arXiv subjects

Partha Sarathi Mandal

Publications and source records attributed to Partha Sarathi Mandal.

At least 19 recordsLinked to original sources

arXiv2026

Approximating Energy-Constrained Drone Delivery Packing Problem for Last-Mile Logistics

Collaboration between drones and trucks in a last-mile delivery system offers numerous benefits and reduces many challenges of the traditional delivery system. Here, we introduce Drone-Delivery Packing Problem, where a set of parcels, associated with delivery intervals and cost, should be delivered to customer locations. The system comprises a set of identical drones and battery stations along truck's route, where drones swap depleted batteries or recharge them. The objective is to find assignment for all parcels by using the minimum number of drones, subject to the battery budget and compatibility of each drone's assignment. We consider three variants of the problem, based on conflicting characteristics and existence of battery service stations. All are NP-hard, and we have proposed approximation algorithms for each. When there are no battery stations, we propose a constant factor approximation algorithm using first fit decreasing bin packing algorithm. When the intervals are non-conflicting, we design a $(2+ ψ)$-approximation algorithm. In the presence of both battery stations and conflicting intervals, we present a $(4+ψ)$-approximation algorithm. The algorithm is later modified into a $(3+ψ)$-approximation algorithm when the battery service stations act as swapping stations. Here $ψ$ is a function of battery budget, minimum and maximum cost of the deliveries. Finally, we validate our results and compare the performance with the optimum on different instances.

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arXiv2026

Black Hole Search: Dynamics, Distribution, and Emergence

A black hole is a malicious node in a graph that destroys resources entering into it without leaving any trace. The problem of Black Hole Search (BHS) using mobile agents requires that at least one agent survives and terminates after locating the black hole. Recently, this problem has been studied on 1-bounded 1-interval connected dynamic graphs \cite{BHS_gen}, where there is a footprint graph, and at most one edge can disappear from the footprint in a round, provided that the graph remains connected. In this setting, the authors in \cite{BHS_gen} proposed an algorithm that solves the BHS problem when all agents start from a single node (rooted initial configuration). They also proved that at least $2δ_{BH} + 1$ agents are necessary to solve the problem when agents are initially placed arbitrarily across the nodes of the graph (scattered initial configuration), where $δ_{BH}$ denotes the degree of the black hole. In this work, we present an algorithm that solves the BHS problem using $2δ_{BH} + 17$ initially scattered agents. Our result matches asymptotically with the rooted algorithm of \cite{BHS_gen} under the same model assumptions. Further, we study the Eventual Black Hole Search (\textsc{Ebhs}) problem, in which the black hole may appear at any node and at any time during the execution of the algorithm, destroying all agents located on that node at the time of its appearance. However, the black hole cannot emerge at the home base in round~0, where the home base is the node at which all agents are initially co-located. Once the black hole appears, it remains active at that node for the rest of the execution. This problem has been studied on static rings~\cite{Bonnet25}; here we extend it to arbitrary static graphs and provide a solution using four agents. Moreover, it does not require any knowledge of global parameters or additional model assumptions.

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arXiv2025

Monotone Decontamination of Arbitrary Dynamic Graphs with Mobile Agents

Network decontamination is a well-known problem, in which the aim of the mobile agents should be to decontaminate the network (i.e., both nodes and edges). This problem comes with an added constraint, i.e., of \emph{monotonicity}, in which whenever a node or an edge is decontaminated, it must not get recontaminated. Hence, the name comes \emph{monotone decontamination}. This problem has been relatively explored in static graphs, but nothing is known yet in dynamic graphs. We, in this paper, study the \emph{monotone decontamination} problem in arbitrary dynamic graphs. We designed two models of dynamicity, based on the time within which a disappeared edge must reappear. In each of these two models, we proposed lower bounds as well as upper bounds on the number of agents, required to fully decontaminate the underlying dynamic graph, monotonically. Our results also highlight the difficulties faced due to the sudden disappearance or reappearance of edges. Our aim in this paper has been to primarily optimize the number of agents required to solve monotone decontamination in these dynamic networks.

