Papers by Niranjan Balachandran
Electronic Journal of Combinatorics, Jun 21, 2019
Let L = { a 1 b 1 , . . . , as bs }, where for every i ∈ In this paper, we introduce and study th... more Let L = { a 1 b 1 , . . . , as bs }, where for every i ∈ In this paper, we introduce and study the notion of fractional L-intersecting families.
arXiv (Cornell University), Aug 24, 2021
Suppose F is a field and let a := (a 1 , a 2 , . . . ) be a sequence of non-zero elements in F. F... more Suppose F is a field and let a := (a 1 , a 2 , . . . ) be a sequence of non-zero elements in F. For a n := (a 1 , . . . , a n ), we consider the family M n (a) of n × n symmetric matrices M over 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 M n (a)-which can be naturally associated with transitive tournaments-have rank at least ⌊2n/3⌋ -1. We also show that if char(F) = 2 and M is a matrix chosen uniformly at random from M n (a), then with high probability rank(M ) ≥ 1 2 -o(1) n.
Almost full rank matrices arising from transitive tournaments
Linear & Multilinear Algebra, Jan 9, 2023
arXiv (Cornell University), Apr 25, 2017
Let B denote a set of bicolorings of [n], where each bicoloring is a mapping of the points in [n]... more Let 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 ∈ B, let YB = (B(1), . . . , B(n)). For each A ⊆ [n], let XA ∈ {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 ∈ B if XA, YB = 0. Given a set B of bicolorings, we study the minimum cardinality of a family A consisting of subsets of [n] such that every bicoloring in B has an unbiased representative in A.
arXiv (Cornell University), Nov 22, 2017
This article studies two problems related to observability and efficient constrained sensor place... more 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 -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.
arXiv (Cornell University), Nov 22, 2017
This article studies two problems related to observability and efficient constrained sensor place... more 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 -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.
arXiv (Cornell University), Oct 1, 2016
Two n-dimensional vectors A and B, A, B ∈ R n , are said to be trivially orthogonal if in every c... more Two n-dimensional vectors A and B, A, B ∈ R n , are said to be trivially orthogonal if in every coordinate i ∈ [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 V of ndimensional {-1, 0, 1} vectors, each containing exactly d non-zero entries, such that every 'possible' point A ∈ {0, 1} n in the Hamming cube has some V ∈ V which is orthogonal, but not trivially orthogonal, to A. We give asymptotically tight lower and (constructive) upper bounds for such a set V except for the even values of d ∈ Ω(n 0.5+ ), for any , 0 < ≤ 0.5.
arXiv (Cornell University), Jun 10, 2012
Let n, r, k be positive integers such that 3 ≤ k < n and 2 ≤ r ≤ k -1. Let m(n, r, k) denote the ... more Let n, r, k be positive integers such that 3 ≤ k < n and 2 ≤ r ≤ k -1. Let m(n, r, k) denote the maximum number of edges an r-uniform hypergraph on n vertices can have under the condition that any collection of i edges, span at least i vertices for all 1 ≤ i ≤ k. We are interested in the asymptotic nature of m(n, r, k) for fixed r and k as n → ∞. This problem is related to the forbidden hypergraph problem introduced by Brown, Erdős, and Sós and very recently discussed in the context of combinatorial batch codes. In this short paper we obtain the following results. (i) Using a result due to Erdős we are able to show m(n, k, r) = o(n r ) for 7 ≤ k, and 3 ≤ r ≤ k -1 -⌈log k⌉. This result is best possible with respect to the upper bound on r as we subsequently show through explicit construction that for 6 ≤ k, and k -⌈log k⌉ ≤ r ≤ k -1, m(n, r, k) = Θ(n r ). This explicit construction improves on the non-constructive general lower bound obtained by Brown, Erdős, and Sós for the considered parameter values. (ii) For 2-uniform CBCs we obtain the following results. (a) We provide exact value of m(n, 2, 5) for n ≥ 5. (b) Using a result of Lazebnik et al. regarding maximum size of graphs with large girth, we improve the existing lower bound on m(n, 2, k) (Ω(n k+1 k-1 )) for all k ≥ 8 and infinitely many values of n. ( ⌋ ) by using a result due to Bondy and Simonovits, and also show m(n, 2, k) = Θ(n 2 ) for k = 6, 7, 8 by using a result of Kövari, Sós, and Turán. We briefly make some definitions, and set up our notation.
