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Sundar Vishwanathan

Publications and source records attributed to Sundar Vishwanathan.

At least 19 recordsLinked to original sources
arXiv2026

Degree Lower Bounds for Torus Polynomials and $MAJORITY$ vs $ACC^0$

The class $ACC^0$ consists of Boolean functions that can be computed by constant-depth circuits of polynomial size with $AND, NOT$ and $MOD_m$ gates, where $m$ is a natural number. At the frontier of our understanding lies a widely believed conjecture asserting that $MAJORITY$ does not belong to $ACC^0$. A few years ago, Bhrushundi, Hosseini, Lovett and Rao (ITCS 2019) introduced torus polynomial approximations as an approach towards this conjecture. Torus polynomials approximate Boolean functions when the fractional part of their value on Boolean points is close to half the value of the function. They reduced the conjecture that $MAJORITY \notin ACC^0$ to a conjecture concerning the non-existence of low degree torus polynomials that approximate $MAJORITY$. We reduce the non-existence problem further, to a statement about finding feasible solutions for an infinite family of linear programs. The main advantage of this statement is that it allows for incremental progress, which means finding feasible solutions for successively larger collections of these programs. As an immediate first step, we find feasible solutions for a large class of these linear programs, leaving only a finite set for further consideration. Our method is inspired by the method of dual polynomials, which is used to study the approximate degree of Boolean functions. Using our method, we also propose a way to progress further. We prove several additional key results with the same method, including lower bounds for approximating the $AND$ function, lower bounds when the approximating polynomial is symmetric, showcasing the power of our machinery.

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arXiv2024

On the Mod-6 Town Rules

This note presents an upper bound of $1.252 n$ on the size of a set system that satisfies the mod-6 town rules. Under these rules the sizes of the sets are not congruent to $0\bmod 6$ while the sizes of all pairwise intersections are congruent to $ 0\bmod 6$.

math.CO↗
arXiv2022

Improved Bounds for Multicovering Hypergraphs

The minimum number of bicliques needed to cover the edge set of the complete graph on $n$ vertices is $\lceil \log_2 n \rceil$. The Graham-Pollak theorem states that at least $n-1$ bicliques are required to partition the edge set of the complete graph on $n$ vertices. In this paper, we provide improvements for the generalizations of coverings of graphs and hypergraphs for some specific multiplicities. We also study an extension of the Katona-Szemer\'edi theorem to $r$-uniform hypergraphs.

math.CO↗
arXiv2018

Maximum Matching on Trees in the Online Preemptive and the Incremental Dynamic Graph Models

We study the Maximum Cardinality Matching (MCM) and the Maximum Weight Matching (MWM) problems, on trees and on some special classes of graphs, in the Online Preemptive and the Incremental Dynamic Graph models. In the {\em Online Preemptive} model, the edges of a graph are revealed one by one and the algorithm is required to always maintain a valid matching. On seeing an edge, the algorithm has to either accept or reject the edge. If accepted, then the adjacent edges are discarded, and all rejections are permanent. In this model, the complexity of the problems is settled for deterministic algorithms. Epstein et al. gave a $5.356$-competitive randomized algorithm for MWM, and also proved a lower bound of $1.693$ for MCM. The same lower bound applies for MWM. In this paper we show that some of the results can be improved in the case of trees and some special classes of graphs. In the online preemptive model, we present a $64/33$-competitive (in expectation) randomized algorithm for MCM on trees. Inspired by the above mentioned algorithm for MCM, we present the main result of the paper, a randomized algorithm for MCM with a "worst case" update time of $O(1)$, in the incremental dynamic graph model, which is $3/2$-approximate (in expectation) on trees, and $1.8$-approximate (in expectation) on general graphs with maximum degree $3$. Note that this algorithm works only against an oblivious adversary. Hence, we derandomize this algorithm, and give a $(3/2 + ε)$-approximate deterministic algorithm for MCM on trees, with an amortized update time of $O(1/ε)$. We also present a minor result for MWM in the online preemptive model, a $3$-competitive (in expectation) randomized algorithm on growing trees (where the input revealed upto any stage is always a tree, i.e. a new edge never connects two disconnected trees).

