arXiv subjects

Arun Kumar Das

Publications and source records attributed to Arun Kumar Das.

11 recordsLinked to original sources

arXiv2026

On the complexity of global Roman domination problem in graphs

A Roman dominating function of a graph $G=(V,E)$ is a labeling $f: V \rightarrow{} \{0 ,1, 2\}$ such that for each vertex $u \in V$ with $f(u) = 0$, there exists a vertex $v \in N(u)$ with $f(v) =2$. A Roman dominating function $f$ is a global Roman dominating function if it is a Roman dominating function for both $G$ and its complement $\overline{G}$. The weight of $f$ is the sum of $f(u)$ over all the vertices $u \in V$. The objective of Global Roman Domination problem is to find a global Roman dominating function with minimum weight. The objective of Global Roman Domination is to compute a global Roman dominating function of minimum weight. In this paper, we study the algorithmic aspects of Global Roman Domination problem on various graph classes and obtain the following results. 1. We prove that Roman domination and Global Roman Domination problems are not computationally equivalent by identifying graph classes on which one is linear-time solvable, while the other is NP-complete. 2. We show that Global Roman Domination problem is NP-complete on split graphs, thereby resolving an open question posed by Panda and Goyal [Discrete Applied Mathematics, 2023]. 3. We prove that Global Roman Domination problem is NP-complete on chordal bipartite graphs, planar bipartite graphs with maximum degree five and circle graphs. 4. On the positive side, we present a linear-time algorithm for Global Roman domination problem on cographs.

math.CO↗

arXiv2025

Exact Algorithms for Resource Reallocation Under Budgetary Constraints

Efficient resource (re-)allocation is a critical challenge in optimizing productivity and sustainability within multi-party supply networks. In this work, we introduce the \textsc{Red-Blue Reinforcement} (R-BR) problem, where a service provider under budgetary constraints must minimize client reallocations to reduce the required number of servers they should maintain by a specified amount. We conduct a systematic algorithmic study, providing three exact algorithms that scale well as the input grows (FPT), which could prove useful in practice. Our algorithms are efficient for topologies that model rural road networks (bounded distance to cluster), modern transportation systems (bounded modular-width), or have bounded clique-width, a parameter that is of great theoretical importance.

cs.DS↗

arXiv2025

Precoloring extension with demands on paths

Let $G$ be a graph with a set of precolored vertices, and let us be given an integer distance parameter $d$ and a set of integer demands $d_1,\dots,d_c$. The Distance Precoloring Extension with Demands (DPED) problem is to compute a vertex $c$-coloring of $G$ such that the following three conditions hold: (i) the resulting coloring respects the colors of the precolored vertices, (ii) the distance of two vertices of the same color is at least $d$, and (iii) the number of vertices colored by color $i$ is exactly $d_i$. This problem is motivated by a program scheduling in commercial broadcast channels with constraints on content repetition and placement, which leads precisely to the DPED problem for paths. In this paper, we study DPED on paths and present a polynomial time exact algorithm when precolored vertices are restricted to the two ends of the path and devise an approximation algorithm for DPED with an additive approximation factor polynomially bounded by $d$ and the number of precolored vertices. Then, we prove that the Distance Precoloring Extension problem on paths, a less restrictive version of DPED without the demand constraints, and then DPED itself, is NP-complete. Motivated by this result, we further study the parameterized complexity of DPED on paths. We establish that the DPED problem on paths is $W[1]$-hard when parameterized by the number of colors and the distance. On the positive side, we devise a fixed parameter tractable (FPT) algorithm for DPED on paths when the number of colors, the distance, and the number of precolored vertices are considered as the parameters. Moreover, we prove that Distance Precoloring Extension is FPT parameterized by the distance. As a byproduct, we also obtain several results for the Distance List Coloring problem on paths.

cs.DS↗

arXiv2025

On Triangular Separation of Bichromatic Point Sets

We address the problem of computing the minimum number of triangles to separate a set of blue points from a set of red points in $\mathbb{R}^2$. A set of triangles is a \emph{separator} of one color from the other if every point of that color is contained in some triangle and no triangle contains points of both colors. We consider several variants of the problem depending on whether the triangles are allowed to overlap or not and whether all points or just the blue points need to be contained in a triangle. We show that computing the minimum cardinality triangular separator of a set of blue points from a set of red points is NP-hard and further investigate worst case bounds on the minimum cardinality of triangular separators for a bichromatic set of $n$ points.

cs.CG↗

arXiv2025

Measurement-device-independent Schmidt number certification of all entangled states

Bipartite quantum states with higher Schmidt numbers have been shown to outperform those with lower Schmidt numbers in various quantum information processing tasks, highlighting the operational advantage of entanglement dimensionality. Certifying the Schmidt number of such states is therefore crucial for efficient resource utilisation. Ideally, this certification should rely as little as possible on the certifying devices to ensure robustness against their potential imperfections. Fully device-independent certification via Bell-nonlocal games offers strong robustness but suffers from fundamental limitations: it cannot certify the Schmidt number of all entangled states. We demonstrate that this insufficiency of Bell-nonlocal games is not limited to entangled states that do not exhibit Bell-nonlocality. Specifically, we prove the existence of Bell-nonlocal states whose Schmidt number cannot be certified by any Bell-nonlocal game when the parties are restricted to local projective measurements. To overcome this, we develop a measurement-device-independent certification method based on semiquantum nonlocal games, which assume trusted preparation devices but treat measurement devices as black boxes. We prove that for any bipartite state with Schmidt number exceeding $r$, there exists a semiquantum nonlocal game that can certify its Schmidt number. Finally, we provide an explicit construction of such a semiquantum nonlocal game based on an optimal Schmidt number witness operator.

