arXiv subjects

Suryajith Chillara

Publications and source records attributed to Suryajith Chillara.

12 recordsLinked to original sources

arXiv2026

Shortest Paths with Linear Edge Weights

We study shortest paths in directed graphs whose edge weights are of the form $$ \mathsf{wt}(e) = a_{e,1} λ_1 + a_{e,2} λ_2 + a_{e,3} λ_3 + \cdots + a_{e,d} λ_d + a_{e,d+1}.$$ Here, each $a_{e,i}\in\mathbb{R}$ is a fixed constant for each edge $e$, whereas each $λ_i\in\mathbb{R}$ is common across the entire graph. So, there could be different shortest paths in the graph for different values of the $λ_i$'s. This is called the Parametric Shortest Paths problem, and has been studied since the 1980s. For $d=1$, Carstensen (1983) showed that the number of shortest paths in $n$-vertex graphs is at most $n^{O(\log n)}$. She also proved a matching lower bound of $n^{Ω(\log n)}$, later refined by Mulmuley & Shah (2001). For $d=2$, Gajjar & Radhakrishnan (2019) showed an upper bound of $n^{O(\log^2 n)}$. Barth, Funke & Proissl (2022) generalized their result to prove an upper bound of $n^{O_d(\log^d n)}$ for all positive integers $d$. The lower bound did not undergo any improvement over the years. In this paper, we close this long line of research by showing an $n^{O(d\log n)}$ upper bound for all positive integers $d$, exponentially improving the previous upper bound. We observe that a matching lower bound of $n^{Ω(d\log n)}$ can be obtained from earlier works. We also show that our proof can be adapted to work for undirected graphs with positive edge weights. Furthermore, for directed graphs whose edge weights are univariate polynomials of degree at most $q$, we prove an upper bound of $n^{O(\log{n}+\log{q})}$. Finally, building upon work on the Point Location problem by Ezra, Har-Peled, Kaplan & Sharir (2020), we construct a Shortest Path Oracle which takes as input a point $\overline{x}\in \mathbb{R}^d$, and outputs a shortest path at $\overlineλ=\overline{x}$ in sublinear time (for a wide regime of $d$).

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arXiv2026

From Submodularity to Matrix Determinants: Strengthening Han's, Szász's, and Fischer's Inequalities

Dembo, Cover, and Thomas (1991) developed an elegant information-theoretic framework for proving determinantal inequalities for positive definite matrices, which relies on the structural inequalities of differential entropy. Submodular functions, which subsume entropy, inherently satisfy these structural inequalities because they obey generalized forms of the fundamental properties of entropy -- a chain rule and the property that conditioning reduces the function's value (under an appropriate definition of conditioning). Applying subadditivity, Han's inequality (1978), and partition subadditivity (i.e., subadditivity over a partition) yields Hadamard's, Szász's, and Fischer's inequalities, respectively. Furthermore, this framework recovers Ky Fan's inequality (1955), a strengthening of Hadamard's inequality. This improvement fundamentally arises because conditional subadditivity yields a tighter upper bound on the joint entropy than the one obtained via unconditional subadditivity. In this paper, we establish conditional strengthenings of Han's inequality and partition subadditivity in the general setting of submodular functions. We derive equality conditions for these strengthened bounds and characterize when they strictly improve their unconditional counterparts. We specialize these results to differential entropy and apply them to establish strengthened versions of Szász's and Fischer's inequalities. The strengthening of Szász's inequality recovers Ky Fan's inequality as a special case, and is strictly stronger than the classical Szász's inequality for any non-diagonal positive definite matrix. We also derive an inequality concerning eigenvalues, which generalizes and strictly strengthens a corresponding eigenvalue inequality of Ky Fan. We provide numerical examples to explicitly illustrate the tightness of our proposed matrix determinantal bounds.

