arXiv subjects

Arindam Biswas

Publications and source records attributed to Arindam Biswas.

At least 19 recordsLinked to original sources

arXiv2026

Fast robbers on abelian Cayley graphs and digraphs

We study the fast-robber version of the Cops and Robbers game on finite strongly connected abelian Cayley digraphs, including undirected Cayley graphs as the symmetric case. For bounded out-degree $D$, we show that $c_{1,\infty}\left(\Gamma\right)=O_D\left(n^{1-\frac{1}{D}}\right)$; in the undirected case with $D\ge2$, this improves to the optimal exponent $1-\frac{1}{\left\lfloor \frac{D}{2}\right\rfloor}$. We also establish the degree-independent bound $c_{1,\infty}\left(\Gamma\right)=O\!\left(\frac{n\left(\log\log n\right)^2}{\log n}\right)$. These estimates follow from an optimized character-theoretic cyclic sweep over subgroup quotients.

math.CO↗

arXiv2026

Lattice point sumsets and asymptotic approximate groups

We establish new quantitative bounds for asymptotic approximate groups arising from finite subsets of lattices and, more generally, semi-linear subsets of abelian groups. Our approach combines Khovanskii's theorem on sumsets with Rogers and Zong bounds for the covering numbers.

math.NT↗

arXiv2026

On asymptotic approximate groups in nilpotent groups

Let $G$ be a group and let $A\subseteq G$ be non-empty. We call $A$ an asymptotic $(r,l)$-approximate group if, for a fixed dilation factor $r$, the larger product sets $A^{hr}$ can, for all sufficiently large $h$, be covered by a bounded number of left translates of $A^h$, with the bound $l$ independent of $h$. We show that, in virtually nilpotent groups, finite sets whose powers contain a symmetric word ball of radius comparable to $h$ are asymptotic approximate groups. We also prove a nonabelian semilinear-set analogue for certain infinite sets in these groups.

math.GR↗

arXiv2026

Uniform two-generator presentations for $SL_n(\mathbb{Z})$ with polynomial complexity bounds

We give a uniform explicit construction of finite two-generator presentations for the special linear groups over the integers in all ranks at least three. The construction builds on the generating-pair work of Conder--Liversidge--Vsemirnov and on a standard Tietze-elimination observation pointed out by Button. It recovers Trott's odd-rank generating pair and extends the same monomial/transvection form uniformly to even rank by a sign correction. After rebalancing, the construction has quadratic transvection words, quartically many relators, and sextic total relator length. We also derive several consequences, including infinite--infinite and finite--finite variants, consequences for congruence quotients, a presentation for the projective quotient, and an exact relator count, valid for both the unbalanced and balanced presentations.

math.GR↗

arXiv2026

On Chromatic Asymptotic Approximate Groups

We study a chromatic theory of asymptotic approximate groups for tuples of subsets of abelian groups, combining Nathanson's chromatic sumset formalism with asymptotic covering ideas from approximate group theory. This framework encodes simultaneous additive growth across several color classes. We show some general lifting and invariance principles, establish chromatic covering theorems for finite tuples and for tuples whose color classes are finite unions of unbounded linear sets, and obtain exact structure theorems for translated submonoids and finite-set-plus-submonoid sets. We also obtain sharper binomial bounds in the finite and unbounded-linear cases than the previous lattice-covering estimates. In the integer setting, we show that for each fixed threshold $t$, the threshold-$t$ chromatic layers form an asymptotic approximate family, using Nathanson's eventual interval-plus-edges description to obtain a uniform bound of size $r+2$ and prove an inhomogeneous extension for certain families.

math.CO↗

arXiv2025

On the cop number and the weak Meyniel conjecture for algebraic graphs

We show that the cop number of the Cayley sum graph of a finite group $G$ with respect to a symmetric subset $S$ is at most twice its degree when the graph is connected, undirected. We also prove that a similar bound holds for the cop number of generalised Cayley graphs and twisted Cayley sum graphs under some conditions. These extend a result of Frankl to such graphs. Using the above bounds and a result of Bollobás--Janson--Riordan, we show that the weak Meyniel conjecture holds for these algebraic graphs.