cs.DS↗
arXiv2017

Bounds for the Graham-Pollak Theorem for Hypergraphs

Let $f_r(n)$ represent the minimum number of complete $r$-partite $r$-graphs required to partition the edge set of the complete $r$-uniform hypergraph on $n$ vertices. The Graham-Pollak theorem states that $f_2(n)=n-1$. An upper bound of $(1+o(1)){n \choose \lfloor{\frac{r}{2}}\rfloor}$ was known. Recently this was improved to $\frac{14}{15}(1+o(1)){n \choose \lfloor{\frac{r}{2}}\rfloor}$ for even $r \geq 4$. A bound of $\bigg[\frac{r}{2}(\frac{14}{15})^{\frac{r}{4}}+o(1)\bigg](1+o(1)){n \choose \lfloor{\frac{r}{2}}\rfloor}$ was also proved recently. The smallest odd $r$ for which $c_r < 1$ that was known was for $r=295$. In this note we improve this to $c_{113}<1$ and also give better upper bounds for $f_r(n)$, for small values of even $r$.

math.CO↗
arXiv2015

On Randomized Algorithms for Matching in the Online Preemptive Model

We investigate the power of randomized algorithms for the maximum cardinality matching (MCM) and the maximum weight matching (MWM) problems in the online preemptive model. In this model, the edges of a graph are revealed one by one and the algorithm is required to always maintain a valid matching. On seeing an edge, the algorithm has to either accept or reject the edge. If accepted, then the adjacent edges are discarded. The complexity of the problem is settled for deterministic algorithms. Almost nothing is known for randomized algorithms. A lower bound of $1.693$ is known for MCM with a trivial upper bound of $2$. An upper bound of $5.356$ is known for MWM. We initiate a systematic study of the same in this paper with an aim to isolate and understand the difficulty. We begin with a primal-dual analysis of the deterministic algorithm due to McGregor. All deterministic lower bounds are on instances which are trees at every step. For this class of (unweighted) graphs we present a randomized algorithm which is $\frac{28}{15}$-competitive. The analysis is a considerable extension of the (simple) primal-dual analysis for the deterministic case. The key new technique is that the distribution of primal charge to dual variables depends on the "neighborhood" and needs to be done after having seen the entire input. The assignment is asymmetric: in that edges may assign different charges to the two end-points. Also the proof depends on a non-trivial structural statement on the performance of the algorithm on the input tree. The other main result of this paper is an extension of the deterministic lower bound of Varadaraja to a natural class of randomized algorithms which decide whether to accept a new edge or not using independent random choices.

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arXiv2014

Approximating the Regular Graphic TSP in near linear time

We present a randomized approximation algorithm for computing traveling salesperson tours in undirected regular graphs. Given an $n$-vertex, $k$-regular graph, the algorithm computes a tour of length at most $\left(1+\frac{7}{\ln k-O(1)}\right)n$, with high probability, in $O(nk \log k)$ time. This improves upon a recent result by Vishnoi (\cite{Vishnoi12}, FOCS 2012) for the same problem, in terms of both approximation factor, and running time. The key ingredient of our algorithm is a technique that uses edge-coloring algorithms to sample a cycle cover with $O(n/\log k)$ cycles with high probability, in near linear time. Additionally, we also give a deterministic $\frac{3}{2}+O\left(\frac{1}{\sqrt{k}}\right)$ factor approximation algorithm running in time $O(nk)$.

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arXiv2014

On Randomized Memoryless Algorithms for the Weighted $k$-server Problem

The weighted $k$-server problem is a generalization of the $k$-server problem in which the cost of moving a server of weight $β_i$ through a distance $d$ is $β_i\cdot d$. The weighted server problem on uniform spaces models caching where caches have different write costs. We prove tight bounds on the performance of randomized memoryless algorithms for this problem on uniform metric spaces. We prove that there is an $α_k$-competitive memoryless algorithm for this problem, where $α_k=α_{k-1}^2+3α_{k-1}+1$; $α_1=1$. On the other hand we also prove that no randomized memoryless algorithm can have competitive ratio better than $α_k$. To prove the upper bound of $α_k$ we develop a framework to bound from above the competitive ratio of any randomized memoryless algorithm for this problem. The key technical contribution is a method for working with potential functions defined implicitly as the solution of a linear system. The result is robust in the sense that a small change in the probabilities used by the algorithm results in a small change in the upper bound on the competitive ratio. The above result has two important implications. Firstly this yields an $α_k$-competitive memoryless algorithm for the weighted $k$-server problem on uniform spaces. This is the first competitive algorithm for $k>2$ which is memoryless. Secondly, this helps us prove that the Harmonic algorithm, which chooses probabilities in inverse proportion to weights, has a competitive ratio of $kα_k$.