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arXiv2025

Separation of Unconscious Robots with Obstructed Visibility

We study a recently introduced \textit{unconscious} mobile robot model, where each robot is associated with a \textit{color}, which is visible to other robots but not to itself. The robots are autonomous, anonymous, oblivious and silent, operating in the Euclidean plane under the conventional \textit{Look-Compute-Move} cycle. A primary task in this model is the \textit{separation problem}, where unconscious robots sharing the same color must separate from others, forming recognizable geometric shapes such as circles, points, or lines. All prior works model the robots as \textit{transparent}, enabling each to know the positions and colors of all other robots. In contrast, we model the robots as \textit{opaque}, where a robot can obstruct the visibility of two other robots, if it lies on the line segment between them. Under this obstructed visibility, we consider a variant of the separation problem in which robots, starting from any arbitrary initial configuration, are required to separate into concentric semicircles. We present a collision-free algorithm that solves the separation problem under a semi-synchronous scheduler in $O(n)$ epochs, where $n$ is the number of robots. The robots agree on one coordinate axis but have no knowledge of $n$.

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arXiv2025

Black Hole Search in Dynamic Graphs

A black hole is considered to be a dangerous node present in a graph that disposes of any resources that enter that node. Therefore, it is essential to find such a node in the graph. Let a group of agents be present on a graph $G$. The Black Hole Search (BHS) problem aims for at least one agent to survive and terminate after {finding} the black hole. This problem is already studied for specific dynamic graph classes such as rings, cactuses, and tori {where finding the black hole means at least one agent needs to survive and terminate after knowing at least one edge associated with the black hole. In this work, we investigate the problem of BHS for general graphs.} In the dynamic graph, adversary may remove edges at each round keeping the graph connected. We consider two cases: (a) at any round at most one edge can be removed (b) at any round at most $f$ edges can be removed. For both scenarios, we study the problem when the agents start from a rooted initial configuration. We consider each agent has $O(\log n)$ memory and each node has $O(\log n)$ storage. For case (a), we present an algorithm with $9$ agents that solves the problem of BHS in $O(|E|^2)$ time where $|E|$ is the number of edges and $δ_v$ is the degree of the node $v$ in $G$. We show it is impossible to solve for $2δ_{BH}$ many agents starting from an arbitrary configuration where $δ_{BH}$ is the degree of the black hole in $G$. We also provide another improved algorithm that uses $6$ agents from a rooted initial configuration to solve the problem of BHS. For case (b), we provide an algorithm using $6f$ agents to solve the problem of BHS, albeit taking exponential time. We also provide an impossibility result for $2f+1$ agents starting from a rooted initial configuration. This result holds even if unlimited storage is available on each node and the agents have infinite memory.

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arXiv2025

Asynchronous Gathering of Opaque Robots with Mobility Faults

We consider the fundamental benchmarking problem of gathering in an $(N,f)$-fault system consisting of $N$ robots, of which at most $f$ might fail at any execution, under asynchrony. Two seminal results established impossibility of a solution in the oblivious robot (OBLOT) model in a $(2,0)$-fault system under semi-synchrony and in a $(3,1)$-Byzantine fault system under asynchrony. Recently, a breakthrough result circumvented the first impossibility result by giving a deterministic algorithm in a $(2,0)$-fault system under asynchrony in the luminous robot (LUMI) model using 2-colored lights. However, a breakthrough result established impossibility of gathering in a $(2,1)$-crash system in the LUMI model under semi-synchrony. In this paper, we consider a {\em mobility fault} model in which a robot crash only impacts it mobility but not the operation of the light. We establish four results under asynchrony in LUMI with the mobility fault model. We show that it is impossible to solve gathering in a $(2,1)$-mobility fault system using 2-colored lights, and then give a solution using 3-colored lights, which is optimal w.r.t. the number of colors. We then consider an $(N,f)$-mobility fault system, $f<N$, both $N,f$ not known, and give two deterministic algorithms that exhibit a nice time-color trade-off: The first with time $O(N)$ using 7-colored lights and the second with time $O(\max\{\ell,f\})$ using 26-colored lights, where $\ell< N$ is the number of distinct convex layers of robot positions in the initial configuration. Interestingly, for $l, f = O(1)$, our result is optimal. Our algorithms for an $(N,f)$-mobility fault system are the first to be analysed time complexity, can withstand obstructed visibility (opaque robot model) and asynchronous scheduling.