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arXiv2026

Generalized Information Inequalities via Submodularity, and Two Combinatorial Problems

It is well known that there is a strong connection between entropy inequalities and submodularity, since the entropy of a collection of random variables is a submodular function. Unifying frameworks for information inequalities arising from submodularity were developed by Madiman and Tetali (2010) and Sason (2022). Madiman and Tetali (2010) established strong and weak fractional inequalities that subsume classical results such as Han's inequality and Shearer's lemma. Sason (2022) introduced a convex-functional framework for generalizing Han's inequality, and derived unified inequalities for submodular and supermodular functions. In this work, we build on these frameworks and make three contributions. First, we establish convex-functional generalizations of the strong and weak Madiman and Tetali inequalities for submodular functions. Second, using a special case of the strong Madiman-Tetali inequality, we derive a new Loomis-Whitney-type projection inequality for finite point sets in $\mathbb{R}^d$, which improves upon the classical Loomis-Whitney bound by incorporating slice-level structural information. Finally, we study an extremal graph theory problem that recovers and extends the previously known results of Sason (2022) and Boucheron et al., employing Shearer's lemma in contrast to the use of Han's inequality in those works.

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arXiv2025

Fractional Subadditivity of Submodular Functions: Equality Conditions and Their Applications

Submodular functions are known to satisfy various forms of fractional subadditivity. This work investigates the conditions for equality to hold exactly or approximately in the fractional subadditivity of submodular functions. We establish that a small gap in the inequality implies that the function is close to being modular, and that the gap is zero if and only if the function is modular. We then present natural implications of these results for special cases of submodular functions, such as entropy, relative entropy, and matroid rank. As a consequence, we characterize the necessary and sufficient conditions for equality to hold in Shearer's lemma, recovering a result of Ellis \emph{et al.} (2016) as a special case. We leverage our results to propose a new multivariate mutual information, which generalizes Watanabe's total correlation (1960), Han's dual total correlation (1978), and Csiszár and Narayan's shared information (2004), and analyze its properties. Among these properties, we extend Watanabe's characterization of total correlation as the maximum correlation over partitions to fractional partitions. When applied to matrix determinantal inequalities for positive definite matrices, our results recover the equality conditions of the classical determinantal inequalities of Hadamard, Szász, and Fischer as special cases.

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arXiv2022

On Hardness of Testing Equivalence to Sparse Polynomials Under Shifts

We say that two given polynomials $f, g \in R[X]$, over a ring $R$, are equivalent under shifts if there exists a vector $a \in R^n$ such that $f(X+a) = g(X)$. Grigoriev and Karpinski (FOCS 1990), Lakshman and Saunders (SICOMP, 1995), and Grigoriev and Lakshman (ISSAC 1995) studied the problem of testing polynomial equivalence of a given polynomial to any $t$-sparse polynomial, over the rational numbers, and gave exponential time algorithms. In this paper, we provide hardness results for this problem. Formally, for a ring $R$, let $\mathrm{SparseShift}_R$ be the following decision problem. Given a polynomial $P(X)$, is there a vector $a$ such that $P(X+a)$ contains fewer monomials than $P(X)$. We show that $\mathrm{SparseShift}_R$ is at least as hard as checking if a given system of polynomial equations over $R[x_1,\ldots, x_n]$ has a solution (Hilbert's Nullstellensatz). As a consequence of this reduction, we get the following results. 1. $\mathrm{SparseShift}_\mathbb{Z}$ is undecidable. 2. For any ring $R$ (which is not a field) such that $\mathrm{HN}_R$ is $\mathrm{NP}_R$-complete over the Blum-Shub-Smale model of computation, $\mathrm{SparseShift}_{R}$ is also $\mathrm{NP}_{R}$-complete. In particular, $\mathrm{SparseShift}_{\mathbb{Z}}$ is also $\mathrm{NP}_{\mathbb{Z}}$-complete. We also study the gap version of the $\mathrm{SparseShift}_R$ and show the following. 1. For every function $β: \mathbb{N}\to\mathbb{R}_+$ such that $β\in o(1)$, $N^β$-gap-$\mathrm{SparseShift}_\mathbb{Z}$ is also undecidable (where $N$ is the input length). 2. For $R=\mathbb{F}_p, \mathbb{Q}, \mathbb{R}$ or $\mathbb{Z}_q$ and for every $β>1$ the $β$-gap-$\mathrm{SparseShift}_R$ problem is $\mathrm{NP}$-hard.

