Somnath Kundu - Academia.edu (original) (raw)

Papers by Somnath Kundu

In this thesis we discuss some novel concepts of stability in bargaining games, over a network se... more In this thesis we discuss some novel concepts of stability in bargaining games, over a network setting. So far, the studies on bargaining games were done as profit sharing problems, whose underlying combinatorial optimization problems are of packing type. In our work, we study bargaining games from a cost sharing perspective, where the underlying combinatorial optimization problems are covering type problems. Unlike previous studies, where bargaining processes are restricted to only two players, we study bargaining games over a more generic hypergraph setting, which allows any bargaining process to be formed among any number of players. In previous studies of bargaining games, the objects that are being negotiated are assumed to be uniform and only the outcomes of the negotiations are allowed to be different. However, in our study, we accommodate possibilities of non-uniform weights of the objects that are being negotiated, which is closer to any real life scenario. Finally we exten...

Lecture Notes in Computer Science, 2021

We study a primitive vehicle routing-type problem in which a fleet of n unit speed robots start f... more We study a primitive vehicle routing-type problem in which a fleet of n unit speed robots start from a point within a non-obtuse triangle Δ, where n ∈ {1, 2, 3}. The goal is to design robots' trajectories so as to visit all edges of the triangle with the smallest visitation time makespan. We begin our study by introducing a framework for subdividing Δ into regions with respect to the type of optimal trajectory that each point P admits, pertaining to the order that edges are visited and to how the cost of the minimum makespan Rn(P) is determined, for n ∈ {1, 2, 3}. These subdivisions are the starting points for our main result, which is to study makespan trade-offs with respect to the size of the fleet. In particular, we define Rn,m(Δ) = maxP ∈Δ Rn(P)/Rm(P), and we prove that, over all non-obtuse triangles Δ: (i) R1,3(Δ) ranges from √ 10 to 4, (ii) R2,3(Δ) ranges from √ 2 to 2, and (iii) R1,2(Δ) ranges from 5/2 to 3. In every case, we pinpoint the starting points within every triangle Δ that maximize Rn,m(Δ), as well as we identify the triangles that determine all infΔ Rn,m(Δ) and sup Δ Rn,m(Δ) over the set of non-obtuse triangles.

ArXiv, 2019

Searching for a line on the plane with nnn unit speed robots is a classic online problem that dat... more Searching for a line on the plane with nnn unit speed robots is a classic online problem that dates back to the 50's, and for which competitive ratio upper bounds are known for every ngeq1n\geq 1ngeq1. In this work we improve the best lower bound known for n=2n=2n=2 robots from 1.5993 to 3. Moreover we prove that the competitive ratio is at least sqrt3\sqrt{3}sqrt3 for n=3n=3n=3 robots, and at least 1/cos(pi/n)1/\cos(\pi/n)1/cos(pi/n) for ngeq4n\geq 4ngeq4 robots. Our lower bounds match the best upper bounds known for ngeq4n\geq 4ngeq4, hence resolving these cases. To the best of our knowledge, these are the first lower bounds proven for the cases ngeq3n\geq 3ngeq3 of this several decades old problem.

The Electronic Journal of Combinatorics, 2021

Let GGG be a graph in which each vertex initially has weight 1. In each step, the unit weight fro... more Let GGG be a graph in which each vertex initially has weight 1. In each step, the unit weight from a vertex uuu to a neighbouring vertex vvv can be moved, provided that the weight on vvv is at least as large as the weight on uuu. The unit acquisition number of GGG, denoted by au(G)a_u(G)au(G), is the minimum cardinality of the set of vertices with positive weight at the end of the process (over all acquisition protocols). In this paper, we investigate the Erdős-Rényi random graph process (mathcalG(n,m))m=0N(\mathcal{G}(n,m))_{m =0}^{N}(mathcalG(n,m))m=0N, where N=nchoose2N = {n \choose 2}N=nchoose2. We show that asymptotically almost surely au(mathcalG(n,m))=1a_u(\mathcal{G}(n,m)) = 1au(mathcalG(n,m))=1 right at the time step the random graph process creates a connected graph. Since trivially au(mathcalG(n,m))ge2a_u(\mathcal{G}(n,m)) \ge 2au(mathcalG(n,m))ge2 if the graphs is disconnected, the result holds in the strongest possible sense.

We introduce and characterize new stability notions in bargaining games over networks. Similar re... more We introduce and characterize new stability notions in bargaining games over networks. Similar results were already known for networks induced by simple graphs, and for bargaining games whose underlying combinatorial optimization problems are packing-type. Our results are threefold. First, we study bargaining games whose underlying combinatorial optimization problems are covering-type. Second, we extend the study of stability notions when the networks are induced by hypergraphs, and we further extend the results to fully weighted instances where the objects that are negotiated have non-uniform value among the agents. Third, we introduce and characterize new stability notions that are naturally derived by polyhedral combinatorics and duality theory for Linear Programming. Interestingly, these new stability notions admit intuitive interpretations touching on socially-aware agents. Overall, our contributions are meant to identify natural and desirable bargaining outcomes as well as to ...