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arXiv2025

Graph Traversal via Connected Mobile Agents

This paper considers the Hamiltonian walk problem in the multi-agent coordination framework, referred to as $k$-agents Hamiltonian walk problem ($k$-HWP). In this problem, a set of $k$ connected agents collectively compute a spanning walk of a given undirected graph in the minimum steps. At each step, the agents are at $k$ distinct vertices and the induced subgraph made by the occupied vertices remains connected. In the next consecutive steps, each agent may remain stationary or move to one of its neighbours.To the best of our knowledge, this problem has not been previously explored in the context of multi-agent systems with connectivity. As a generalization of the well-known Hamiltonian walk problem (when $k=1$), $k$-HWP is NP-hard. We propose a $(3-\frac{1}{21})$-approximation algorithm for 2-HWP on arbitrary graphs. For the tree, we define a restricted version of the problem and present an optimal algorithm for arbitrary values of $k$. Finally, we formalize the problem for $k$-uniform hypergraphs and present a $2(1+\ln k)$-approximation algorithm. This result is also adapted to design an approximation algorithm for $k$-HWP on general graphs when $k = O(1)$.

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arXiv2025

Time-optimal Asynchronous Minimal Vertex Covering by Myopic Robots

In a connected graph with an autonomous robot swarm with limited visibility, it is natural to ask whether the robots can be deployed to certain vertices satisfying a given property using only local knowledge. This paper affirmatively answers the question with a set of \emph{myopic} (finite visibility range) luminous robots with the aim of \emph{filling a minimal vertex cover} (MVC) of a given graph $G = (V, E)$. The graph has special vertices, called \emph{doors}, through which robots enter sequentially. Starting from the doors, the goal of the robots is to settle on a set of vertices that forms a minimal vertex cover of $G$ under the asynchronous ($\mathcal{ASYNC}$) scheduler. We are also interested in achieving the \emph{minimum vertex cover} (MinVC, which is NP-hard \cite{Karp1972} for general graphs) for a specific graph class using the myopic robots. We establish lower bounds on the visibility range for the robots and on the time complexity (which is $Ω(|E|)$). We present two algorithms for trees: one for single door, which is both time and memory-optimal, and the other for multiple doors, which is memory-optimal and achieves time-optimality when the number of doors is a constant. Interestingly, our technique achieves MinVC on trees with a single door. We then move to the general graph, where we present two algorithms, one for the single door and the other for the multiple doors with an extra memory of $O(\log Δ)$ for the robots, where $Δ$ is the maximum degree of $G$. All our algorithms run in $O(|E|)$ epochs.

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arXiv2025

Perpetual exploration in anonymous synchronous networks with a Byzantine black hole

In this paper, we investigate: ``How can a group of initially co-located mobile agents perpetually explore an unknown graph, when one stationary node occasionally behaves maliciously, under an adversary's control?'' We call this node a ``Byzantine black hole (BBH)'' and at any given round it may choose to destroy all visiting agents, or none. This subtle power can drastically undermine classical exploration strategies designed for an always active black hole. We study this perpetual exploration problem in the presence of at most one BBH, without initial knowledge of the network size. Since the underlying graph may be 1-connected, perpetual exploration of the entire graph may be infeasible. We thus define two variants: \pbmPerpExpl\ and \pbmPerpExplHome. In the former, the agents are tasked to perform perpetual exploration of at least one component, obtained after the exclusion of the BBH. In the latter, the agents are tasked to perform perpetual exploration of the component which contains the \emph{home} node, where agents are initially co-located. Naturally, \pbmPerpExplHome\ is a special case of \pbmPerpExpl. Agents operate under a synchronous scheduler and communicate in a face-to-face model. Our goal is to determine the minimum number of agents necessary and sufficient to solve these problems. In acyclic networks, we obtain optimal algorithms that solve \pbmPerpExpl\ with $4$ agents, and \pbmPerpExplHome\ with $6$ agents in trees. The lower bounds hold even in path graphs. In general graphs, we give a non-trivial lower bound of $2Δ-1$ agents for \pbmPerpExpl, and an upper bound of $3Δ+3$ agents for \pbmPerpExplHome. To our knowledge, this is the first study of a black-hole variant in arbitrary networks without initial topological knowledge.

