arXiv subjects

Manas Jyoti Kashyop

Publications and source records attributed to Manas Jyoti Kashyop.

4 recordsLinked to original sources

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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arXiv2020

Trade-offs in dynamic coloring for bipartite and general graphs

We present trade-offs in the incremental and fully dynamic settings to maintian a proper coloring. For any fully dynamic $2$-coloring algorithm, the maximum of the update time, number of recolorings, and query time is $Ω(\log n)$. We present a deterministic fully dynamic $2$-coloring algorithm with $O(\log^2 n)$ amortized update time, $O(\log n)$ amortized query time, and one recoloring in the worst case. For any incremental $2$-coloring algorithm which explicitly maintains the color of every vertex after each update, the amortized update time and the amortized number of recolorings is $Ω(\log n)$. For such an algorithm, in the worst case the update time and the number of recolorings is $Ω(n)$. We then design a deterministic incremental $2$-coloring algorithm which explicitly maintains the color of every vertex after each update, with amortized $O(\log n)$ update time and amortized $O(\log n)$ many recolorings. Further, in the worst case the update time and the number of recolorings is $O(n)$. Further, we present a deterministic incremental $(1+2 \log n)$-coloring algorithm which explicitly maintains the color of every vertex after each update, with $O(α(n))$ amortized update time, at most one recoloring and $O(1)$ query time. We then show a deterministic incremental $2$-coloring algorithm which does not maintain color of every vertex after each update, with amortized $O(α(n))$ update time, amortized $O(α(n))$ recolorings, and amortized $O(α(n))$ query time. For general graphs and graphs of bounded arboricity $γ$ and maximum degree $Δ$ we present a deterministic $(Δ+1)$-coloring algorithm with $O(\sqrt{m})$ update time, $O(1)$ query time, and one recoloring. Finally, we show a deterministic $(Δ+1)$-coloring algorithm with amortized $O(γ+ \log{n})$ update time, $O(1)$ query time, and one recoloring.

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arXiv2020

Dynamic Data Structures for Interval Coloring

We consider the dynamic graph coloring problem restricted to the class of interval graphs. At each update step the algorithm is presented with an interval to be colored, or a previously colored interval to delete. The goal of the algorithm is to efficiently maintain a proper coloring of the intervals with as few colors as possible by an online algorithm. In the incremental model, each update step presents the algorithm with an interval to be colored. The problem is closely connected to the online vertex coloring problem of interval graphs for which the Kierstead-Trotter (KT) algorithm achieves the best possible competitive ratio. We first show that a sub-quadratic time direct implementation of the KT-algorithm is unlikely to exist conditioned on the correctness of the Online Boolean Matrix Vector multiplication conjecture due to Henzinger et al. \cite{DBLP:conf/stoc/HenzingerKNS15}. We then design an incremental algorithm that is subtly different from the KT-algorithm and uses at most $3 ω- 2$ colors, where $ω$ is the maximum clique in the interval graph associated with the set of intervals. Our incremental data structure maintains a proper coloring in amortized $O(\log n + Δ)$ update time where $n$ is the total number of intervals inserted and $Δ$ is the maximum degree of a vertex in the interval graph. We then consider the fully dynamic framework involving insertions and deletions. On each update, our aim is to maintain a $3 ω- 2$ coloring of the remaining set of intervals, where $ω$ is the maximum clique in the interval graph associated with the remaining set of intervals. Our fully dynamic algorithm supports insertion of an interval in $O(\log n + Δ\log ω)$ worst case update time and deletion of an interval in $O(Δ^2 \log n)$ worst case update time.

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arXiv2018

Fully dynamic $3/2$ approximate maximum cardinality matching in $O(\sqrt{n})$ update time

We present a randomized algorithm to maintain a maximal matching without 3 length augmenting paths in the fully dynamic setting. Consequently, we maintain a $3/2$ approximate maximum cardinality matching. Our algorithm takes expected amortized $O(\sqrt{n})$ time where $n$ is the number of vertices in the graph when the update sequence is generated by an oblivious adversary. Over any sequence of $t$ edge insertions and deletions presented by an oblivious adversary, the total update time of our algorithm is $O(t\sqrt{n})$ in expectation and $O(t\sqrt{n} + n \log n)$ with high probability. To the best of our knowledge, our algorithm is the first one to maintain an approximate matching in which all augmenting paths are of length at least $5$ in $o(\sqrt{m})$ update time.

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