameterized by $k$ and $d$. Finally, we complement the result about polynomial kernelization by showing a lower bound that rules out the existence of a kernel whose size is polynomial in $k$ alone, unless $\mathsf{NP} \subseteq \mathsf{coNP}/\text{poly}$.

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arXiv2023

Fixed-Parameter Algorithms for Fair Hitting Set Problems

Selection of a group of representatives satisfying certain fairness constraints, is a commonly occurring scenario. Motivated by this, we initiate a systematic algorithmic study of a \emph{fair} version of \textsc{Hitting Set}. In the classical \textsc{Hitting Set} problem, the input is a universe $\mathcal{U}$, a family $\mathcal{F}$ of subsets of $\mathcal{U}$, and a non-negative integer $k$. The goal is to determine whether there exists a subset $S \subseteq \mathcal{U}$ of size $k$ that \emph{hits} (i.e., intersects) every set in $\mathcal{F}$. Inspired by several recent works, we formulate a fair version of this problem, as follows. The input additionally contains a family $\mathcal{B}$ of subsets of $\mathcal{U}$, where each subset in $\mathcal{B}$ can be thought of as the group of elements of the same \emph{type}. We want to find a set $S \subseteq \mathcal{U}$ of size $k$ that (i) hits all sets of $\mathcal{F}$, and (ii) does not contain \emph{too many} elements of each type. We call this problem \textsc{Fair Hitting Set}, and chart out its tractability boundary from both classical as well as multivariate perspective. Our results use a multitude of techniques from parameterized complexity including classical to advanced tools, such as, methods of representative sets for matroids, FO model checking, and a generalization of best known kernels for \textsc{Hitting Set}.

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arXiv2023

FPT Approximations for Capacitated/Fair Clustering with Outliers

Clustering problems such as $k$-Median, and $k$-Means, are motivated from applications such as location planning, unsupervised learning among others. In such applications, it is important to find the clustering of points that is not ``skewed'' in terms of the number of points, i.e., no cluster should contain too many points. This is modeled by capacity constraints on the sizes of clusters. In an orthogonal direction, another important consideration in clustering is how to handle the presence of outliers in the data. Indeed, these clustering problems have been generalized in the literature to separately handle capacity constraints and outliers. To the best of our knowledge, there has been very little work on studying the approximability of clustering problems that can simultaneously handle both capacities and outliers. We initiate the study of the Capacitated $k$-Median with Outliers (C$k$MO) problem. Here, we want to cluster all except $m$ outlier points into at most $k$ clusters, such that (i) the clusters respect the capacity constraints, and (ii) the cost of clustering, defined as the sum of distances of each non-outlier point to its assigned cluster-center, is minimized. We design the first constant-factor approximation algorithms for C$k$MO. In particular, our algorithm returns a (3+ε)-approximation for C$k$MO in general metric spaces, and a (1+ε)-approximation in Euclidean spaces of constant dimension, that runs in time in time $f(k, m, ε) \cdot |I_m|^{O(1)}$, where $|I_m|$ denotes the input size. We can also extend these results to a broader class of problems, including Capacitated k-Means/k-Facility Location with Outliers, and Size-Balanced Fair Clustering problems with Outliers. For each of these problems, we obtain an approximation ratio that matches the best known guarantee of the corresponding outlier-free problem.

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arXiv2023

Coresets for Clustering in Geometric Intersection Graphs

Designing coresets--small-space sketches of the data preserving cost of the solutions within $(1\pm ε)$-approximate factor--is an important research direction in the study of center-based $k$-clustering problems, such as $k$-means or $k$-median. Feldman and Langberg [STOC'11] have shown that for $k$-clustering of $n$ points in general metrics, it is possible to obtain coresets whose size depends logarithmically in $n$. Moreover, such a dependency in $n$ is inevitable in general metrics. A significant amount of recent work in the area is devoted to obtaining coresests whose sizes are independent of $n$ (i.e., ``small'' coresets) for special metrics, like $d$-dimensional Euclidean spaces, doubling metrics, metrics of graphs of bounded treewidth, or those excluding a fixed minor. In this paper, we provide the first constructions of small coresets for $k$-clustering in the metrics induced by geometric intersection graphs, such as Euclidean-weighted Unit Disk/Square Graphs. These constructions follow from a general theorem that identifies two canonical properties of a graph metric sufficient for obtaining small coresets. The proof of our theorem builds on the recent work of Cohen-Addad, Saulpic, and Schwiegelshohn [STOC '21], which ensures small-sized coresets conditioned on the existence of an interesting set of centers, called ``centroid set''. The main technical contribution of our work is the proof of the existence of such a small-sized centroid set for graphs that satisfy the two canonical geometric properties. The new coreset construction helps to design the first $(1+ε)$-approximation for center-based clustering problems in UDGs and USGs, that is fixed-parameter tractable in $k$ and $ε$ (FPT-AS).