math.CO↗

arXiv2024

Counting of Shortest Paths in Cubic Grid

The enumeration of shortest paths in cubic grid is presented herein, which could have importance in image processing and also in the network sciences. The cubic grid considers three neighborhoods - namely, 6-, 18- and 26-neighborhood related to face connectivity, edge connectivity and vertex connectivity, respectively. The formulation for distance metrics is given. L1, D18, and L_$\infty$ are the three metrics for 6-neighborhood, 18-neighborhood and 26-neighborhood. The task is to count the number of minimal paths, based on given neighborhood relations, from any given point to any other, in the three-dimensional cubic grid. Based on the coordinate triplets describing the grid, the formulations for the three neighborhoods are presented in this work. The problem both of theoretical importance and has several practical aspects.

cs.DM↗

arXiv2024

Graph Expansion in Pruned Recurrent Neural Network Layers Preserve Performance

Expansion property of a graph refers to its strong connectivity as well as sparseness. It has been reported that deep neural networks can be pruned to a high degree of sparsity while maintaining their performance. Such pruning is essential for performing real time sequence learning tasks using recurrent neural networks in resource constrained platforms. We prune recurrent networks such as RNNs and LSTMs, maintaining a large spectral gap of the underlying graphs and ensuring their layerwise expansion properties. We also study the time unfolded recurrent network graphs in terms of the properties of their bipartite layers. Experimental results for the benchmark sequence MNIST, CIFAR-10, and Google speech command data show that expander graph properties are key to preserving classification accuracy of RNN and LSTM.

cs.LG↗

arXiv2023

Kernelizing Problems on Planar Graphs in Sublinear Space and Polynomial Time

In this paper, we devise a scheme for kernelizing, in sublinear space and polynomial time, various problems on planar graphs. The scheme exploits planarity to ensure that the resulting algorithms run in polynomial time and use O((sqrt(n) + k) log n) bits of space, where n is the number of vertices in the input instance and k is the intended solution size. As examples, we apply the scheme to Dominating Set and Vertex Cover. For Dominating Set, we also show that a well-known kernelization algorithm due to Alber et al. (JACM 2004) can be carried out in polynomial time and space O(k log n). Along the way, we devise restricted-memory procedures for computing region decompositions and approximating the aforementioned problems, which might be of independent interest.

cs.DS↗

arXiv2022

Logarithmic girth expander graphs of $SL_n(\mathbb F_p)$

We provide an explicit construction of finite 4-regular graphs $(Γ_k)_{k\in \mathbb N}$ with ${girth Γ_k\to\infty}$ as $k\to\infty$ and $\frac{diam Γ_k}{girth Γ_k}\leqslant D$ for some $D>0$ and all $k\in\mathbb{N}$. For each fixed dimension $n\geqslant 2,$ we find a pair of matrices in $SL_{n}(\mathbb{Z})$ such that (i) they generate a free subgroup, (ii)~their reductions $\bmod\, p$ generate $SL_{n}(\mathbb{F}_{p})$ for all sufficiently large primes $p$, (iii) the corresponding Cayley graphs of $SL_{n}(\mathbb{F}_{p})$ have girth at least $c_n\log p$ for some $c_n>0$. Relying on growth results (with no use of expansion properties of the involved graphs), we observe that the diameter of those Cayley graphs is at most $O(\log p)$. This gives infinite sequences of finite $4$-regular Cayley graphs of $SL_n(\mathbb F_p)$ as $p\to\infty$ with large girth and bounded diameter-by-girth ratio. These are the first explicit examples in all dimensions $n\geqslant 2$ (all prior examples were in $n=2$). Moreover, they happen to be expanders. Together with Margulis' and Lubotzky-Phillips-Sarnak's classical constructions, these new graphs are the only known explicit logarithmic girth Cayley graph expanders.

