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Niranjan Balachandran

Publications and source records attributed to Niranjan Balachandran.

At least 19 recordsLinked to original sources
arXiv2026

Caged subsequences in permutations

Given a sequence $\mathfrak{a}:=(a_1,\ldots,a_n)$ of reals, a subsequence $\mathfrak{b}=(a_{i_1},\ldots,a_{i_k})$ is said to be "caged" if the largest and smallest among the members of $\mathfrak{b}$ are $a_{i_1}$ and $a_{i_k}$, though not necessarily in that order. In this paper, we consider the problem of maximal caged sequences in permutations $π\in S_n$. We also consider the same problem for a random permutation, both when the permutation is chosen uniformly at random and also when it is picked uniformly at random from among the permutations of rectangular shape, via the RSK correspondence.

math.CO↗
arXiv2026

Low-rank matrices, tournaments, and symmetric designs

Let $\mathbf{a} = (a_{i})_{i \geq 1}$ be a sequence in a field $\mathbb{F}$, and $f \colon \mathbb{F} \times \mathbb{F} \to \mathbb{F}$ be a function such that $f(a_{i},a_{i}) \neq 0$ for all $i \geq 1$. For any tournament $T$ over $[n]$, consider the $n \times n$ symmetric matrix $M_{T}(f; \mathbf{a})$ with zero diagonal whose $(i,j)$th entry (for $i < j$) is $f(a_{i},a_{j})$ if $i \to j$ in $T$, and $f(a_{j},a_{i})$ if $j \to i$ in $T$. It is known (cf. Balachandran et al., Linear Algebra Appl. 658 (2023), 310-318) that if $T$ is a uniformly random tournament over $[n]$, then $\operatorname{rank}(M_{T}(f; \mathbf{a})) \geq (\frac{1}{2}-o(1))n$ with high probability when $\operatorname{char}(\mathbb{F}) \neq 2$ and $f$ is a linear function. In this paper, we investigate the other extremal question: how low can the ranks of such matrices be? We work with sequences $\mathbf{a}$ that take only two distinct values, so the rank of any such $n \times n$ matrix is at least $n/2$. First, we show that the rank of any such matrix depends on whether an associated bipartite graph has certain eigenvalues of high multiplicity. Using this, we show that if $f$ is linear, then there are $n \times n$ real matrices $M_{T}(f; \mathbf{a})$ of rank at most $\frac{n}{2} + O(1)$. For rational matrices, we show that for each $\varepsilon > 0$ we can find a sequence $\mathbf{a}(\varepsilon)$ for which there are $n \times n$ matrices $M_{T}(f; \mathbf{a}(\varepsilon))$ of rank at most $(\frac{1}{2} + \varepsilon)n + O(1)$. These matrices are constructed from symmetric designs, and we also use them to produce bisection-closed families of size greater than $\lfloor 3n/2 \rfloor - 2$ for $n \leq 15$, which improves the previously best known bound (cf. Balachandran et al., Electron J. Combin. 26 (2019), #P2.40).

math.CO↗
arXiv2025

An ensemble of high rank matrices arising from tournaments

Suppose $\mathbb{F}$ is a field and let $\mathbf{a} := (a_1, a_2, \dotsc)$ be a sequence of non-zero elements in $\mathbb{F}$. For $\mathbf{a}_n := (a_1, \dotsc, a_n)$, we consider the family $\mathcal{M}_n(\mathbf{a})$ of $n \times n$ symmetric matrices $M$ over $\mathbb{F}$ with all diagonal entries zero and the $(i, j)$th element of $M$ either $a_i$ or $a_j$ for $i < j$. In this short paper, we show that all matrices in a certain subclass of $\mathcal{M}_n(\mathbf{a})$ -- which can be naturally associated with transitive tournaments -- have rank at least $\lfloor 2n/3 \rfloor - 1$. We also show that if $\operatorname{char}(\mathbb{F}) \neq 2$ and $M$ is a matrix chosen uniformly at random from $\mathcal{M}_n(\mathbf{a})$, then with high probability $\operatorname{rank}(M) \geq \bigl(\frac{1}{2} - o(1)\bigr)n$.

math.CO↗
arXiv2025

Bounded fractional intersecting families are linear in size

Using the sunflower method, we show that if $θ\in (0,1) \cap \mathbb{Q}$ and $\mathcal{F}$ is a $O(n^{1/3})$-bounded $θ$-intersecting family over $[n]$, then $\lvert \mathcal{F} \rvert = O(n)$, and that if $\mathcal{F}$ is $o(n^{1/3})$-bounded, then $\lvert \mathcal{F} \rvert \leq (\frac{3}{2} + o(1))n$. This partially solves a conjecture of Balachandran, Mathew and Mishra that any $θ$-intersecting family over $[n]$ has size at most linear in $n$, in the regime where we have no very large sets.