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arXiv2021

Functional lower bounds for restricted arithmetic circuits of depth four

Recently, Forbes, Kumar and Saptharishi [CCC, 2016] proved that there exists an explicit $d^{O(1)}$-variate and degree $d$ polynomial $P_{d}\in VNP$ such that if any depth four circuit $C$ of bounded formal degree $d$ which computes a polynomial of bounded individual degree $O(1)$, that is functionally equivalent to $P_d$, then $C$ must have size $2^{Ω(\sqrt{d}\log{d})}$. The motivation for their work comes from Boolean Circuit Complexity. Based on a characterization for $ACC^0$ circuits by Yao [FOCS, 1985] and Beigel and Tarui [CC, 1994], Forbes, Kumar and Saptharishi [CCC, 2016] observed that functions in $ACC^0$ can also be computed by algebraic $Σ\mathord{\wedge}ΣΠ$ circuits (i.e., circuits of the form -- sums of powers of polynomials) of $2^{\log^{O(1)}n}$ size. Thus they argued that a $2^{ω(\log^{O(1)}{n})}$ "functional" lower bound for an explicit polynomial $Q$ against $Σ\mathord{\wedge}ΣΠ$ circuits would imply a lower bound for the "corresponding Boolean function" of $Q$ against non-uniform $ACC^0$. In their work, they ask if their lower bound be extended to $Σ\mathord{\wedge}ΣΠ$ circuits. In this paper, for large integers $n$ and $d$ such that $ω(\log^2n)\leq d\leq n^{0.01}$, we show that any $Σ\mathord{\wedge}ΣΠ$ circuit of bounded individual degree at most $O\left(\frac{d}{k^2}\right)$ that functionally computes Iterated Matrix Multiplication polynomial $IMM_{n,d}$ ($\in VP$) over $\{0,1\}^{n^2d}$ must have size $n^{Ω(k)}$. Since Iterated Matrix Multiplication $IMM_{n,d}$ over $\{0,1\}^{n^2d}$ is functionally in $GapL$, improvement of the afore mentioned lower bound to hold for quasipolynomially large values of individual degree would imply a fine-grained separation of $ACC^0$ from $GapL$.

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arXiv2021

New Exponential Size Lower Bounds against Depth Four Circuits of Bounded Individual Degree

Kayal, Saha and Tavenas [Theory of Computing, 2018] showed that for all large enough integers $n$ and $d$ such that $d\geq ω(\log{n})$, any syntactic depth four circuit of bounded individual degree $δ= o(d)$ that computes the Iterated Matrix Multiplication polynomial ($IMM_{n,d}$) must have size $n^{Ω\left(\sqrt{d/δ}\right)}$. Unfortunately, this bound deteriorates as the value of $δ$ increases. Further, the bound is superpolynomial only when $δ$ is $o(d)$. It is natural to ask if the dependence on $δ$ in the bound could be weakened. Towards this, in an earlier result [STACS, 2020], we showed that for all large enough integers $n$ and $d$ such that $d = Θ(\log^2{n})$, any syntactic depth four circuit of bounded individual degree $δ\leq n^{0.2}$ that computes $IMM_{n,d}$ must have size $n^{Ω(\log{n})}$. In this paper, we make further progress by proving that for all large enough integers $n$ and $d$, and absolute constants $a$ and $b$ such that $ω(\log^2n)\leq d\leq n^{a}$, any syntactic depth four circuit of bounded individual degree $δ\leq n^{b}$ that computes $IMM_{n,d}$ must have size $n^{Ω(\sqrt{d})}$. Our bound is obtained by carefully adapting the proof of Kumar and Saraf [SIAM J. Computing, 2017] to the complexity measure introduced in our earlier work [STACS, 2020].