cs.DS↗
arXiv2014

Metrical Service Systems with Multiple Servers

We study the problem of metrical service systems with multiple servers (MSSMS), which generalizes two well-known problems -- the $k$-server problem, and metrical service systems. The MSSMS problem is to service requests, each of which is an $l$-point subset of a metric space, using $k$ servers, with the objective of minimizing the total distance traveled by the servers. Feuerstein initiated a study of this problem by proving upper and lower bounds on the deterministic competitive ratio for uniform metric spaces. We improve Feuerstein's analysis of the upper bound and prove that his algorithm achieves a competitive ratio of $k({{k+l}\choose{l}}-1)$. In the randomized online setting, for uniform metric spaces, we give an algorithm which achieves a competitive ratio $\mathcal{O}(k^3\log l)$, beating the deterministic lower bound of ${{k+l}\choose{l}}-1$. We prove that any randomized algorithm for MSSMS on uniform metric spaces must be $Ω(\log kl)$-competitive. We then prove an improved lower bound of ${{k+2l-1}\choose{k}}-{{k+l-1}\choose{k}}$ on the competitive ratio of any deterministic algorithm for $(k,l)$-MSSMS, on general metric spaces. In the offline setting, we give a pseudo-approximation algorithm for $(k,l)$-MSSMS on general metric spaces, which achieves an approximation ratio of $l$ using $kl$ servers. We also prove a matching hardness result, that a pseudo-approximation with less than $kl$ servers is unlikely, even for uniform metric spaces. For general metric spaces, we highlight the limitations of a few popular techniques, that have been used in algorithm design for the $k$-server problem and metrical service systems.

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arXiv2013

Improved bounds on the sandpile diffusions on Grid graphs

The Abelian Sandpile Model is a discrete diffusion process defined on graphs (Dhar [10], Dhar et al. [11]) which serves as the standard model of self-organized criticality. The transience class of a sandpile is defined as the maximum number of particles that can be added without making the system recurrent ([3]). Using elementary combinatorial arguments and symmetry properties, Babai and Gorodezky (SODA 2007,[2]) demonstrated a bound of O(n^30) on the transience class of an nxn grid. This was later improved by Choure and Vishwanathan (SODA 2012,[7]) to O(n^7) using techniques based on harmonic functions on graphs. We improve this bound to O(n^7 log n). We also demonstrate tight bounds on certain resistance ratios over grid networks. The tools used for deriving these bounds may be of independent interest.

math-ph↗
arXiv2012

On graph parameters guaranteeing fast Sandpile diffusion

The Abelian Sandpile Model is a discrete diffusion process defined on graphs (Dhar \cite{DD90}, Dhar et al. \cite{DD95}) which serves as the standard model of self-organized criticality. The transience class of a sandpile is defined as the maximum number of particles that can be added without making the system recurrent (\cite{BT05}). We demonstrate a class of sandpile which have polynomially bound transience classes by identifying key graph properties that play a role in the rapid diffusion process. These are the volume growth parameters, boundary regularity type properties and non-empty interior type constraints. This generalizes a previous result by Babai and Gorodezky (SODA 2007,\cite{LB07}), in which they establish polynomial bounds on $n \times n$ grid. Indeed the properties we show are based on ideas extracted from their proof as well as the continuous analogs in complex analysis. We conclude with a discussion on the notion of degeneracy and dimensions in graphs.