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arXiv2024

Perpetual Exploration of a Ring in Presence of Byzantine Black Hole

Perpetual exploration is a fundamental problem in the domain of mobile agents, where an agent needs to visit each node infinitely often. This issue has received lot of attention, mainly for ring topologies, presence of black holes adds more complexity. A black hole can destroy any incoming agent without any observable trace. In \cite{BampasImprovedPeriodicDataRetrieval,KralovivcPeriodicDataRetrievalFirst}, the authors considered this problem in the context of \textit{ Periodic data retrieval}. They introduced a variant of black hole called gray hole (where the adversary chooses whether to destroy an agent or let it pass) among others and showed that 4 asynchronous and co-located agents are essential to solve this problem (hence perpetual exploration) in presence of such a gray hole if each node of the ring has a whiteboard. This paper investigates the exploration of a ring in presence of a ``byzantine black hole''. In addition to the capabilities of a gray hole, in this variant, the adversary chooses whether to erase any previously stored information on that node. Previously, one particular initial scenario (i.e., agents are co-located) and one particular communication model (i.e., whiteboard) are investigated. Now, there can be other initial scenarios where all agents may not be co-located. Also, there are many weaker models of communications (i.e., Face-to-Face, Pebble) where this problem is yet to be investigated. The agents are synchronous. The main results focus on minimizing the agent number while ensuring that perpetual exploration is achieved even in presence of such a node under various communication models and starting positions. Further, we achieved a better upper and lower bound result (i.e., 3 agents) for this problem (where the malicious node is a generalized version of a gray hole), by trading-off scheduler capability, for co-located and in presence of a whiteboard.

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arXiv2024

Uniform Partitioning of a Bounded Region using Opaque ASYNC Luminous Mobile Robots

We are given $N$ autonomous mobile robots inside a bounded region. The robots are opaque which means that three collinear robots are unable to see each other as one of the robots acts as an obstruction for the other two. They operate in classical \emph{Look-Compute-Move} (LCM) activation cycles. Moreover, the robots are oblivious except for a persistent light (which is why they are called \emph{Luminous robots}) that can determine a color from a fixed color set. Obliviousness does not allow the robots to remember any information from past activation cycles. The Uniform Partitioning problem requires the robots to partition the whole region into sub-regions of equal area, each of which contains exactly one robot. Due to application-oriented motivation, we, in this paper consider the region to be well-known geometric shapes such as rectangle, square and circle. We investigate the problem in \emph{asynchronous} setting where there is no notion of common time and any robot gets activated at any time with a fair assumption that every robot needs to get activated infinitely often. To the best of our knowledge, this is the first attempt to study the Uniform Partitioning problem using oblivious opaque robots working under asynchronous settings. We propose three algorithms considering three different regions: rectangle, square and circle. The algorithms proposed for rectangular and square regions run in $O(N)$ epochs whereas the algorithm for circular regions runs in $O(N^2)$ epochs, where an epoch is the smallest unit of time in which all robots are activated at least once and execute their LCM cycles. The algorithms for the rectangular, square and circular regions require $2$ (which is optimal), $5$ and $8$ colors, respectively.