math.CO↗
arXiv2025

Temperatures of Robin Hood

Cumulative Games were introduced by Larsson, Meir, and Zick (2020) to bridge some conceptual and technical gaps between Combinatorial Game Theory (CGT) and Economic Game Theory. The partizan ruleset {\sc Robin Hood} is an instance of a Cumulative Game, viz., {\sc Wealth Nim}. It is played on multiple heaps, each associated with a pair of cumulations, interpreted here as wealth. Each player chooses one of the heaps, removes tokens from that heap not exceeding their own wealth, while simultaneously diminishing the other player's wealth by the same amount. In CGT, the {\em temperature} of a {\em disjunctive sum} game component is an estimate of the urgency of moving first in that component. It turns out that most of the positions of {\sc Robin Hood} are {\em hot}. The temperature of {\sc Robin Hood} on a single large heap shows a dichotomy in behavior depending on the ratio of the wealths of the players. Interestingly, this bifurcation is related to Pingala (Fibonacci) sequences and the Golden Ratio $ϕ$: when the ratio of the wealths lies in the interval $(ϕ^{-1},ϕ)$, the temperature increases linearly with the heap size, and otherwise it remains constant, and the mean values has a reciprocal property. It turns out that despite {\sc Robin Hood} displaying high temperatures, playing in the hottest component might be a sub-optimal strategy.

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arXiv2024

Cascaded Group Testing

In this paper, we introduce a variation of the group testing problem where each test is specified by an ordered subset of items and returns the first defective item in the specified order or returns null if there are no defectives. We refer to this as cascaded group testing and the goal is to identify a small set of $K$ defective items amongst a collection of size $N$, using as few tests as possible for perfect recovery. For the adaptive testing regime, we show that a simple scheme can find all defective items in at most $K$ tests, which is optimal. For the non-adaptive setting, we first come up with a necessary and sufficient condition for any collection of tests to be feasible for recovering all the defectives. Using this, we show that any feasible non-adaptive strategy requires at least $Ω(K^2)$ tests. In terms of achievability, it is easy to show the existence of a feasible collection of $O(K^2 \log (N/K))$ tests. We show via carefully constructed explicit designs that one can do significantly better for constant $K$. While the cases $K = 1, 2$ are straightforward, the case $K=3$ is already non-trivial and we come up with an iterative design that is asymptotically optimal and requires $Θ(\log \log N)$ tests. Note that this is in contrast to standard binary group testing, where at least $Ω(\log N)$ tests are required. For constant $K \ge 3$, our iterative design requires only poly$(\log \log N)$ tests.

cs.IT↗
arXiv2024

On hierarchically closed fractional intersecting families

For a set $L$ of positive proper fractions and a positive integer $r \geq 2$, a fractional $r$-closed $L$-intersecting family is a collection $\mathcal{F} \subset \mathcal{P}([n])$ with the property that for any $2 \leq t \leq r$ and $A_1, \dotsc, A_t \in \mathcal{F}$ there exists $θ\in L$ such that $\lvert A_1 \cap \dotsb \cap A_t \rvert \in \{ θ\lvert A_1 \rvert, \dotsc, θ\lvert A_t \rvert\}$. In this paper we show that for $r \geq 3$ and $L = \{θ\}$ any fractional $r$-closed $θ$-intersecting family has size at most linear in $n$, and this is best possible up to a constant factor. We also show that in the case $θ= 1/2$ we have a tight upper bound of $\lfloor \frac{3n}{2} \rfloor - 2$ and that a maximal $r$-closed $(1/2)$-intersecting family is determined uniquely up to isomorphism.

math.CO↗
arXiv2022

Cyclability, Connectivity and Circumference

In a graph $G$, a subset of vertices $S \subseteq V(G)$ is said to be cyclable if there is a cycle containing the vertices in some order. $G$ is said to be $k$-cyclable if any subset of $k \geq 2$ vertices is cyclable. If any $k$ \textit{ordered} vertices are present in a common cycle in that order, then the graph is said to be $k$-ordered. We show that when $k \leq \sqrt{n+3}$, $k$-cyclable graphs also have circumference $c(G) \geq 2k$, and that this is best possible. Furthermore when $k \leq \frac{3n}{4} -1$, $c(G) \geq k+2$, and for $k$-ordered graphs we show $c(G) \geq \min\{n,2k\}$. We also generalize a result by Byer et al. on the maximum number of edges in nonhamiltonian $k$-connected graphs, and show that if $G$ is a $k$-connected graph of order $n \geq 2(k^2+k)$ with $|E(G)| > \binom{n-k}{2} + k^2$, then the graph is hamiltonian, and moreover the extremal graphs are unique.