cs.DM↗
arXiv2012

Random Walks, Electric Networks and The Transience Class problem of Sandpiles

The Abelian Sandpile Model is a discrete diffusion process defined on graphs (Dhar \cite{DD90}, Dhar et al. \cite{DD95}) which serves as the standard model of \textit{self-organized criticality}. The transience class of a sandpile is defined as the maximum number of particles that can be added without making the system recurrent (\cite{BT05}). We develop the theory of discrete diffusions in contrast to continuous harmonic functions on graphs and establish deep connections between standard results in the study of random walks on graphs and sandpiles on graphs. Using this connection and building other necessary machinery we improve the main result of Babai and Gorodezky (SODA 2007,\cite{LB07}) of the bound on the transience class of an $n \times n$ grid, from $O(n^{30})$ to $O(n^{7})$. Proving that the transience class is small validates the general notion that for most natural phenomenon, the time during which the system is transient is small. In addition, we use the machinery developed to prove a number of auxiliary results. We exhibit an equivalence between two other tessellations of plane, the honeycomb and triangular lattices. We give general upper bounds on the transience class as a function of the number of edges to the sink. Further, for planar sandpiles we derive an explicit algebraic expression which provably approximates the transience class of $G$ to within $O(|E(G)|)$. This expression is based on the spectrum of the Laplacian of the dual of the graph $G$. We also show a lower bound of $Ω(n^{3})$ on the transience class on the grid improving the obvious bound of $Ω(n^{2})$.

cs.DM↗
arXiv2009

Biclique Coverings and the Chromatic Number

Consider a graph $G$ with chromatic number $k$ and a collection of complete bipartite graphs, or bicliques, that cover the edges of $G$. We prove the following two results: \medskip \noindent $\bullet$ If the bicliques partition the edges of $G$, then their number is at least $2^{\sqrt{\log_2 k}}$. This is the first improvement of the easy lower bound of $\log_2 k$, while the Alon-Saks-Seymour conjecture states that this can be improved to $k-1$. \medskip \noindent $\bullet$ The sum of the orders of the bicliques is at least $(1-o(1))k\log_2 k$. This generalizes, in asymptotic form, a result of Katona and Szemerédi who proved that the minimum is $k\log_2 k$ when $G$ is a clique.

math.CO↗
arXiv2008

Circumference, Chromatic Number and Online Coloring

Erdös conjectured that if $G$ is a triangle free graph of chromatic number at least $k\geq 3$, then it contains an odd cycle of length at least $k^{2-o(1)}$ \cite{sudakovverstraete, verstraete}. Nothing better than a linear bound (\cite{gyarfas}, Problem 5.1.55 in \cite{West}) was so far known. We make progress on this conjecture by showing that $G$ contains an odd cycle of length at least $O(k\log\log k)$. Erdös' conjecture is known to hold for graphs with girth at least 5. We show that if a girth 4 graph is $C_5$ free, then Erdös' conjecture holds. When the number of vertices is not too large we can prove better bounds on $χ$. We also give bounds on the chromatic number of graphs with at most $r$ cycles of length $1\bmod k$, or at most $s$ cycles of length $2\bmod k$, or no cycles of length $3\bmod k$. Our techniques essentially consist of using a depth first search tree to decompose the graph into ordered paths, which are then fed to an online coloring algorithm. Using this technique we give simple proofs of some old results, and also obtain several simpler results. We also obtain a lower bound on the number of colors an online coloring algorithm needs to use on triangle free graphs.

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arXiv2006

The Common Prefix Problem On Trees

We present a theoretical study of a problem arising in database query optimization, which we call as The Common Prefix Problem. We present a $(1-o(1))$ factor approximation algorithm for this problem, when the underlying graph is a binary tree. We then use a result of Feige and Kogan to show that even on stars, the problem is hard to approximate.

cs.DS↗
arXiv2006

Approximation Algorithms for the Bipartite Multi-cut Problem

We introduce the {\it Bipartite Multi-cut} problem. This is a generalization of the {\it st-Min-cut} problem, is similar to the {\it Multi-cut} problem (except for more stringent requirements) and also turns out to be an immediate generalization of the {\it Min UnCut} problem. We prove that this problem is {\bf NP}-hard and then present LP and SDP based approximation algorithms. While the LP algorithm is based on the Garg-Vazirani-Yannakakis algorithm for {\it Multi-cut}, the SDP algorithm uses the {\it Structure Theorem} of $\ell_2^2$ Metrics.

cs.CC↗