Niranjan Balachandran

On an extremal hypergraph problem related to combinatorial batch codes

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 spans 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, Erd?s, and Sos and very recently discussed in the context of combinatorial batch codes (CBCs). In this short paper we obtain the following results. (i)Using a result due to Erd?s we are able to show m ( n , r , k ) = 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 ) = ? ( n r ) .This explicit construction improves on the non-constructive general lower bound obtained by Brown, Erd?s, and Sos 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 ) ( ? ( n k + 1 k - 1 ) ) for all k ? 8 and infinitely many values of n .(c)We show m ( n , 2 , k ) = O ( n 1 + 1 ? k 4 ? ) by using a result due to Bondy and Simonovits, and also show m ( n , 2 , k ) = ? ( n 3 2 ) for k = 6 , 7 , 8 by using a result of K?vari, Sos, and Turan.

Persistence based convergence rate analysis of consensus protocols for dynamic graph networks

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.

Randomized algorithms for stabilizing switching signals

In this article we study algorithmic synthesis of the class of stabilizing switching signals for discrete-time switched linear systems proposed in [12]. 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 [12]. 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.

Induced-bisecting families of bicolorings for hypergraphs

Abstract 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 n -dimensional { − 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 case where d ∈ Ω ( n 0 . 5 + ϵ ) and d is even, for any ϵ , 0 ϵ ≤ 0 . 5 .

The list distinguishing number of Kneser graphs

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Deterministic and probabilistic algorithms for stabilizing discrete-time switched linear systems ?

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Bisecting and D-secting families for set systems

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Bisecting families for set systems.

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Algorithmic synthesis of stabilizing switching signals for discrete-time switched linear systems.

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

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