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arXiv2024

Online Drone Scheduling for Last-mile Delivery

Delivering a parcel from the distribution hub to the customer's doorstep is called the \textit{last-mile delivery} step in delivery logistics. In this paper, we study a hybrid {\it truck-drones} model for the last-mile delivery step, in which a truck moves on a predefined path carrying parcels and drones deliver the parcels. We define the \textsc{online drone scheduling} problem, where the truck moves in a predefined path, and the customer's requests appear online during the truck's movement. The objective is to schedule a drone associated with every request to minimize the number of drones used subject to the battery budget of the drones and compatibility of the schedules. We propose a 3-competitive deterministic algorithm using the next-fit strategy and 2.7-competitive algorithms using the first-fit strategy for the problem with $O(\log n)$ worst-case time complexity per request, where $n$ is the maximum number of active requests at any time. We also introduce \textsc{online variable-size drone scheduling} problem (OVDS). Here, we know all the customer's requests in advance; however, the drones with different battery capacities appear online. The objective is to schedule customers' requests for drones to minimize the number of drones used. We propose a $(2α+ 1)$-competitive algorithm for the OVDS problem with total running time $O(n \log n)$ for $n$ customer requests, where $α$ is the ratio of the maximum battery capacity to the minimum battery capacity of the drones. Finally, we address how to generate intervals corresponding to each customer request when there are discrete stopping points on the truck's route, from where the drone can fly and meet with the truck.

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arXiv2024

Black Hole Search in Dynamic Tori

We investigate the black hole search problem by a set of mobile agents in a dynamic torus. Black hole is defined to be a dangerous stationary node which has the capability to destroy any number of incoming agents without leaving any trace of its existence. A torus of size $n\times m$ ($3\leq n \leq m$) is a collection of $n$ row rings and $m$ column rings, and the dynamicity is such that each ring is considered to be 1-interval connected, i.e., in other words at most one edge can be missing from each ring at any round. The parameters which define the efficiency of any black hole search algorithm are: the number of agents and the number of rounds (or \textit{time}) for termination. We consider two initial configurations of mobile agents: first, the agents are co-located and second, the agents are scattered. In each case, we establish lower and upper bounds on the number of agents and on the amount of time required to solve the black hole search problem.

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arXiv2023

Black Hole Search in Dynamic Cactus Graph

We study the problem of black hole search by a set of mobile agents, where the underlying graph is a dynamic cactus. A black hole is a dangerous vertex in the graph that eliminates any visiting agent without leaving any trace behind. Key parameters that dictate the complexity of finding the black hole include: the number of agents required (termed as \textit{size}), the number of moves performed by the agents in order to determine the black hole location (termed as \textit{move}) and the \textit{time} (or round) taken to terminate. This problem has already been studied where the underlying graph is a dynamic ring \cite{di2021black}. In this paper, we extend the same problem to a dynamic cactus. We introduce two categories of dynamicity, but still the underlying graph needs to be connected: first, we examine the scenario where, at most, one dynamic edge can disappear or reappear at any round. Secondly, we consider the problem for at most $k$ dynamic edges. In both scenarios, we establish lower and upper bounds for the necessary number of agents, moves and rounds.

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arXiv2023

Pebble guided Treasure Hunt in Plane

We study the problem of treasure hunt in a Euclidean plane by a mobile agent with the guidance of pebbles. The initial position of the agent and position of the treasure are modeled as special points in the Euclidean plane. The treasure is situated at a distance at most $D>0$ from the initial position of the agent. The agent has a perfect compass, but an adversary controls the speed of the agent. Hence, the agent can not measure how much distance it traveled for a given time. The agent can find the treasure only when it reaches the exact position of the treasure. The cost of the treasure hunt is defined as the total distance traveled by the agent before it finds the treasure. The agent has no prior knowledge of the position of the treasure or the value of $D$. An Oracle, which knows the treasure's position and the agent's initial location, places some pebbles to guide the agent towards the treasure. Once decided to move along some specified angular direction, the agent can decide to change its direction only when it encounters a pebble or a special point. We ask the following central question in this paper: ``For given $k \ge 0$, What is cheapest treasure hunt algorithm if at most $k$ pebbles are placed by the Oracle?" We show that for $k=1$, there does not exist any treasure hunt algorithm that finds the treasure with finite cost. We show the existence of an algorithm with cost $O(D)$ for $k=2$. For $k>8$ we have designed an algorithm that uses $k$ many pebbles to find the treasure with cost $O(k^{2}) + D(\sinθ' + \cosθ')$, where $θ'=\fracπ{2^{k-8}}$. The second result shows the existence of an algorithm with cost arbitrarily close to $D$ for sufficiently large values of $D$.