math.GR↗

arXiv2021

Space-Efficient FPT Algorithms

We prove algorithmic results showing that a number of natural parameterized problems are in the restricted-space parameterized classes Para-L and FPT+XL. The first class comprises problems solvable in f(k) n^{O(1)} time using g(k) + O(log n)) bits of space (k is the parameter and n is the input size; f and g are computable functions). The second class comprises problems solvable under the same time bound, but using g(k) log n bits of space instead. Earlier work on these classes has focused largely on their structural aspects and their relationships with various other classes. We complement this with Para-L and FPT+XL algorithms for a restriction of Hitting Set, some graph deletion problems where the target class has an infinite forbidden set characterization, a number of problems parameterized by vertex cover number, and Feedback Vertex Set.

cs.DS↗

arXiv2021

Sublinear-Space Approximation Algorithms for Max r-SAT

In the Max $r$-SAT problem, the input is a CNF formula with $n$ variables where each clause is a disjunction of at most $r$ literals. The objective is to compute an assignment which satisfies as many of the clauses as possible. While there are a large number of polynomial-time approximation algorithms for this problem, we take the viewpoint of space complexity following [Biswas et al., Algorithmica 2021] and design sublinear-space approximation algorithms for the problem. We show that the classical algorithm of [Lieberherr and Specker, JACM 1981] can be implemented to run in $n^{O(1)}$ time while using $(\log{n})$ bits of space. The more advanced algorithms use linear or semi-definite programming, and seem harder to carry out in sublinear space. We show that a more recent algorithm with approximation ratio $\sqrt{2}/2$ [Chou et al., FOCS 2020], designed for the streaming model, can be implemented to run in time $n^{O(r)}$ using $O(r \log{n})$ bits of space. While known streaming algorithms for the problem approximate optimum values and use randomization, our algorithms are deterministic and can output the approximately optimal assignments in sublinear space. For instances of Max $r$-SAT with planar incidence graphs, we devise a factor-$(1 - ε)$ approximation scheme which computes assignments in time $n^{O(r / ε)}$ and uses $\max\{\sqrt{n} \log{n}, (r / ε) \log^2{n}\}$ bits of space.

cs.DS↗

arXiv2021

A spectral bound for vertex-transitive graphs

For any finite, undirected, non-bipartite, vertex-transitive graph, we establish an explicit lower bound for the smallest eigenvalue of its normalised adjacency operator, which depends on the graph only through its degree and its isoperimetric constant.

math.CO↗

arXiv2021

Expansion in Cayley graphs, Cayley sum graphs and their twists

The Cayley graphs of finite groups are known to provide several examples of families of expanders, and some of them are Ramanujan graphs. Babai studied isospectral non-isomorphic Cayley graphs of the dihedral groups. Lubotzky, Samuels and Vishne proved that there are isospectral non-isomorphic Cayley graphs of $\mathrm{PSL}_d(\mathbb F_q)$ for every $d\geq 5$ ($d \neq 6$) and prime power $q> 2$. In this article, we focus on three variants of Cayley graphs, viz., the Cayley sum graphs, the twisted Cayley graphs, and the twisted Cayley sum graphs. We prove the existence of non-isomorphic expander families of bounded degree, whose spectra are related by the values of certain characters. We also provide several new examples of expander families, and examples of non-expanders and Ramanujan graphs formed by these three variants.