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arXiv2018

A Near-Optimal Depth-Hierarchy Theorem for Small-Depth Multilinear Circuits

We study the size blow-up that is necessary to convert an algebraic circuit of product-depth $Δ+1$ to one of product-depth $Δ$ in the multilinear setting. We show that for every positive $Δ= Δ(n) = o(\log n/\log \log n),$ there is an explicit multilinear polynomial $P^{(Δ)}$ on $n$ variables that can be computed by a multilinear formula of product-depth $Δ+1$ and size $O(n)$, but not by any multilinear circuit of product-depth $Δ$ and size less than $\exp(n^{Ω(1/Δ)})$. This result is tight up to the constant implicit in the double exponent for all $Δ= o(\log n/\log \log n).$ This strengthens a result of Raz and Yehudayoff (Computational Complexity 2009) who prove a quasipolynomial separation for constant-depth multilinear circuits, and a result of Kayal, Nair and Saha (STACS 2016) who give an exponential separation in the case $Δ= 1.$ Our separating examples may be viewed as algebraic analogues of variants of the Graph Reachability problem studied by Chen, Oliveira, Servedio and Tan (STOC 2016), who used them to prove lower bounds for constant-depth Boolean circuits.

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arXiv2017

Small-depth Multilinear Formula Lower Bounds for Iterated Matrix Multiplication, with Applications

In this paper, we study the algebraic formula complexity of multiplying $d$ many $2\times 2$ matrices, denoted $\mathrm{IMM}_{d}$, and show that the well-known divide-and-conquer algorithm cannot be significantly improved at any depth, as long as the formulas are multilinear. Formally, for each depth $Δ\leq \log d$, we show that any product-depth $Δ$ multilinear formula for $\mathrm{IMM}_d$ must have size $\exp(Ω(Δd^{1/Δ})).$ It also follows from this that any multilinear circuit of product-depth $Δ$ for the same polynomial of the above form must have a size of $\exp(Ω(d^{1/Δ})).$ In particular, any polynomial-sized multilinear formula for $\mathrm{IMM}_d$ must have depth $Ω(\log d)$, and any polynomial-sized multilinear circuit for $\mathrm{IMM}_d$ must have depth $Ω(\log d/\log \log d).$ Both these bounds are tight up to constant factors. 1. Depth-reduction: A well-known result of Brent (JACM 1974) implies that any formula of size $s$ can be converted to one of size $s^{O(1)}$ and depth $O(\log s)$; further, this reduction continues to hold for multilinear formulas. Our lower bound implies that any depth-reduction in the multilinear setting cannot reduce the depth to $o(\log s)$ without a superpolynomial blow-up in size. 2. Separations from general formulas: Our result, along with a non-trivial upper bound for $\mathrm{IMM}_{d}$ implied by a result of Gupta, Kamath, Kayal and Saptharishi (SICOMP 2016), shows that for any size $s$ and product-depth $Δ= o(\log s),$ general formulas of size $s$ and product-depth $Δ$ cannot be converted to multilinear formulas of size $s^{ω(1)}$ and product-depth $Δ,$ when the underlying field has characteristic zero.

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arXiv2017

The Chasm at Depth Four, and Tensor Rank : Old results, new insights

Agrawal and Vinay [AV08] showed how any polynomial size arithmetic circuit can be thought of as a depth four arithmetic circuit of subexponential size. The resulting circuit size in this simulation was more carefully analyzed by Korian [Koiran] and subsequently by Tavenas [Tav13]. We provide a simple proof of this chain of results. We then abstract the main ingredient to apply it to formulas and constant depth circuits, and show more structured depth reductions for them. In an apriori surprising result, Raz [Raz10] showed that for any $n$ and $d$, such that $ ω(1) \leq d \leq O\left(\frac{\log n}{\log\log n}\right)$, constructing explicit tensors $T:[n]^d \rightarrow F$ of high enough rank would imply superpolynomial lower bounds for arithmetic formulas over the field $F$. Using the additional structure we obtain from our proof of the depth reduction for arithmetic formulas, we give a new and arguably simpler proof of this connection. We also extend this result for homogeneous formulas to show that, in fact, the connection holds for any $d$ such that $ω(1) \leq d \leq n^{o(1)}$.