quant-ph↗

arXiv2024

Multipacking in Euclidean Metric Space

Here we study the multipacking problems for geometric point sets with respect to their Euclidean distances. We consider a set of $n$ points $P$ and define $N_s[v]$ as the subset of $P$ that includes the $s$ nearest points of $v \in P$ and the point $v$ itself. We assume that the \emph{$s$-th neighbor} of each point is unique, for every $s \in \{0, 1, 2, \dots , n-1\}$. For a natural number $r \leq n-1$, an $r$-multipacking is a set $ M \subseteq P $ such that for each point $ v \in P $ and for every integer $ 1\leq s \leq r $, $|N_s[v]\cap M|\leq (s+1)/2$. The $r$-multipacking number of $ P $ is the maximum cardinality of an $r$-multipacking of $ P $ and is denoted by $ \MP_{r}(P) $. For $r=n-1$, an $r$-multipacking is called a multipacking and $r$-multipacking number is called as multipacking number. For $r=1 \text{ and } 2$, we study the problem of computing a maximum $r$-multipacking of the point sets in $\mathbb{R}^2$. We show that a maximum $1$-multipacking can be computed in polynomial time but computing a maximum $2$-multipacking is \textsc{NP-hard}. Further, we provide approximation and parameterized solutions to the $2$-multipacking problem.

cs.CG↗

arXiv2024

Classifying Measurement Incompatibility under Classical Pre- and Post-Processing Operations

Measurement incompatibility has proved to be an important resource for quantum information processing. In this work, we present an operational approach that leverages classical operations on the inputs (pre-processing) and outputs (post-processing) of measurement devices to explore different layers of incompatibility among the measurements performed by the device. We study classifications of measurement incompatibility with respect to these two types of classical operations, viz., post-processing or coarse-graining of measurement outcomes and pre-processing or convex-mixing of different measurements. We derive analytical criteria for determining when a set of projective measurements is fully incompatible with respect to coarse-graining or convex-mixing. Robustness against white noise for different layers of incompatibility for mutually unbiased bases is investigated. Furthermore, we study operational witnesses for incompatibility subject to these classical operations, using the input-output statistics of Bell-type experiments as well as experiments in the prepare-and-measure scenario.

quant-ph↗

arXiv2023

Measurement incompatibility and quantum advantage in communication

Measurement incompatibility stipulates the existence of quantum measurements that cannot be carried out simultaneously on single systems. We show that the set of input-output probabilities obtained from d-dimensional classical systems assisted with shared randomness is the same as the set obtained from d-dimensional quantum strategies restricted to compatible measurements with shared randomness in any communication scenario. Thus, measurement incompatibility is necessary for quantum advantage in communication, and any quantum advantage (with or without shared randomness) in communication acts as a witness to the incompatibility of the measurements at the receiver's end in a semi-device-independent way. We introduce a class of communication tasks - a general version of random access codes - to witness incompatibility of an arbitrary number of quantum measurements with arbitrary outcomes acting on d-dimensional systems, and provide generic upper bounds on the success metric of these tasks for compatible measurements. We identify all sets of three incompatible rank-one projective qubit measurements that random access codes can witness. Finally, we present the generic relationship between different sets of probability distributions - classical, quantum with or without shared randomness, and quantum restricted to compatible measurements with or without shared randomness - produced in communication scenarios.

quant-ph↗

arXiv2022

Complexity Results on Untangling Red-Blue Matchings

Given a matching between n red points and n blue points by line segments in the plane, we consider the problem of obtaining a crossing-free matching through flip operations that replace two crossing segments by two non-crossing ones. We first show that (i) it is NP-hard to alpha-approximate the shortest flip sequence, for any constant alpha. Second, we show that when the red points are colinear, (ii) given a matching, a flip sequence of length at most n(n-1)/2 always exists, and (iii) the number of flips in any sequence never exceeds (n(n-1)/2) (n+4)/6. Finally, we present (iv) a lower bounding flip sequence with roughly 1.5 n(n-1)/2 flips, which shows that the n(n-1)/2 flips attained in the convex case are not the maximum, and (v) a convex matching from which any flip sequence has roughly 1.5 n flips. The last four results, based on novel analyses, improve the constants of state-of-the-art bounds.

cs.CG↗

arXiv2022

Resource theoretic efficacy of the single copy of a two-qubit entangled state in a sequential network

How best one can recycle a given quantum resource, mitigating the various difficulties involved in its preparation and preservation, is of considerable importance for ensuring efficient applications in quantum technology. Here we demonstrate quantitatively the resource theoretic advantage of reusing a single copy of a two-qubit entangled state towards information processing. To this end, we consider a scenario of sequential entanglement detection of a given two-qubit state by multiple independent observers on each of the two spatially separated wings. In particular, we consider equal numbers of sequential observers on the two wings. We first determine the upper bound on the number of observers who can detect entanglement employing suitable entanglement witness operators. In terms of the parameters characterizing the entanglement consumed and the robustness of measurements, we then compare the above scenario with the corresponding scenario involving multiple pairs of entangled qubits shared among the two wings. This reveals a clear resource theoretic advantage of recycling a single copy of a two-qubit entangled state in the sequential network.

quant-ph↗

arXiv2021

Finding a Largest-Area Triangle in a Terrain in Near-Linear Time

A terrain is an $x$-monotone polygon whose lower boundary is a single line segment. We present an algorithm to find in a terrain a triangle of largest area in $O(n \log n)$ time, where $n$ is the number of vertices defining the terrain. The best previous algorithm for this problem has a running time of $O(n^2)$.

cs.CG↗