arXiv subjects

Sudebkumar Prasant Pal

Publications and source records attributed to Sudebkumar Prasant Pal.

At least 19 recordsLinked to original sources

arXiv2025

Quantum Approximate and Quantum Walk Optimization Approaches to Set Balancing

We explore the application of variational quantum algorithms to the NP-hard set balancing problem, a critical challenge in clinical trial design and experimental scheduling. The problem is mapped to an Ising model, with tailored Quadratic Unconstrained Binary Optimization (QUBO) formulations and cost Hamiltonians expressed in Pauli-Z form. We implement both the Quantum Approximate Optimization Algorithm (QAOA) and the Quantum Walk Optimization Algorithm (QWOA), evaluating them in separate experimental settings. For QAOA, we perform a comparative analysis of six mixer Hamiltonians (X, XY, Full-SWAP, Ring-SWAP, Grover, and Warm-Started), employing scaled-exponential Pauli-string realizations of the mixer unitaries, which yield superior performance over conventional circuit decompositions. Additionally, we introduce a Shannon-entropy-based post-processing technique that refines solutions by maximizing feature-distribution uniformity across partitions. These results underscore the importance of mixer choice and circuit implementation in enhancing QAOA performance for combinatorial optimization.

quant-ph↗

arXiv2025

Discrete-Time Quantum Random Walk for Epidemiological Modeling

We introduce a discrete-time quantum random walk (QRW) framework for spatial epidemic modelling on a two-dimensional square lattice and compare its dynamics to classical random-walk SIR models. In our model, each infected site spawns a quantum walker whose coherent evolution (controlled by an amplitude-splitting coin and conditional shifts) can infect visited susceptible sites with probability $p$ and persists for a lifetime of $\tau$ steps. We perform extensive quantum simulations on finite lattices and compute the basic reproduction number $R_0$ across a broad grid of $(p,\tau)$ values. Results show that QRW dynamics interpolate between diffusive and super-diffusive regimes: at low $p$ the QRW reproduces classical-like $R_0$, while at higher $p$ and $\tau$ ballistic propagation and interference produce markedly larger $R_0$ and non-Gaussian spatial profiles. We compare the QRW $R_0$ range to empirical estimates from historical outbreaks and discuss parameter regimes where QRW offers a closer qualitative match than classical diffusion. We conclude that QRWs provide a flexible, conceptually novel toy model for exploring rapid or heavy-tailed epidemic spread.

quant-ph↗

arXiv2024

Forbidden induced subgraphs in iterative higher order line graphs

Let $G$ be a simple finite connected graph. The line graph $L(G)$ of graph $G$ is the graph whose vertices are the edges of $G$, where $ef \in E(L(G))$ when $e \cap f \neq \emptyset$. Iteratively, the higher order line graphs are defined inductively as $L^1(G) = L(G)$ and $L^n(G) = L(L^{n-1}(G))$ for $n \geq 2$. In [Derived graphs and digraphs, Beitrage zur Graphentheorie (Teubner, Leipzig 1968), 17--33 (1968)], Beineke characterize line graphs in terms of nine forbidden subgraphs. Inspired by this result, in this paper, we characterize second order line graphs in terms of pure forbidden induced subgraphs. We also give a sufficient list of forbidden subgraphs for a graph $G$ such that $G$ is a higher order line graph. We characterize all order line graphs of graph $G$ with $\Delta(G) = 3$ and $4$.

math.CO↗

arXiv2019

Bisecting and D-secting families for set systems

Let $n$ be any positive integer and $\mathcal{F}$ be a family of subsets of $[n]$. A family $\mathcal{F}'$ is said to be $D$-\emph{secting} for $\mathcal{F}$ if for every $A \in \mathcal{F}$, there exists a subset $A' \in \mathcal{F}'$ such that $|A \cap A'| - |A \cap ([n] \setminus A')|=i$, where $i \in D$, $D \subseteq \{-n,-n+1,\ldots,0,\ldots,n\}$. A $D$-\emph{secting} family $\mathcal{F}'$ of $\mathcal{F}$, where $D=\{-1,0,1\}$, is a \emph{bisecting} family ensuring the existence of a subset $A' \in \mathcal{F}'$ such that $|A \cap A'| \in \{\lceil \frac{|A|}{2}\rceil,\lfloor \frac{|A|}{2}\rfloor\}$, for each $A \in \mathcal{F}$. In this paper, we study $D$-secting families for $\mathcal{F}$ with restrictions on $D$, and the cardinalities of $\mathcal{F}$ and the subsets of $\mathcal{F}$.

