arXiv subjects

Sanjiv Kapoor

Publications and source records attributed to Sanjiv Kapoor.

11 recordsLinked to original sources

arXiv2026

Equilibrium and Selfish Behavior in Network Contagion

In this paper we consider non-atomic games in populations that are provided with a choice of preventive policies to act against a contagion spreading amongst interacting populations, be it biological organisms or connected computing devices. The spreading model of the contagion is the standard SIR model. Each participant of the population has a choice from amongst a set of precautionary policies with each policy presenting a payoff or utility, which we assume is the same within each group, the risk being the possibility of infection. The policy groups interact with each other. We also define a network model to model interactions between different population sets. The population sets reside at nodes of the network and follow policies available at that node. We define game-theoretic models and study the inefficiency of allowing for individual decision making, as opposed to centralized control. We study the computational aspects as well.

cs.GT↗

arXiv2019

Approximate Euclidean shortest paths in polygonal domains

Given a set $\mathcal{P}$ of $h$ pairwise disjoint simple polygonal obstacles in $\mathbb{R}^2$ defined with $n$ vertices, we compute a sketch $Ω$ of $\mathcal{P}$ whose size is independent of $n$, depending only on $h$ and the input parameter $ε$. We utilize $Ω$ to compute a $(1+ε)$-approximate geodesic shortest path between the two given points in $O(n + h((\lg{n}) + (\lg{h})^{1+δ} + (\frac{1}ε\lg{\frac{h}ε})))$ time. Here, $ε$ is a user parameter, and $δ$ is a small positive constant (resulting from the time for triangulating the free space of $\cal P$ using the algorithm in \cite{journals/ijcga/Bar-YehudaC94}). Moreover, we devise a $(2+ε)$-approximation algorithm to answer two-point Euclidean distance queries for the case of convex polygonal obstacles.

cs.CG↗

arXiv2019

Price of Anarchy with Heterogeneous Latency Functions

In this paper we consider the price of anarchy (PoA) in multi-commodity flows where the latency or delay function on an edge has a heterogeneous dependency on the flow commodities, i.e. when the delay on each link is dependent on the flow of individual commodities, rather than on the aggregate flow. An application of this study is the performance analysis of a network with differentiated traffic that may arise when traffic is prioritized according to some type classification. This study has implications in the debate on net-neutrality. We provide price of anarchy bounds for networks with $k$ (types of) commodities where each link is associated with heterogeneous polynomial delays, i.e. commodity $i$ on edge $e$ faces delay specified by $g_{i1}(e)f^θ_1(e) + g_{i2}(e)f^θ_2(e) + \ldots + g_{ik}(e)f^θ_k(e) + c_i(e), $ where $f_i(e)$ is the flow of the $i$th commodity through edge $e$, $θ\in {\cal N}$, $g_{i1}(e), g_{i2}(e), \ldots, g_{ik}(e)$ and $c_i(e)$ are nonnegative constants. We consider both atomic and non-atomic flows. For networks with decomposable delay functions where the delay induced by a particular commodity is the same, i.e. delays on edge $e$ are defined by $a_1(e)f_1^θ(e) + a_2(e)f_2^θ(e) + \ldots + c(e)$ where $\forall j , \forall e: g_{1j}(e) = g_{2j}(e) = \ldots = a_j(e)$, we show an improved bound on the price of anarchy. Further, we show bounds on the price of anarchy for uniform latency functions where each edge of the network has the same delay function.

cs.GT↗

arXiv2016

A Polynomial Time Algorithm to Compute an Approximate Weighted Shortest Path

We devise a polynomial-time approximation scheme for the classical geometric problem of finding an approximate short path amid weighted regions. In this problem, a triangulated region P comprising of n vertices, a positive weight associated with each triangle, and two points s and t that belong to P are given as the input. The objective is to find a path whose cost is at most (1+epsilon)OPT where OPT is the cost of an optimal path between s and t. Our algorithm initiates a discretized-Dijkstra wavefront from source s and progresses the wavefront till it strikes t. This result is about a cubic factor (in n) improvement over the Mitchell and Papadimitriou '91 result, which is the only known polynomial time algorithm for this problem to date. Further, with polynomial time preprocessing of P, a set of data structures are computed which allow answering approximate weighted shortest path queries in polynomial time.

cs.CG↗

arXiv2014

ANN queries: covering Voronoi diagram with hyperboxes

Given a set $S$ of $n$ points in $d$-dimensional Euclidean metric space $X$ and a small positive real number $ε$, we present an algorithm to preprocess $S$ and answer queries that require finding a set $S' \subseteq S$ of $ε$-approximate nearest neighbors (ANNs) to a given query point $q \in X$. The following are the characteristics of points belonging to set $S'$: - $\forall s \in S'$, $\exists$ a point $p \in X$ such that $|pq| \le ε$ and the nearest neighbor of $p$ is $s$, and - $\exists$ a $s' \in S'$ such that $s'$ is a nearest neighbor of $q$. During the preprocessing phase, from the Voronoi diagram of $S$ we construct a set of box trees of size $O(4^d\frac{V}δ(\fracπε)^{d-1})$ which facilitate in querying ANNs of any input query point in $O(\frac{1}{d}lg \frac{V}δ + (\fracπε)^{d-1})$ time. Here $δ$ equals to $(\fracε{2\sqrt{d}})^d$, and $V$ is the volume of a large bounding box that contains all the points of set $S$. The average case cardinality of $S'$ is shown to rely on $S$ and $ε$.