math.CO↗

arXiv2021

Approximation in (Poly-) Logarithmic Space

We develop new approximation algorithms for classical graph and set problems in the RAM model under space constraints. As one of our main results, we devise an algorithm for d-Hitting Set that runs in time n^{O(d^2 + d/ε})}, uses O((d^2 + d/ε) log n) bits of space, and achieves an approximation ratio of O((d/ε) n^ε) for any positive ε\leq 1 and any natural number d. In particular, this yields a factor-O(log n) approximation algorithm which runs in time n^{O(log n)} and uses O(log^2 n) bits of space (for constant d). As a corollary, we obtain similar bounds for Vertex Cover and several graph deletion problems. For bounded-multiplicity problem instances, one can do better. We devise a factor-2 approximation algorithm for Vertex Cover on graphs with maximum degree Δ, and an algorithm for computing maximal independent sets which both run in time n^{O(Δ)} and use O(Δlog n) bits of space. For the more general d-Hitting Set problem, we devise a factor-d approximation algorithm which runs in time n^{O(d δ^2)} and uses O(d δ^2 log n) bits of space on set families where each element appears in at most δsets. For Independent Set restricted to graphs with average degree d, we give a factor-(2d) approximation algorithm which runs in polynomial time and uses O(log n) bits of space. We also devise a factor-O(d^2) approximation algorithm for Dominating Set on d-degenerate graphs which runs in time n^{O(log n)} and uses O(log^2 n) bits of space. For d-regular graphs, we show how a known randomized factor-O(log d) approximation algorithm can be derandomized to run in time n^{O(1)} and use O(log n) bits of space. Our results use a combination of ideas from the theory of kernelization, distributed algorithms and randomized algorithms.

cs.DS↗

arXiv2020

On non-surjective word maps on $\mathrm{PSL}_{2}(\mathbb{F}_{q})$

Jambor--Liebeck--O'Brien showed that there exist non-proper-power word maps which are not surjective on $\mathrm{PSL}_{2}(\mathbb{F}_{q})$ for infinitely many $q$. This provided the first counterexamples to a conjecture of Shalev which stated that if a two-variable word is not a proper power of a non-trivial word, then the corresponding word map is surjective on $\mathrm{PSL}_2(\mathbb{F}_{q})$ for all sufficiently large $q$. Motivated by their work, we construct new examples of these types of non-surjective word maps. As an application, we obtain non-surjective word maps on the absolute Galois group of $\mathbb Q$.

math.GR↗

arXiv2020

Spectrum of twists of Cayley and Cayley sum graphs

Let $G$ be a finite group with $|G|\geq 4$ and $S$ be a subset of $G$. Given an automorphism $σ$ of $G$, the twisted Cayley graph $C(G, S)^σ$ (resp. the twisted Cayley sum graph $C_Σ(G, S)^σ$) is defined as the graph having $G$ as its set of vertices and the adjacent vertices of a vertex $g\in G$ are of the form $σ(gs)$ (resp. $σ(g^{-1} s)$) for some $s\in S$. If the twisted Cayley graph $C(G, S)^σ$ is undirected and connected, then we prove that the nontrivial spectrum of its normalised adjacency operator is bounded away from $-1$ and this bound depends only on its degree, the order of $σ$ and the vertex Cheeger constant of $C(G, S)^σ$. Moreover, if the twisted Cayley sum graph $C_Σ(G, S)^σ$ is undirected and connected, then we prove that the nontrivial spectrum of its normalised adjacency operator is bounded away from $-1$ and this bound depends only on its degree and the vertex Cheeger constant of $C_Σ(G, S)^σ$. We also study these twisted graphs with respect to anti-automorphisms, and obtain similar results. Further, we prove an analogous result for the Schreier graphs satisfying certain conditions.

math.CO↗

arXiv2020

Asymptotic behaviour of minimal complements

The notion of minimal complements was introduced by Nathanson in 2011 as a natural group-theoretic analogue of the metric concept of nets. Given two non-empty subsets $W,W'$ in a group $G$, the set $W'$ is said to be a complement to $W$ if $W\cdot W'=G$ and it is minimal if no proper subset of $W'$ is a complement to $W$. The inverse problem asks which sets may or not occur as minimal complements. We show some new results on the inverse problem and investigate how the study of the inverse problem naturally gives rise to questions about the asymptotic behaviour of these sets, providing partial answers to some of them.

math.CO↗