math.CO↗

arXiv2017

System of unbiased representatives for a collection of bicolorings

Let $\mathcal{B}$ denote a set of bicolorings of $[n]$, where each bicoloring is a mapping of the points in $[n]$ to $\{-1,+1\}$. For each $B \in \mathcal{B}$, let $Y_B=(B(1),\ldots,B(n))$. For each $A \subseteq [n]$, let $X_A \in \{0,1\}^n$ denote the incidence vector of $A$. A non-empty set $A$ is said to be an `unbiased representative' for a bicoloring $B \in \mathcal{B}$ if $\left\langle X_A,Y_B\right\rangle =0$. Given a set $\mathcal{B}$ of bicolorings, we study the minimum cardinality of a family $\mathcal{A}$ consisting of subsets of $[n]$ such that every bicoloring in $\mathcal{B}$ has an unbiased representative in $\mathcal{A}$.

math.CO↗

arXiv2017

Induced bisecting families for hypergraphs

Two $n$-dimensional vectors $A$ and $B$, $A,B \in \mathbb{R}^n$, are said to be \emph{trivially orthogonal} if in every coordinate $i \in [n]$, at least one of $A(i)$ or $B(i)$ is zero. Given the $n$-dimensional Hamming cube $\{0,1\}^n$, we study the minimum cardinality of a set $\mathcal{V}$ of $n$-dimensional $\{-1,0,1\}$ vectors, each containing exactly $d$ non-zero entries, such that every `possible' point $A \in \{0,1\}^n$ in the Hamming cube has some $V \in \mathcal{V}$ which is orthogonal, but not trivially orthogonal, to $A$. We give asymptotically tight lower and (constructive) upper bounds for such a set $\mathcal{V}$ except for the even values of $d \in Ω(n^{0.5+ε})$, for any $ε$, $0< ε\leq 0.5$.

math.CO↗

arXiv2016

A National Effort for Motivating Indian Students and Teachers towards Algorithmic Research

During 2008-2015, twenty-two introductory workshops on graph and geometric algorithms were organized for teachers and students (undergraduate, post-graduate and doctoral) of engineering colleges and universities at different states and union territories of India. The lectures were meant to provide exposure to the field of graph and geometric algorithms and to motivate the participants towards research. Fifty-eight professors from TIFR, IITs, IISc, IMSc, CMI, ISI Kolkata, and other institutes and universities delivered invited lectures on different topics in the design and analysis of algorithms, discrete applied mathematics, computer graphics, computer vision, and robotics. The first four workshops were funded by TIFR, BRNS and IIT Kharagpur, and the remaining workshops were funded by the NBHM. In this paper, we present the salient features of these workshops, and state our observations on the national impact of these workshops.

cs.CY↗

arXiv2016

Bicoloring covers for graphs and hypergraphs

Let the {\it bicoloring cover number $χ^c(G)$} for a hypergraph $G(V,E)$ be the minimum number of bicolorings of vertices of $G$ such that every hyperedge $e\in E$ of $G$ is properly bicolored in at least one of the $χ^c(G)$ bicolorings. We investigate the relationship between $χ^c(G)$, matchings, hitting sets, $α(G)$(independence number) and $χ(G)$ (chromatic number). We design a factor $O(\frac{\log n}{\log \log n-\log \log \log n})$ approximation algorithm for computing a bicoloring cover. We define a new parameter for hypergraphs - "cover independence number $γ(G)$" and prove that $\log \frac{|V|}{γ(G)}$ and $\frac{|V|}{2γ(G)}$ are lower bounds for $χ^c(G)$ and $χ(G)$, respectively. We show that $χ^c(G)$ can be approximated by a polynomial time algorithm achieving approximation ratio $\frac{1}{1-t}$, if $γ(G)=n^t$, where $t<1$. We also construct a particular class of hypergraphs $G(V,E)$ called {\it cover friendly} hypergraphs where the ratio of $α(G)$ to $γ(G)$ can be arbitrarily large.We prove that for any $t\geq 1$, there exists a $k$-uniform hypergraph $G$ such that the {\it clique number} $ω(G)=k$ and $χ^c(G) > t$. Let $m(k,x)$ denote the minimum number of hyperedges %in a $k$-uniform hypergraph $G$ such that some $k$-uniform hypergraph $G$ with $m(k,x)$ hyperedges does not have a bicoloring cover of size $x$. We show that $ 2^{(k-1)x-1} < m(k,x) \leq x \cdot k^2 \cdot 2^{(k+1)x+2}$. Let the {\it dependency $d(G)$} of $G$ be the maximum number of hyperedge neighbors of any hyperedge in $G$. We propose an algorithm for computing a bicoloring cover of size $x$ for $G$ if $d(G) \leq(\frac{2^{x(k-1)}}{e}-1)$ using $nx+kx\frac{m}{d}$ random bits.