In this thesis we discuss some novel concepts of stability in bargaining games, over a network se... more In this thesis we discuss some novel concepts of stability in bargaining games, over a network setting. So far, the studies on bargaining games were done as profit sharing problems, whose underlying combinatorial optimization problems are of packing type. In our work, we study bargaining games from a cost sharing perspective, where the underlying combinatorial optimization problems are covering type problems. Unlike previous studies, where bargaining processes are restricted to only two players, we study bargaining games over a more generic hypergraph setting, which allows any bargaining process to be formed among any number of players. In previous studies of bargaining games, the objects that are being negotiated are assumed to be uniform and only the outcomes of the negotiations are allowed to be different. However, in our study, we accommodate possibilities of non-uniform weights of the objects that are being negotiated, which is closer to any real life scenario. Finally we exten...

Lecture Notes in Computer Science, 2021

We study a primitive vehicle routing-type problem in which a fleet of n unit speed robots start f... more We study a primitive vehicle routing-type problem in which a fleet of n unit speed robots start from a point within a non-obtuse triangle Δ, where n ∈ {1, 2, 3}. The goal is to design robots' trajectories so as to visit all edges of the triangle with the smallest visitation time makespan. We begin our study by introducing a framework for subdividing Δ into regions with respect to the type of optimal trajectory that each point P admits, pertaining to the order that edges are visited and to how the cost of the minimum makespan Rn(P) is determined, for n ∈ {1, 2, 3}. These subdivisions are the starting points for our main result, which is to study makespan trade-offs with respect to the size of the fleet. In particular, we define Rn,m(Δ) = maxP ∈Δ Rn(P)/Rm(P), and we prove that, over all non-obtuse triangles Δ: (i) R1,3(Δ) ranges from √ 10 to 4, (ii) R2,3(Δ) ranges from √ 2 to 2, and (iii) R1,2(Δ) ranges from 5/2 to 3. In every case, we pinpoint the starting points within every triangle Δ that maximize Rn,m(Δ), as well as we identify the triangles that determine all infΔ Rn,m(Δ) and sup Δ Rn,m(Δ) over the set of non-obtuse triangles.

ArXiv, 2019

Searching for a line on the plane with nnn unit speed robots is a classic online problem that dat... more Searching for a line on the plane with nnn unit speed robots is a classic online problem that dates back to the 50's, and for which competitive ratio upper bounds are known for every ngeq1n\geq 1ngeq1. In this work we improve the best lower bound known for n=2n=2n=2 robots from 1.5993 to 3. Moreover we prove that the competitive ratio is at least sqrt3\sqrt{3}sqrt3 for n=3n=3n=3 robots, and at least 1/cos(pi/n)1/\cos(\pi/n)1/cos(pi/n) for ngeq4n\geq 4ngeq4 robots. Our lower bounds match the best upper bounds known for ngeq4n\geq 4ngeq4, hence resolving these cases. To the best of our knowledge, these are the first lower bounds proven for the cases ngeq3n\geq 3ngeq3 of this several decades old problem.

The Electronic Journal of Combinatorics, 2021

Let GGG be a graph in which each vertex initially has weight 1. In each step, the unit weight fro... more Let GGG be a graph in which each vertex initially has weight 1. In each step, the unit weight from a vertex uuu to a neighbouring vertex vvv can be moved, provided that the weight on vvv is at least as large as the weight on uuu. The unit acquisition number of GGG, denoted by au(G)a_u(G)au(G), is the minimum cardinality of the set of vertices with positive weight at the end of the process (over all acquisition protocols). In this paper, we investigate the Erdős-Rényi random graph process (mathcalG(n,m))m=0N(\mathcal{G}(n,m))_{m =0}^{N}(mathcalG(n,m))m=0N, where N=nchoose2N = {n \choose 2}N=nchoose2. We show that asymptotically almost surely au(mathcalG(n,m))=1a_u(\mathcal{G}(n,m)) = 1au(mathcalG(n,m))=1 right at the time step the random graph process creates a connected graph. Since trivially au(mathcalG(n,m))ge2a_u(\mathcal{G}(n,m)) \ge 2au(mathcalG(n,m))ge2 if the graphs is disconnected, the result holds in the strongest possible sense.

We introduce and characterize new stability notions in bargaining games over networks. Similar re... more We introduce and characterize new stability notions in bargaining games over networks. Similar results were already known for networks induced by simple graphs, and for bargaining games whose underlying combinatorial optimization problems are packing-type. Our results are threefold. First, we study bargaining games whose underlying combinatorial optimization problems are covering-type. Second, we extend the study of stability notions when the networks are induced by hypergraphs, and we further extend the results to fully weighted instances where the objects that are negotiated have non-uniform value among the agents. Third, we introduce and characterize new stability notions that are naturally derived by polyhedral combinatorics and duality theory for Linear Programming. Interestingly, these new stability notions admit intuitive interpretations touching on socially-aware agents. Overall, our contributions are meant to identify natural and desirable bargaining outcomes as well as to ...