Almost full rank matrices arising from transitive tournaments
Linear and Multilinear Algebra
arXiv (Cornell University), Oct 1, 2016
Two n-dimensional vectors A and B, A, B ∈ R n , are said to be trivially orthogonal if in every c... more Two n-dimensional vectors A and B, A, B ∈ R n , are said to be trivially orthogonal if in every coordinate i ∈ [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 V of ndimensional {-1, 0, 1} vectors, each containing exactly d non-zero entries, such that every 'possible' point A ∈ {0, 1} n in the Hamming cube has some V ∈ V which is orthogonal, but not trivially orthogonal, to A. We give asymptotically tight lower and (constructive) upper bounds for such a set V except for the even values of d ∈ Ω(n 0.5+ ), for any , 0 < ≤ 0.5.
arXiv (Cornell University), Nov 4, 2022
For a set L of positive proper fractions and a positive integer r ≥ 2, a fractional r-closed L-in... more For a set L of positive proper fractions and a positive integer r ≥ 2, a fractional r-closed L-intersecting family is a collection F ⊂ P([n]) with the property that for any 2 ≤ t ≤ r and A 1 , . . . , A t ∈ F there exists θ ∈ L such that In this paper we show that for r ≥ 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 ⌊ 3n 2 ⌋ -2 and that a maximal r-closed (1/2)-intersecting family is determined uniquely up to isomorphism.
arXiv (Cornell University), Nov 22, 2017
This article studies two problems related to observability and efficient constrained sensor place... more 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 -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.
The Harborth constant of a finite group G, denoted g(G), is the smallest integer k such that the ... more The Harborth constant of a finite group G, denoted g(G), is the smallest integer k such that the following holds: For A ⊆ G with |A| = k, there exists B ⊆ 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 g(G) of this combinatorial invariant for nonabelian groups and prove that if G is a dihedral group of order 2n with n ≥ 3, then g(G) = n + 2 if n is even and g(G) = 2n + 1 otherwise.
ArXiv, 2021
A bipartite graph $G(X,Y,E)$ with vertex partition $(X,Y)$ is said to have the Normalized Matchin... more A bipartite graph $G(X,Y,E)$ with vertex partition $(X,Y)$ is said to have the Normalized Matching Property (NMP) if for any subset $S\subseteq X$ we have $\frac{|N(S)|}{|Y|}\geq\frac{|S|}{|X|}$. In this paper, we prove the following results about the Normalized Matching Property. The random bipartite graph $\mathbb{G}(k,n,p)$ with $|X|=k,|Y|=n$, and $k\leq n<\exp(k)$, and each pair $(x,y)\in X\times Y$ being an edge in $\mathbb{G}$ independently with probability $p$ has $p=\frac{\log n}{k}$ as the threshold for NMP. This generalizes a classic result of Erdős-Rényi on the $\frac{\log n}{n}$ threshold for the existence of a perfect matching in $\mathbb{G}(n,n,p)$. 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 every $x\in X$ has degree at least $pn$ and every pair of distinct $x, x'\in X$ have at most $(1+\varepsilon)p^2n$ common neighbours. We show that...