cs.DM↗

arXiv2015

Maximum weighted independent sets with a budget

Given a graph $G$, a non-negative integer $k$, and a weight function that maps each vertex in $G$ to a positive real number, the \emph{Maximum Weighted Budgeted Independent Set (MWBIS) problem} is about finding a maximum weighted independent set in $G$ of cardinality at most $k$. A special case of MWBIS, when the weight assigned to each vertex is equal to its degree in $G$, is called the \emph{Maximum Independent Vertex Coverage (MIVC)} problem. In other words, the MIVC problem is about finding an independent set of cardinality at most $k$ with maximum coverage. Since it is a generalization of the well-known Maximum Weighted Independent Set (MWIS) problem, MWBIS too does not have any constant factor polynomial time approximation algorithm assuming $P \neq NP$. In this paper, we study MWBIS in the context of bipartite graphs. We show that, unlike MWIS, the MIVC (and thereby the MWBIS) problem in bipartite graphs is NP-hard. Then, we show that the MWBIS problem admits a $\frac{1}{2}$-factor approximation algorithm in the class of bipartite graphs, which matches the integrality gap of a natural LP relaxation.

cs.CC↗

arXiv2015

Strong $(r,p)$ Cover for Hypergraphs

We introduce the notion of the { \it strong $(r,p)$ cover} number $χ^c(G,k,r,p)$ for $k$-uniform hypergraphs $G(V,E)$, where $χ^c(G,k,r,p)$ denotes the minimum number of $r$-colorings of vertices in $V$ such that each hyperedge in $E$ contains at least $min(p,k)$ vertices of distinct colors in at least one of the $χ^c(G,k,r,p)$ $r$-colorings. We derive the exact values of $χ^c(K_n^k,k,r,p)$ for small values of $n$, $k$, $r$ and $p$, where $K_n^k$ denotes the complete $k$-uniform hypergraph of $n$ vertices. We study the variation of $χ^c(G,k,r,p)$ with respect to changes in $k$, $r$, $p$ and $n$; we show that $χ^c(G,k,r,p)$ is at least (i) $χ^c(G,k,r-1,p-1)$, and, (ii) $χ^c(G',k-1,r,p-1)$, where $G'$ is any $(n-1)$-vertex induced sub-hypergraph of $G$. We establish a general upper bound for $χ^c(K_n^k,k,r,p)$ for complete $k$-uniform hypergraphs using a divide-and-conquer strategy for arbitrary values of $k$, $r$ and $p$. We also relate $χ^c(G,k,r,p)$ to the number $|E|$ of hyperedges, and the maximum {\it hyperedge degree (dependency)} $d(G)$, as follows. We show that $χ^c(G,k,r,p)\leq x$ for integer $x>0$, if $|E|\leq \frac{1}{2}({\frac{r^k}{(t-1)^k \binom{r}{t-1}}})^x $, for any $k$-uniform hypergraph. We prove that a { \it strong $(r,p)$ cover} of size $x$ can be computed in randomized polynomial time if $d(G)\leq \frac{1}{e}({\frac{r^k}{(p-1)^k \binom{r}{p-1}}})^x-1$.

