arXiv subjects

Mursalin Habib

Publications and source records attributed to Mursalin Habib.

8 recordsLinked to original sources

arXiv2026

Ulam Rank Aggregation Is Hard to Approximate for Four Rankings

We study the approximability of rank aggregation under the Ulam metric. In the \emph{Ulam median} problem, the goal is to find a ranking (permutation) minimizing the sum of its Ulam distances to the input rankings, while in the \emph{Ulam center} problem, the objective is to minimize the maximum such distance. We prove that, for every $0<\varepsilon< 1/34$, it is $\mathrm{NP}$-hard to approximate either Ulam median or Ulam center within a factor of $35/34-\varepsilon$, even when the input consists of only four rankings. We further show that unless P = NP, neither problem admits a polynomial-time additive approximation scheme. Prior to our work, only the exact versions of both problems were known to be $\mathrm{NP}$-hard, and that too only when the number of input rankings is unbounded [Fischer et al., ESA'25 and Bachmaier et al., J. of Discrete Algorithms'15]. Furthermore, our inapproximability results are optimal in terms of the number of input rankings since for three inputs it is already known to be polynomial-time solvable [Chakraborty, Das, Krauthgamer, SODA'21]. En route, we introduce a new general framework for reducing Boolean constraint satisfaction problems (CSP) to the Ulam median with only four inputs. As a specific instantiation of the reduction framework, we obtain our hardness-of-approximation results. The corresponding hardness for the Ulam center follows from a reduction from the Ulam median.

cs.CC↗

arXiv2026

Ulam Median is NP-hard for Four Permutations

We show that computing a median under the Ulam distance is NP-hard even when the input consists of exactly four permutations. Previously, NP-hardness was known only for an unbounded number of input permutations (Fischer et al., ESA '25). Our result is tight, since an Ulam median of three permutations can be computed in polynomial time (Chakraborty--Das--Krauthgamer, SODA '21).

cs.CC↗

arXiv2026

An Optimal Algorithm for Binary Closest String

We revisit the Binary Closest String problem, which asks, given a set of binary strings $X \subseteq \{0, 1\}^n$, to compute a string minimizing the maximum Hamming distance to $X$. A long line of work has focused on parameterized algorithms with respect to the optimal distance $d$, yielding a sequence of improvements from $O^*(d^d)$ through $O^*(16^d)$, $O^*(9.513^d)$, $O^*(8^d)$, $O^*(6.731^d)$ to the current best-known running time of $O^*(5^d)$ [Chen, Ma, Wang; Algorithmica '16]. We present a faster randomized algorithm running in time $O^*(4^d)$. Our result matches a recent fine-grained lower bound [Abboud, Fischer, Goldenberg, Karthik C.S., Safier; ESA '23], and is therefore conditionally optimal. As an extra benefit, our algorithm is remarkably simple.

cs.DS↗

arXiv2025

Algorithmic Improvements to List Decoding of Folded Reed-Solomon Codes

Folded Reed-Solomon (FRS) codes are a well-studied family of codes, known for achieving list decoding capacity. In this work, we give improved deterministic and randomized algorithms for list decoding FRS codes of rate $R$ up to radius $1-R-\varepsilon$. We present a deterministic decoder that runs in near-linear time $\widetilde{O}_{\varepsilon}(n)$, improving upon the best-known runtime $n^{\Omega(1/\varepsilon)}$ for decoding FRS codes. Prior to our work, no capacity achieving code was known whose deterministic decoding could be done in time $\widetilde{O}_{\varepsilon}(n)$. We also present a randomized decoder that runs in fully polynomial time $\mathrm{poly}(1/\varepsilon) \cdot \widetilde{O}(n)$, improving the best-known runtime $\mathrm{exp}(1/\varepsilon)\cdot \widetilde{O}(n)$ for decoding FRS codes. Again, prior to our work, no capacity achieving code was known whose decoding time depended polynomially on $1/\varepsilon$. Our results are based on improved pruning procedures for finding the list of codewords inside a constant-dimensional affine subspace.

cs.IT↗

arXiv2025

Hardness of Median and Center in the Ulam Metric

The classical rank aggregation problem seeks to combine a set X of n permutations into a single representative "consensus" permutation. In this paper, we investigate two fundamental rank aggregation tasks under the well-studied Ulam metric: computing a median permutation (which minimizes the sum of Ulam distances to X) and computing a center permutation (which minimizes the maximum Ulam distance to X) in two settings. $\bullet$ Continuous Setting: In the continuous setting, the median/center is allowed to be any permutation. It is known that computing a center in the Ulam metric is NP-hard and we add to this by showing that computing a median is NP-hard as well via a simple reduction from the Max-Cut problem. While this result may not be unexpected, it had remained elusive until now and confirms a speculation by Chakraborty, Das, and Krauthgamer [SODA '21]. $\bullet$ Discrete Setting: In the discrete setting, the median/center must be a permutation from the input set. We fully resolve the fine-grained complexity of the discrete median and discrete center problems under the Ulam metric, proving that the naive $\widetilde{O}(n^2 L)$-time algorithm (where L is the length of the permutation) is conditionally optimal. This resolves an open problem raised by Abboud, Bateni, Cohen-Addad, Karthik C. S., and Seddighin [APPROX '23]. Our reductions are inspired by the known fine-grained lower bounds for similarity measures, but we face and overcome several new highly technical challenges.