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arXiv2023

Clustering What Matters: Optimal Approximation for Clustering with Outliers

Clustering with outliers is one of the most fundamental problems in Computer Science. Given a set $X$ of $n$ points and two integers $k$ and $m$, the clustering with outliers aims to exclude $m$ points from $X$ and partition the remaining points into $k$ clusters that minimizes a certain cost function. In this paper, we give a general approach for solving clustering with outliers, which results in a fixed-parameter tractable (FPT) algorithm in $k$ and $m$, that almost matches the approximation ratio for its outlier-free counterpart. As a corollary, we obtain FPT approximation algorithms with optimal approximation ratios for $k$-Median and $k$-Means with outliers in general metrics. We also exhibit more applications of our approach to other variants of the problem that impose additional constraints on the clustering, such as fairness or matroid constraints.

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arXiv2023

Proportionally Fair Matching with Multiple Groups

The study of fair algorithms has become mainstream in machine learning and artificial intelligence due to its increasing demand in dealing with biases and discrimination. Along this line, researchers have considered fair versions of traditional optimization problems including clustering, regression, ranking and voting. However, most of the efforts have been channeled into designing heuristic algorithms, which often do not provide any guarantees on the quality of the solution. In this work, we study matching problems with the notion of proportional fairness. Proportional fairness is one of the most popular notions of group fairness where every group is represented up to an extent proportional to the final selection size. Matching with proportional fairness or more commonly, proportionally fair matching, was introduced in [Chierichetti et al., AISTATS, 2019], where the problem was studied with only two groups. However, in many practical applications, the number of groups -- although often a small constant -- is larger than two. In this work, we make the first step towards understanding the computational complexity of proportionally fair matching with more than two groups. We design exact and approximation algorithms achieving reasonable guarantees on the quality of the matching as well as on the time complexity. Our algorithms are also supported by suitable hardness bounds.

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arXiv2022

(Re)packing Equal Disks into Rectangle

The problem of packing of equal disks (or circles) into a rectangle is a fundamental geometric problem. (By a packing here we mean an arrangement of disks in a rectangle without overlapping.) We consider the following algorithmic generalization of the equal disk packing problem. In this problem, for a given packing of equal disks into a rectangle, the question is whether by changing positions of a small number of disks, we can allocate space for packing more disks. More formally, in the repacking problem, for a given set of $n$ equal disks packed into a rectangle and integers $k$ and $h$, we ask whether it is possible by changing positions of at most $h$ disks to pack $n+k$ disks. Thus the problem of packing equal disks is the special case of our problem with $n=h=0$. While the computational complexity of packing equal disks into a rectangle remains open, we prove that the repacking problem is NP-hard already for $h=0$. Our main algorithmic contribution is an algorithm that solves the repacking problem in time $(h+k)^{O(h+k)}\cdot |I|^{O(1)}$, where $I$ is the input size. That is, the problem is fixed-parameter tractable parameterized by $k$ and $h$.

cs.CG↗

arXiv2022

On strong chains of sets and functions

Shelah has shown that there are no chains of length $ω_3$ increasing modulo finite in ${}^{ω_2}ω_2$. We improve this result to sets. That is, we show that there are no chains of length $ω_3$ in $[ω_2]^{\aleph_2}$ increasing modulo finite. This contrasts with results of Koszmider who has shown that there are, consistently, chains of length $ω_2$ increasing modulo finite in $[ω_1]^{\aleph_1}$ as well as in ${}^{ω_1}ω_1$. More generally, we study the depth of function spaces ${}^κμ$ quotiented by the ideal $[κ]^{< θ}$ where $θ< κ$ are infinite cardinals.