cs.DM↗

arXiv2015

An extremal problem in proper $(r,p)$-coloring of hypergraphs

Let $G(V,E)$ be a $k$-uniform hypergraph. A hyperedge $e \in E$ is said to be properly $(r,p)$ colored by an $r$-coloring of vertices in $V$ if $e$ contains vertices of at least $p$ distinct colors in the $r$-coloring. An $r$-coloring of vertices in $V$ is called a {\it strong $(r,p)$ coloring} if every hyperedge $e \in E$ is properly $(r,p)$ colored by the $r$-coloring. We study the maximum number of hyperedges that can be properly $(r,p)$ colored by a single $r$-coloring and the structures that maximizes number of properly $(r,p)$ colored hyperedges.

cs.DM↗

arXiv2014

An Algorithm for Computing Constrained Reflection Paths in Simple Polygon

Let $s$ be a source point and $t$ be a destination point inside an $n$-vertex simple polygon $P$. Euclidean shortest paths and minimum-link paths between $s$ and $t$ inside $P$ have been well studied. Both these kinds of paths are simple and piecewise-convex. However, computing optimal paths in the context of diffuse or specular reflections does not seem to be an easy task. A path from a light source $s$ to $t$ inside $P$ is called a diffuse reflection path if the turning points of the path lie in the interiors of the boundary edges of $P$. A diffuse reflection path is said to be optimal if it has the minimum number of turning points amongst all diffuse reflection paths between $s$ and $t$. The minimum diffuse reflection path may not be simple. The problem of computing the minimum diffuse reflection path in low degree polynomial time has remained open. In our quest for understanding the geometric structure of the minimum diffuse reflection paths vis-a-vis shortest paths and minimum link paths, we define a new kind of diffuse reflection path called a constrained diffuse reflection path where (i) the path is simple, (ii) it intersects only the eaves of the Euclidean shortest path between $s$ and $t$, and (iii) it intersects each eave exactly once. For computing a minimum constrained diffuse reflection path from $s$ to $t$, we present an $O(n(n+β))$ time algorithm, where $β=Θ(n^2)$ in the worst case. Here, $β$ depends on the shape of the polygon. We also establish some properties relating minimum constrained diffuse reflection paths and minimum diffuse reflection paths. Constrained diffuse reflection paths introduced in this paper provide new geometric insights into the hitherto unknown structures and shapes of optimal reflection paths.

cs.CG↗

arXiv2007

Hypergraph-theoretic charaterizations for LOCC incomparable ensembles of multipartite CAT states

Using graphs and hypergraphs to systematically model collections of arbitrary subsets of parties representing {\it ensembles (or collections)} of shared multipartite CAT states, we study transformations between such {\it ensembles} under {\it local operations and classical communication (LOCC)}. We show using partial entropic criteria, that any two such distinct ensembles represented by {\it $r$-uniform hypergraphs} with the same number of hyperedges (CAT states), are LOCC incomparable for even integers $r\geq 2$, generalizing results in \cite{mscthesis,sin:pal:kum:sri}. We show that the cardinality of the largest set of mutually LOCC incomparable ensembles represented by $r$-uniform hypergraphs for even $r\geq 2$, is exponential in the number of parties. We also demonstrate LOCC incomparability between two ensembles represented by 3-uniform hypergraphs where partial entropic criteria do not help in establishing incomparability. Further we characterize LOCC comparability of EPR graphs in a model where LOCC is restricted to teleportation and edge destruction. We show that this model is equivalent to one in which LOCC transformations are carried out through a sequence of operations where each operation adds at most one new EPR pair.

quant-ph↗

arXiv2006

Faultfree Tromino Tilings of Rectangles

In this paper we consider faultfree tromino tilings of rectangles and characterize rectangles that admit such tilings. We introduce the notion of {\it crossing numbers} for tilings and derive bounds on the crossing numbers of faultfree tilings. We develop an iterative scheme for generating faultfree tromino tilings for rectangles and derive the closed form expression for the exact number of faultfree tromino tilings for $4\times3t$ rectangles and the exact generating function for $5\times 3t$ rectangles, $t\geq 1$. Our iterative scheme generalizes to arbitrary rectangles; for $6\times 6t$ and $7\times 6t$ rectangles, $t\geq 1$, we derive generating functions for estimating lower bounds on the number of faultfree tilings. We also derive an upper bound on the number of tromino tilings of an $m\times n$ rectangle, where $3|mn$ and $m,n>0$.