cs.DS↗

arXiv2025

Constant Rate Isometric Embeddings of Hamming Metric into Edit Metric

A function $\varphi:\{0,1\}^n \to \{0,1\}^N$ is called an isometric embedding of the $n$-dimensional Hamming metric space to the $N$-dimensional edit metric space if, for all $x,y\in\{0,1\}^n$, the Hamming distance between $x$ and $y$ is equal to the edit distance between $\varphi(x)$ and $\varphi(y)$. The rate of such an embedding is defined as the ratio $n/N$. It is well known in the literature how to construct isometric embeddings with rate $\Omega(1/\log n)$. However, achieving even near-isometric embeddings with positive constant rate has remained elusive until now. In this paper, we present an isometric embedding with rate $1/8$ by discovering connections to synchronization strings, which were studied in the context of insertion-deletion codes (Haeupler-Shahrasbi [JACM'21]). At a technical level, we introduce a framework for obtaining high-rate isometric embeddings using a novel object called misaligners. As an immediate consequence of our constant-rate isometric embedding, we improve known conditional lower bounds for various optimization problems in the edit metric, now with optimal dependence on the dimension. We complement our results by showing that no isometric embedding $\varphi:\{0,1\}^n \to \{0,1\}^N$ can have rate greater than $15/32$ for all positive integers $n$. En route to proving this upper bound, we uncover fundamental structural properties necessary for every Hamming-to-edit isometric embedding. We also prove similar upper and lower bounds for embeddings over larger alphabets. Finally, we consider embeddings $\varphi:\Sigma_{\mathrm{in}}^n \to \Sigma_{\mathrm{out}}^N$ between different input and output alphabets, where the rate is given by $\frac{n\log|\Sigma_{\mathrm{in}}|}{N\log|\Sigma_{\mathrm{out}}|}$. In this setting, we show that the rate can be made arbitrarily close to $1$.

cs.DM↗

arXiv2024

Explicit Good Codes Approaching Distance 1 in Ulam Metric

The Ulam distance of two permutations on $[n]$ is $n$ minus the length of their longest common subsequence. In this paper, we show that for every $\varepsilon>0$, there exists some $\alpha>0$, and an infinite set $\Gamma\subseteq \mathbb{N}$, such that for all $n\in\Gamma$, there is an explicit set $C_n$ of $(n!)^{\alpha}$ many permutations on $[n]$, such that every pair of permutations in $C_n$ has pairwise Ulam distance at least $(1-\varepsilon)\cdot n$. Moreover, we can compute the $i^{\text{th}}$ permutation in $C_n$ in poly$(n)$ time and can also decode in poly$(n)$ time, a permutation $\pi$ on $[n]$ to its closest permutation $\pi^*$ in $C_n$, if the Ulam distance of $\pi$ and $\pi^*$ is less than $ \frac{(1-\varepsilon)\cdot n}{4} $. Previously, it was implicitly known by combining works of Goldreich and Wigderson [Israel Journal of Mathematics'23] and Farnoud, Skachek, and Milenkovic [IEEE Transactions on Information Theory'13] in a black-box manner, that it is possible to explicitly construct $(n!)^{\Omega(1)}$ many permutations on $[n]$, such that every pair of them have pairwise Ulam distance at least $\frac{n}{6}\cdot (1-\varepsilon)$, for any $\varepsilon>0$, and the bound on the distance can be improved to $\frac{n}{4}\cdot (1-\varepsilon)$ if the construction of Goldreich and Wigderson is directly analyzed in the Ulam metric.

cs.IT↗

arXiv2021

Counting and Verifying Abelian Border Arrays of Binary Words

In this note, we consider the problem of counting and verifying abelian border arrays of binary words. We show that the number of valid abelian border arrays of length $n$ is $2^{n-1}$. We also show that verifying whether a given array is the abelian border array of some binary word reduces to computing the abelian border array of a specific binary word. Thus, assuming the word-RAM model, we present an $O\left(\frac{n^2}{\log^2n}\right)$ time algorithm for the abelian border array verification problem.

cs.DS↗