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arXiv2022

Exact Exponential Algorithms for Clustering Problems

In this paper we initiate a systematic study of exact algorithms for well-known clustering problems, namely $k$-Median and $k$-Means. In $k$-Median, the input consists of a set $X$ of $n$ points belonging to a metric space, and the task is to select a subset $C \subseteq X$ of $k$ points as centers, such that the sum of the distances of every point to its nearest center is minimized. In $k$-Means, the objective is to minimize the sum of squares of the distances instead. It is easy to design an algorithm running in time $\max_{k\leq n} {n \choose k} n^{O(1)} = O^*(2^n)$ ($O^*(\cdot)$ notation hides polynomial factors in $n$). We design first non-trivial exact algorithms for these problems. In particular, we obtain an $O^*((1.89)^n)$ time exact algorithm for $k$-Median that works for any value of $k$. Our algorithm is quite general in that it does not use any properties of the underlying (metric) space -- it does not even require the distances to satisfy the triangle inequality. In particular, the same algorithm also works for $k$-Means. We complement this result by showing that the running time of our algorithm is asymptotically optimal, up to the base of the exponent. That is, unless ETH fails, there is no algorithm for these problems running in time $2^{o(n)} \cdot n^{O(1)}$. Finally, we consider the "supplier" versions of these clustering problems, where, in addition to the set $X$ we are additionally given a set of $m$ candidate centers $F$, and objective is to find a subset of $k$ centers from $F$. The goal is still to minimize the $k$-Median/$k$-Means/$k$-Center objective. For these versions we give a $O(2^n (mn)^{O(1)})$ time algorithms using subset convolution. We complement this result by showing that, under the Set Cover Conjecture, the supplier versions of these problems do not admit an exact algorithm running in time $2^{(1-ε) n} (mn)^{O(1)}$.

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arXiv2022

A club guessing toolbox I

Club guessing principles were introduced by Shelah as a weakening of Jensen's diamond. Most spectacularly, they were used to prove Shelah's ZFC bound on the power of the first singular cardinal. These principles have found many other applications: in cardinal arithmetic and PCF theory; in the construction of combinatorial objects on uncountable cardinals such as Jonsson algebras, strong colourings, Souslin trees, and pathological graphs; to the non-existence of universals in model theory; to the non-existence of forcing axioms at higher uncountable cardinals; and many more. In this paper, the first part of a series, we survey various forms of club-guessing that have appeared in the literature, and then systematically study the various ways in which a club-guessing sequences can be improved, especially in the way the frequency of guessing is calibrated. We include an expository section intended for those unfamiliar with club-guessing and which can be read independently of the rest of the article.

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arXiv2021

Was Ulam right? I: Basic theory and subnormal ideals

We introduce various colouring principles which generalise the so-called "onto mapping principle" of Sierpinski to larger cardinals and general ideals. We prove that these principles capture the notion of an Ulam matrix and allow to characterise large cardinals, most notably weakly compact and ineffable cardinals. We also develop the basic theory of these colouring principles, connecting them to the classical negative square bracket partition relations, proving pumping-up theorems, and deciding various instances of theirs. We also demonstrate that our principles provide a uniform way of obtaining non-saturation results for ideals satisfying a property we call subnormality in contexts where Ulam matrices might not be available.

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arXiv2021

Non-Uniform $k$-Center and Greedy Clustering

In the Non-Uniform $k$-Center problem, a generalization of the famous $k$-center clustering problem, we want to cover the given set of points in a metric space by finding a placement of balls with specified radii. In $t$-NU$k$C Problem, we assume that the number of distinct radii is equal to $t$, and we are allowed to use $k_i$ balls of radius $r_i$, for $1 \le i \le t$. This problem was introduced by Chakrabarty et al. [ACM Trans. Alg. 16(4):46:1-46:19], who showed that a constant approximation for $t$-NU$k$C is not possible if $t$ is unbounded. On the other hand, they gave a bicriteria approximation that violates the number of allowed balls as well as the given radii by a constant factor. They also conjectured that a constant approximation for $t$-NU$k$C should be possible if $t$ is a fixed constant. Since then, there has been steady progress towards resolving this conjecture -- currently, a constant approximation for $3$-NU$k$C is known via the results of Chakrabarty and Negahbani [IPCO 2021], and Jia et al. [To appear in SOSA 2022]. We push the horizon by giving an $O(1)$-approximation for the Non-Uniform $k$-Center for $4$ distinct types of radii. Our result is obtained via a novel combination of tools and techniques from the $k$-center literature, which also demonstrates that the different generalizations of $k$-center involving non-uniform radii, and multiple coverage constraints (i.e., colorful $k$-center), are closely interlinked with each other. We hope that our ideas will contribute towards a deeper understanding of the $t$-NU$k$C problem, eventually bringing us closer to the resolution of the CGK conjecture.

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