math.CO↗

arXiv2006

Constant communication complexity protocols for multiparty accumulative boolean functions

Generalizing a boolean function from Cleve and Buhrman \cite{cb:sqec}, we consider the class of {\it accumulative boolean functions} of the form $f_B(X_1,X_2,..., X_m)=\bigoplus_{i=1}^n t_B(x_i^1x_i^2... x_i^m)$, where $X_j=(x^j_1,x^j_2,..., x^j_n), 1\leq j\leq m$ and $t_B(x_i^1x_i^2... x_i^m)=1$ for input $m$-tuples $x_i^1x_i^2...x_i^m\in B\subseteq A\subseteq \{0,1\}^n$, and 0, if $x_i^1x_i^2...x_i^m\in A\setminus B$. Here the set $A$ is the input {\it promise} set for function $f_B$. The input vectors $X_j, 1\leq j\leq m$ are given to the $m\geq 3$ parties respectively, who communicate cbits in a distributed environment so that one of them (say Alice) comes up with the value of the function. We algebraically characterize entanglement assisted LOCC protocols requiring only $m-1$ cbits of communication for such multipartite boolean functions $f_B$, for certain sets $B\subseteq \{0,1\}^n$, for $m\geq 3$ parties under appropriate uniform parity promise restrictions on input $m$-tuples $x_i^1x_i^2...x_i^m, 1\leq i\leq n$. We also show that these functions can be computed using $2m-3$ cbits in a purely classical deterministic setup. In contrast, for certain $m$-party accumulative boolean functions ($m\geq 2$), we characterize promise sets of mixed parity for input $m$-tuples so that $m-1$ cbits of communication suffice in computing the functions in the absence of any a priori quantum entanglement. We compactly represent all these protocols and the corresponding input promise restrictions using uniform group theoretic and hamming distance characterizations.

quant-ph↗

arXiv2005

A combinatorial approach for studying LOCC transformations of multipartite states

We develop graph theoretic methods for analysing maximally entangled pure states distributed between a number of different parties. We introduce a technique called {\it bicolored merging}, based on the monotonicity feature of entanglement measures, for determining combinatorial conditions that must be satisfied for any two distinct multiparticle states to be comparable under local operations and classical communication (LOCC). We present several results based on the possibility or impossibility of comparability of pure multipartite states. We show that there are exponentially many such entangled multipartite states among $n$ agents. Further, we discuss a new graph theoretic metric on a class of multi-partite states, and its implications.

quant-ph↗

arXiv2004

Characterizing the combinatorics of distributed EPR pairs for multi-partite entanglement

We develop protocols for preparing a GHZ state and, in general,a pure multi-partite maximally entangled state in a distributed network with apriori quantum entanglement between agents using classical communication and local operations. We investigate and characterize the minimal combinatorics of the sharing of EPR pairs required amongst agents in a network for the creation of multi-partite entanglement. We also characterize the minimal combinatorics of agents in the creation of pure maximal multi-partite entanglement amongst the set $N$ of $n$ agents in a network using apriori multi-partite entanglement states amongst subsets of $N$. We propose protocols for establishing multi-partite entanglement in the above cases.

quant-ph↗

arXiv2003

Multi-partite Quantum Entanglement versus Randomization: Fair and Unbiased Leader Election in Networks

In this paper we show that sufficient multi-partite quantum entanglement helps in fair and unbiased election of a leader in a distributed network of processors with only linear classical communication complexity. We show that a total of $O(\log n)$ distinct multi-partite maximally entanglement sets (ebits) are capable of supporting such a protocol in the presence of nodes that may lie and thus be biased. Here, $n$ is the number of nodes in the network. We also demonstrate the difficulty of performing unbiased and fair election of a leader with linear classical communication complexity in the absence of quantum entanglement even if all nodes have perfect random bit generators. We show that the presence of a sufficient number $O(n/\log n)$ of biased agents leads to a non-zero limiting probability of biased election of the leader, whereas, the presence of a smaller number $O(\log n)$ of biased agents matters little. We define two new related complexity classes motivated by the our leader election problem and discuss a few open questions.

quant-ph↗