cs.CG↗

arXiv2013

Approximating Quadratic 0-1 Programming via SOCP

We consider the problem of approximating Quadratic O-1 Integer Programs with bounded number of constraints and non-negative constraint matrix entries, which we term as PIQP. We describe and analyze a randomized algorithm based on a program with hyperbolic constraints (a Second-Order Cone Programming -SOCP- formulation) that achieves an approximation ratio of $O(a_{max} \frac{n}{β(n)})$, where $a_{max}$ is the maximum size of an entry in the constraint matrix and $β(n) \leq \min_i{W_i} $, where $W_i$ are the constant terms that define the constraint inequalities. We note that by appropriately choosing $β(n)$ the randomized algorithm, when combined with other algorithms that achieve good approximations for smaller values of $ W_i$, allows better algorithms for the complete range of $W_i$. This, together with a greedy algorithm, provides a $O^*(a_{max} n^{1/2} )$ factor approximation, where $O^*$ hides logarithmic terms. Our solution is achieved by a randomization of the optimal solution to the relaxed version of the hyperbolic program. We show that this solution provides the approximation bounds using concentration bounds provided by Chernoff-Hoeffding and Kim-Vu.

cs.CC↗

arXiv2013

Auction Algorithm for Production Models

We show an auction-based algorithm to compute market equilibrium prices in a production model, where consumers purchase items under separable nonlinear utility concave functions which satisfy W.G.S(Weak Gross Substitutes); producers produce items with multiple linear production constraints. Our algorithm differs from previous approaches in that the prices are allowed to both increase and decrease to handle changes in the production. This provides a t^atonnement style algorithm which converges and provides a PTAS. The algorithm can also be extended to arbitrary convex production regions and the Arrow-Debreu model. The convergence is dependent on the behavior of the marginal utility of the concave function.

cs.GT↗

arXiv2013

Efficient Construction of Spanners in $d$-Dimensions

In this paper we consider the problem of efficiently constructing $k$-vertex fault-tolerant geometric $t$-spanners in $\dspace$ (for $k \ge 0$ and $t >1$). Vertex fault-tolerant spanners were introduced by Levcopoulus et. al in 1998. For $k=0$, we present an $O(n \log n)$ method using the algebraic computation tree model to find a $t$-spanner with degree bound O(1) and weight $O(\weight(MST))$. This resolves an open problem. For $k \ge 1$, we present an efficient method that, given $n$ points in $\dspace$, constructs $k$-vertex fault-tolerant $t$-spanners with the maximum degree bound O(k) and weight bound $O(k^2 \weight(MST))$ in time $O(n \log n)$. Our method achieves the best possible bounds on degree, total edge length, and the time complexity, and solves the open problem of efficient construction of (fault-tolerant) $t$-spanners in $\dspace$ in time $O(n \log n)$.

cs.CG↗

arXiv2010

A near optimal algorithm for finding Euclidean shortest path in polygonal domain

We present an algorithm to find an {\it Euclidean Shortest Path} from a source vertex $s$ to a sink vertex $t$ in the presence of obstacles in $\Re^2$. Our algorithm takes $O(T+m(\lg{m})(\lg{n}))$ time and $O(n)$ space. Here, $O(T)$ is the time to triangulate the polygonal region, $m$ is the number of obstacles, and $n$ is the number of vertices. This bound is close to the known lower bound of $O(n+m\lg{m})$ time and $O(n)$ space. Our approach involve progressing shortest path wavefront as in continuous Dijkstra-type method, and confining its expansion to regions of interest.

cs.CG↗

arXiv2010

Approximate Shortest Path through a Weighted Planar Subdivision

This paper presents an approximation algorithm for finding a shortest path between two points $s$ and $t$ in a weighted planar subdivision $\PS$. Each face $f$ of $\PS$ is associated with a weight $w_f$, and the cost of travel along a line segment on $f$ is $w_f$ multiplied by the Euclidean norm of that line segment. The cost of a path which traverses across several faces of the subdivision is the sum of the costs of travel along each face. Our algorithm progreeses the discretized shortest path wavefront from source $s$, and takes polynomial time in finding an $ε$-approximate shortest path.

cs.CG↗

arXiv1999

A Quantum Algorithm for finding the Maximum

This paper describes a quantum algorithm for finding the maximum among N items. The classical method for the same problem takes O(N) steps because we need to compare two numbers in one step. This algorithm takes O(sqrt(N)) steps by exploiting the property of quantum states to exist in a superposition of states and hence performing an operation on a number of elements in one go. A tight upper bound of 6.8(sqrt(N)) for the number of steps needed using this algorithm was found. These steps are the number of queries made to the oracle.

quant-ph↗