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arXiv2021

The Normalized Matching Property in Random and Pseudorandom Bipartite Graphs

A simple generalization of the Hall's condition in bipartite graphs, the Normalized Matching Property (NMP) in a graph $G(X,Y,E)$ with vertex partition $(X,Y)$ states that for any subset $S\subseteq X$, we have $\frac{|N(S)|}{|Y|}\ge\frac{|S|}{|X|}$. In this paper, we show the following results about having the Normalized Matching Property in random and pseudorandom graphs. 1. We establish $p=\frac{\log n}{k}$ as a sharp threshold for having NMP in $\mathbb{G}(k,n,p)$, which is the graph with $|X|=k,|Y|=n$ (assuming $k\le n\leq \exp(o(k))$), and in which each pair $(x,y)\in X\times Y$ is an edge independently with probability $p$. This generalizes a classic result of Erdős-Rényi on the $\frac{\log n}{n}$ threshold for having a perfect matching in $\mathbb{G}(n,n,p)$. 2. We also show that a pseudorandom bipartite graph - upon deletion of a vanishingly small fraction of vertices - admits NMP, provided it is not too sparse. More precisely, a bipartite graph $G(X,Y)$, with $k=|X|\le |Y|=n$, is said to be Thomason pseudorandom (following A. Thomason (Discrete Math., 1989)) with parameters $(p,\varepsilon)$ if each $x\in X$ has degree at least $pn$ and each pair of distinct $x, x'\in X$ has at most $(1+\varepsilon)p^2n$ common neighbors. We show that for any large enough $(p,\varepsilon)$-Thomason pseudorandom graph $G(X,Y)$, there are "tiny" subsets $\mathrm{Del}_X\subset X, \ \mathrm{Del}_Y\subset Y$ such that the subgraph $G(X\setminus \mathrm{Del}_X,Y\setminus \mathrm{Del}_Y)$ has NMP, provided $p \gg\tfrac{1}{k}$. En route, we prove an "almost" vertex decomposition theorem: Every such Thomason pseudorandom graph admits - excluding a negligible portion of its vertex set - a partition of its vertex set into graphs that we call Euclidean trees. These are trees that have NMP, and which arise organically through the Euclidean GCD algorithm.

math.CO↗
arXiv2021

The choice number versus the chromatic number for graphs embeddable on orientable surfaces

We show that for loopless $6$-regular triangulations on the torus the gap between the choice number and chromatic number is at most $2$. We also show that the largest gap for graphs embeddable in an orientable surface of genus $g$ is of the order $Θ(\sqrt{g})$, and moreover for graphs with chromatic number of the order $o(\sqrt{g}/\log_{2}(g))$ the largest gap is of the order $o(\sqrt{g})$.

math.CO↗
arXiv2021

Efficient constrained sensor placement for observability of linear systems

This article studies two problems related to observability and efficient constrained sensor placement in linear time-invariant discrete-time systems with partial state observations. (i) We impose the condition that both the set of outputs and the state that each output can measure are pre-specified. We establish that for any fixed \(k > 2\), the problem of placing the minimum number of sensors/outputs required to ensure that the structural observability index is at most \(k\), is NP-complete. Conversely, we identify a subclass of systems whose structures are directed trees with self-loops at every state vertex, for which the problem can be solved in linear time. (ii) Assuming that the set of states that each given output can measure is given, we prove that the problem of selecting a pre-assigned number of sensors in order to maximize the number of states of the system that are structurally observable is also NP-hard. As an application, we identify suitable conditions on the system structure under which there exists an efficient greedy strategy, which we provide, to obtain a \((1-\frac{1}{e})\)-approximate solution. An illustration of the techniques developed for this problem is given on the benchmark IEEE 118-bus power network containing roughly \(400\) states in its linearized model.

math.OC↗
arXiv2021

Zero sums in restricted sequences

A sequence $\bfx=(x_1,\ldots,x_m)$ of elements of $\Z_n$ is called an \textit{$A$-weighted Davenport Z-sequence} if there exists $\bfa:=(a_1,\ldots,a_m)\in (A\cup\{0\})^m\setminus\bfzero_m$ such that $\sum_i a_ix_i=0$. Here $\bfzero_m=(0,\ldots,0)\in\Z_n^m$. Similarly, the sequence $\bfx$ is called an \textit{$A$-weighted Erdős Z-sequence} if there exists $\bfa:=(a_1,\ldots,a_m)\in (A\cup\{0\})^m\setminus\{\bfzero_m\}$ with $|Supp(\bfa)|=n$, such that $\sum_i a_ix_i=0$, where $Supp(\bfa):=\{i: a_i\ne 0\}$. A $\Z_n$-sequence $\bfx$ is called $k$-restricted if no element of $\Z_n$ appears more than $k$ times in $\bfx$. In this paper, we study the problem of determining the least value of $m$ for which a $k$-restricted $\Z_n$-sequence of length $m$ is an $A$-weighted Davenport Z-sequence (resp. an$A$-weighted Erdős Z-sequence). We also consider the same problem for random $\Z_n$ sequences, for certain very natural choices for the set $A$.