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arXiv2022

Approximation Algorithms for Drone Delivery Scheduling Problem

The coordination among drones and ground vehicles for last-mile delivery has gained significant interest in recent years. In this paper, we study \textit{multiple drone delivery scheduling problem(MDSP) \cite{Betti_ICDCN22} for last-mile delivery, where we have a set of drones with an identical battery budget and a set of delivery locations, along with reward or profit for delivery, cost and delivery time intervals. The objective of the MDSP is to find a collection of conflict-free schedules for each drone such that the total profit for delivery is maximum subject to the battery constraint of the drones. Here we propose a fully polynomial time approximation scheme (FPTAS) for the single drone delivery scheduling problem (SDSP) and a $\frac{1}{4}$-approximation algorithm for MDSP with a constraint on the number of drones.

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arXiv2022

Filling MIS Vertices by Myopic Luminous Robots

We present the problem of finding a maximal independent set (MIS) (named as \emph{MIS Filling problem}) of an arbitrary connected graph having $n$ vertices with luminous myopic mobile robots. The robots enter the graph one after another from a particular vertex called the \emph{Door} and disperse along the edges of the graph without collision to occupy vertices such that the set of vertices occupied by the robots is a maximal independent set. We assume the robots have knowledge only about the maximum degree of the graph, denoted by $Δ$. In this paper, we explore two versions of the problem: the solution to the first version, named as \emph{MIS Filling with Single Door}, works under an asynchronous scheduler using robots with 3 hops of visibility range, $Δ+ 6$ number of colors and $O(\log Δ)$ bits of persistent storage. The time complexity is measured in terms of epochs and it can be solved in $O(n^2)$ epochs. An epoch is the smallest time interval in which each participating robot gets activated and executes the algorithm at least once. For the second version with $k~ ( > 1)$ \textit{Doors}, named as \emph{MIS Filling with Multiple Doors}, the solution works under a semi-synchronous scheduler using robots with 5 hops of visibility range, $Δ+ k + 6$ number of colors and $O(\log (Δ+ k))$ bits of persistent storage. The problem with multiple Doors can be solved in $O(n^2)$ epochs.

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arXiv2022

Treasure Hunt in Graph using Pebbles

In this paper, we study the treasure hunt problem in a graph by a mobile agent. The nodes in the graph $G=(V,E)$ are anonymous and the edges incident to a vertex $v\in V$ whose degree is $deg(v)$ are labeled arbitrarily as $0,1,\ldots, deg(v)-1$. At a node $t$ in $G$ a stationary object, called {\it treasure} is located. The mobile agent that is initially located at a node $s$ in $G$, the starting point of the agent, must find the treasure by reaching the node $t$. The distance from $s$ to $t$ is $D$. The {\it time} required to find the treasure is the total number of edges the agent visits before it finds the treasure. The agent does not have any prior knowledge about the graph or the position of the treasure. An oracle, that knows the graph, the initial position of the agent, and the position of the treasure, places some pebbles on the nodes, at most one per node, of the graph to guide the agent towards the treasure. This paper aims to study the trade-off between the number of pebbles provided and the time required to find the treasure. To be specific, we aim to answer the following question. ``What is the minimum time for treasure hunt in a graph with maximum degree $Δ$ and diameter $D$ if $k$ pebbles are placed? " We answer the above question when $k<D$ or $k=cD$ for some positive integer $c$. We design efficient algorithms for the agent for different values of $k$. We also propose an almost matching lower bound result for $k<D$.

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