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arXiv2013

On the Limits of Depth Reduction at Depth 3 Over Small Finite Fields

Recently, Gupta et.al. [GKKS2013] proved that over Q any $n^{O(1)}$-variate and $n$-degree polynomial in VP can also be computed by a depth three $ΣΠΣ$ circuit of size $2^{O(\sqrt{n}\log^{3/2}n)}$. Over fixed-size finite fields, Grigoriev and Karpinski proved that any $ΣΠΣ$ circuit that computes $Det_n$ (or $Perm_n$) must be of size $2^{Ω(n)}$ [GK1998]. In this paper, we prove that over fixed-size finite fields, any $ΣΠΣ$ circuit for computing the iterated matrix multiplication polynomial of $n$ generic matrices of size $n\times n$, must be of size $2^{Ω(n\log n)}$. The importance of this result is that over fixed-size fields there is no depth reduction technique that can be used to compute all the $n^{O(1)}$-variate and $n$-degree polynomials in VP by depth 3 circuits of size $2^{o(n\log n)}$. The result [GK1998] can only rule out such a possibility for depth 3 circuits of size $2^{o(n)}$. We also give an example of an explicit polynomial ($NW_{n,ε}(X)$) in VNP (not known to be in VP), for which any $ΣΠΣ$ circuit computing it (over fixed-size fields) must be of size $2^{Ω(n\log n)}$. The polynomial we consider is constructed from the combinatorial design. An interesting feature of this result is that we get the first examples of two polynomials (one in VP and one in VNP) such that they have provably stronger circuit size lower bounds than Permanent in a reasonably strong model of computation. Next, we prove that any depth 4 $ΣΠ^{[O(\sqrt{n})]}ΣΠ^{[\sqrt{n}]}$ circuit computing $NW_{n,ε}(X)$ (over any field) must be of size $2^{Ω(\sqrt{n}\log n)}$. To the best of our knowledge, the polynomial $NW_{n,ε}(X)$ is the first example of an explicit polynomial in VNP such that it requires $2^{Ω(\sqrt{n}\log n)}$ size depth four circuits, but no known matching upper bound.

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arXiv2013

Depth-4 Lower Bounds, Determinantal Complexity : A Unified Approach

Tavenas has recently proved that any n^{O(1)}-variate and degree n polynomial in VP can be computed by a depth-4 circuit of size 2^{O(\sqrt{n}\log n)}. So to prove VP not equal to VNP, it is sufficient to show that an explicit polynomial in VNP of degree n requires 2^{ω(\sqrt{n}\log n)} size depth-4 circuits. Soon after Tavenas's result, for two different explicit polynomials, depth-4 circuit size lower bounds of 2^{Ω(\sqrt{n}\log n)} have been proved Kayal et al. and Fournier et al. In particular, using combinatorial design Kayal et al.\ construct an explicit polynomial in VNP that requires depth-4 circuits of size 2^{Ω(\sqrt{n}\log n)} and Fournier et al.\ show that iterated matrix multiplication polynomial (which is in VP) also requires 2^{Ω(\sqrt{n}\log n)} size depth-4 circuits. In this paper, we identify a simple combinatorial property such that any polynomial f that satisfies the property would achieve similar circuit size lower bound for depth-4 circuits. In particular, it does not matter whether f is in VP or in VNP. As a result, we get a very simple unified lower bound analysis for the above mentioned polynomials. Another goal of this paper is to compare between our current knowledge of depth-4 circuit size lower bounds and determinantal complexity lower bounds. We prove the that the determinantal complexity of iterated matrix multiplication polynomial is Ω(dn) where d is the number of matrices and n is the dimension of the matrices. So for d=n, we get that the iterated matrix multiplication polynomial achieves the current best known lower bounds in both fronts: depth-4 circuit size and determinantal complexity. To the best of our knowledge, a Θ(n) bound for the determinantal complexity for the iterated matrix multiplication polynomial was known only for constant d>1 by Jansen.

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