Discrete Mathematics, 2021
Suppose A ⊂ Z n \{0}. A sequence x = (x 1 , . . . , x m ) of elements of Z n is called an A-weigh... more Suppose A ⊂ Z n \{0}. A sequence x = (x 1 , . . . , x m ) of elements of Z n is called an A-weighted Davenport Z-sequence if there exists a := (a 1 , . . . , a m ) ∈ (A ∪ {0}) m \ 0 m such that i a i x i = 0, where 0 m = (0, . . . , 0) ∈ Z m n . Similarly, the sequence x is called an A-weighted Erdős Z-sequence if there exists a = (a 1 , . . . , a m ) ∈ (A ∪ {0}) m with |Supp(a)| = n, such that i a i x i = 0, where Supp(a) := {i : a i = 0}. A Z n -sequence x is called k-restricted if no element of Z n appears more than k times in x. 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 and some very natural choices of the set A.
The Electronic Journal of Combinatorics, 2019
Let $L = \{\frac{a_1}{b_1}, \ldots , \frac{a_s}{b_s}\}$, where for every $i \in [s]$, $\frac{a_i}... more Let $L = \{\frac{a_1}{b_1}, \ldots , \frac{a_s}{b_s}\}$, where for every $i \in [s]$, $\frac{a_i}{b_i} \in [0,1)$ is an irreducible fraction. Let $\mathcal{F} = \{A_1, \ldots , A_m\}$ be a family of subsets of $[n]$. We say $\mathcal{F}$ is a fractional $L$-intersecting family if for every distinct $i,j \in [m]$, there exists an $\frac{a}{b} \in L$ such that $|A_i \cap A_j| \in \{ \frac{a}{b}|A_i|, \frac{a}{b} |A_j|\}$. In this paper, we introduce and study the notion of fractional $L$-intersecting families.
Discrete Applied Mathematics, 2019
Let B denote a set of bicolorings of [n], where each bicoloring is a mapping of the points in [n]... more Let 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 ∈ B, let YB = (B(1), . . . , B(n)). For each A ⊆ [n], let XA ∈ {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 ∈ B if XA, YB = 0. Given a set B of bicolorings, we study the minimum cardinality of a family A consisting of subsets of [n] such that every bicoloring in B has an unbiased representative in A.
Persistence based convergence rate analysis of consensus protocols for dynamic graph networks
European Journal of Control, 2016
Abstract This paper deals with the consensus problem of agents communicating via time-varying com... more Abstract This paper deals with the consensus problem of agents communicating via time-varying communication links in undirected graph networks. The highlight of the current work is to provide practically computable rates of convergence to consensus that hold for a large class of time-varying edge weights. A novel analysis technique based on classical notions of persistence of excitation and uniform complete observability is proposed. The new analysis technique for consensus laws under time-varying graphs provides explicit bounds on rate of convergence to consensus for single integrator dynamics. In the case of double integrators a minor modification to the standard relative state feedback law is shown to guarantee exponential convergence to consensus under similar assumptions as the single integrator case. The consensus problem is re-formulated in the edge agreement framework to which persistence of excitation based results apply. A novel application of results from Ramsey theory allows for proof of consensus and convergence rate estimation under switching graph topology. The current work connects classical results from nonlinear adaptive control and combinatorics to the modern theory of consensus over multi-agent networks.
Note: On an extremal hypergraph problem related to combinatorial batch codes
Discrete Applied Mathematics, 2014
In this article we study algorithmic synthesis of the class of stabilizing switching signals for ... more In this article we study algorithmic synthesis of the class of stabilizing switching signals for discrete-time switched linear systems proposed in . A weighted digraph is associated in a natural way to a switched system, and the switching signal is expressed as an infinite walk on this weighted digraph. We employ graph-theoretic tools and discuss different algorithms for designing walks whose corresponding switching signals satisfy the stabilizing switching conditions proposed in . We also address the issue of how likely/generic it is for a family of systems to admit stabilizing switching signals, and under mild assumptions give sufficient conditions for the same. Our solutions have both deterministic and probabilistic flavours.
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Papers by Niranjan Balachandran