math.NT↗
arXiv2019

The Weighted Davenport constant of a group and a related extremal problem II

For a finite abelian group $G$ with $\exp(G)=n$ and an integer $k\ge 2$, Balachandran and Mazumdar \cite{BM} introduced the extremal function $\fD_G(k)$ which is defined to be $\min\{|A|: \emptyset \neq A\subseteq[1,n-1]\textrm{\ with\ }D_A(G)\le k\}$ (and $\infty$ if there is no such $A$), where $D_A(G)$ denotes the $A$-weighted Davenport constant of the group $G$. Denoting $\fD_G(k)$ by $\fD(p,k)$ when $G=\bF_p$ (for $p$ prime), it is known (\cite{BM}) that $p^{1/k}-1\le \fD(p,k)\le O_k(p\log p)^{1/k}$ holds for each $k\ge 2$ and $p$ sufficiently large, and that for $k=2,4$, we have the sharper bound $\fD(p,k)\le O(p^{1/k})$. It was furthermore conjectured that $\fD(p,k)=Θ(p^{1/k})$. In this short paper we prove that $\fD(p,k)\le 4^{k^2}p^{1/k}$ for sufficiently large primes $p$.

math.CO↗
arXiv2019

On Minimum Cost Sparsest Input-Connectivity for Controllability of Linear Systems

We deal with algorithmic techniques for minimal cost input-connectivity while maintaining controllability of linear systems. The input matrix is assumed to be constrained in the sense that the set of states that each input (if present) can influence is known a priori, and that each interconnection between an input and a state is associated with a certain cost. In this setting we determine a set of input-connections that lead to the minimum cost and ensures that the resulting system is structurally controllable. We also identify a sparsest set of input-connections with minimum cost while maintaining structural controllability of the system. A large class of systems are identified for which these problems are solvable in polynomial time using efficient algorithms. A 2-approximation solution is presented for the general case. Graph-theoretic tools are employed to tackle the above class of constrained design problems. Illustrative examples are included to demonstrate the efficacy of the techniques developed here.

math.OC↗
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↗
arXiv2019

The Harborth Constant of Dihedral Groups

The Harborth constant of a finite group $G$, denoted $\gs(G)$, is the smallest integer $k$ such that the following holds: For $A\subseteq G$ with $|A|=k$, there exists $B\subseteq A$ with $|B|=\exp(G)$ such that the elements of $B$ can be rearranged into a sequence whose product equals $1_G$, the identity element of $G$. The Harborth constant is a well studied combinatorial invariant in the case of abelian groups. In this paper, we consider a generalization $\gs(G)$ of this combinatorial invariant for nonabelian groups and prove that if $G$ is a dihedral group of order $2n$ with $n\ge 3$, then $\gs(G) = n + 2$ if $n$ is even and $\gs(G) = 2n + 1$ otherwise.

math.CO↗
arXiv2018

The Weighted Davenport Constant of a group and a related extremal problem

For a finite abelian group $G$ written additively, and a non-empty subset $A\subset [1,\exp(G)-1]$ the weighted Davenport Constant of $G$ with respect to the set $A$, denoted $D_A(G)$, is the least positive integer $k$ for which the following holds: Given an arbitrary $G$-sequence $(x_1,\ldots,x_k)$, there exists a non-empty subsequence $(x_{i_1},\ldots,x_{i_t})$ along with $a_{j}\in A$ such that $\sum_{j=1}^t a_jx_{i_j}=0$. In this paper, we pose and study a natural new extremal problem that arises from the study of $D_A(G)$: For an integer $k\ge 2$, determine $\fD_G(k):=\min\{|A|: D_A(G)\le k\}$ (if the problem posed makes sense). It turns out that for $k$ `not-too-small', this is a well-posed problem and one of the most interesting cases occurs for $G=\Z_p$, the cyclic group of prime order, for which we obtain near optimal bounds for all $k$ (for sufficiently large primes $p$), and asymptotically tight (up to constants) bounds for $k=2,4$.

math.CO↗