arXiv subjects

Mohit Gurumukhani

Publications and source records attributed to Mohit Gurumukhani.

12 recordsLinked to original sources

arXiv2026

Sumset Structure in Local Computation

We introduce and study a new model of decision trees that lies at the frontier of provable circuit lower bounds. We study $\ell$-local decision trees, in which each internal node queries an $\ell$-local function of the input. We show that sufficiently strong lower bounds for $\ell$-local decision trees would imply several breakthrough circuit lower bounds, including super-linear size lower bounds for log-depth circuits and improved bounds for unrestricted-depth circuits. Previously, such consequences were known to follow from strong lower bounds for depth-$3$ circuits, which has been the main prior approach in attempting to prove these results. Since local decision trees are strictly weaker than depth-$3$ circuits, this provides a formally easier route to the same circuit lower-bound consequences. We prove essentially optimal lower bounds for a weaker variant that we call oblivious $\ell$-local decision trees, where all nodes at the same depth query the same function. Our lower bounds follow from a new technique that uncovers sumset structure in local maps, and our hard functions are sumset dispersers, sumset condensers, and directional affine dispersers. Along the way, we give an explicit construction of a sumset condenser with small entropy loss. As an additional contribution, we initiate the study of degree-$2$ decision trees, in which each query is a quadratic polynomial of the input. We show that proving lower bounds in this model is a natural stepping stone toward constructing dispersers for degree-$2$ variety sources, which imply improved circuit lower bounds for unrestricted depth circuits. We prove nearly maximal lower bounds for the oblivious variant of the model.

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arXiv2026

A Resolution of Friedgut's Conjecture on Influential Coalitions

We prove that, for every constant $\varepsilon>0$ and every function $f:Σ^n\to\{0, 1\}$, there is a coalition of $O(n/\sqrt{\log n})$ coordinates and a target output $b\in\{0, 1\}$ such that, after the remaining coordinates are sampled uniformly and independently, the coalition can choose its values to make the output equal to $b$ with probability at least $1-\varepsilon$. The bound is independent of the alphabet size and also holds for monotone Boolean functions on $[0,1]^n$, resolving a conjecture of Friedgut (Combinatorics, Probability and Computing, 2004). Unlike the Boolean cube setting, where Kahn, Kalai, and Linial (FOCS, 1988) give a coalition bound of $O(n/\log n)$, no sublinear bound independent of the alphabet size was previously known. In collective coin flipping, our result gives the first sublinear bound on the number of bad players needed to force a fixed output with probability at least $1-\varepsilon$ in any one-round protocol with independent uniform messages, regardless of the message length. A key ingredient in our proof is an encoding that lets us relate the influence of a function on a product space to the $p$-biased influence of the encoded function. We then rely on a structure theorem of Hatami (Annals of Mathematics, 2012) for functions with small $p$-biased influence to bias the encoded function.

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arXiv2026

Improved Bounds for Coin Flipping, Leader Election, and Random Selection

Random selection, leader election, and collective coin flipping are fundamental tasks in fault-tolerant distributed computing. We study these problems in the full-information model where despite decades of study, key gaps remain in our understanding of the trade-offs between round complexity, communication per player in each round, and adversarial resilience. We make progress by proving improved bounds for these problems. We first show that any $k$-round coin flipping protocol over $\ell$ players, each player sending one bit per round, can be biased by $O(\ell/\log^{(k)}(\ell))$ bad players. We obtain a similar lower bound for leader election. This strengthens prior best bounds [RSZ, SICOMP 2002] of $O(\ell/\log^{(2k-1)}(\ell))$ for coin flipping protocols and $O(\ell/\log^{(2k+1)}(\ell))$ for leader election protocols. Our result implies that any (1-bit per player) protocol tolerating linear fraction of bad players requires at least $\log^* \ell$ rounds, showing existing protocols [RZ, JCSS 2001; F, FOCS 1999] are near-optimal. We next initiate the study of one-round, (1-bit per player) random selection. For all $m\ge (\log(\ell))^2$, we obtain an optimal protocol (a first in the full information model for any task): We construct a protocol resilient to $O(\ell / m)$ bad players that outputs $m$ uniform random bits. And, we show that any protocol that outputs $m$ uniform random bits can be corrupted using $O(\ell / m)$ bad players. This also implies a one-round leader election protocol resilient to $\ell / (\log \ell)^2$ bad players, improving the prior best protocol [RZ, JCSS 2001] which was resilient to $\ell / (\log \ell)^3$ bad players. Our resilience matches that of the best one-round coin flipping protocol by Ajtai & Linial. To obtain our lower bound, we introduce multi-output influence, an extension of influence of boolean functions to the multi-output setting.

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arXiv2026

Optimal Depth-Three Circuits for Inner Product

We show that Inner Product in $2n$ variables, $\mathbf{IP}_n(x, y) = x_1y_1 \oplus \ldots \oplus x_ny_n$, can be computed by depth-3 bottom fan-in 2 circuits of size $\mathsf{poly}(n)\cdot (9/5)^n$, matching the lower bound of Göös, Guan, and Mosnoi (Inform. Comput.'24). Our construction is obtained via the following steps. - We provide a general template for constructing optimal depth-3 circuits with bottom fan-in $k$ for an arbitrary function $f$. We do this in two steps. First, we partition $f^{-1}(1)$ into orbits of its automorphism group. Second, for each orbit, we construct one $k$-CNF that (a) accepts the largest number of inputs from that orbit and (b) rejects all inputs rejected by $f$. - We instantiate the template for $\mathbf{IP}_n$ and $k = 2$. Guided by the intuition (which we call modularity principle) that optimal 2-CNFs can be constructed by taking the conjunction of variable-disjoint copies of smaller $2$-CNFs, we use computer search to identify a small set of building block 2-CNFs over at most 4 variables. - We again use computer search to discover appropriate combinations (disjoint conjunctions) of building blocks to arrive at optimal 2-CNFs and analyze them using techniques from analytic combinatorics. We believe that the approach outlined in this paper can be applied to a wide range of functions to determine their depth-3 complexity.

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arXiv2026

Optimal Monotone Depth-Three Circuit Lower Bounds for Majority

Gurumuhkani et al. (CCC'24) introduced the local enumeration problem $Enum(k, t)$ as follows: for a natural number $k$ and a parameter $t$, given an $n$-variate $k$-CNF with no satisfying assignment with Hamming weight less than $t(n)$, enumerate all satisfying assignments of Hamming weight exactly $t(n)$. They showed that efficient algorithms for local enumeration yield new $k$-SAT algorithms and depth-3 lower bounds for Majority function. As the first non-trivial case, they gave an algorithm for $k = 3$ which in particular gave a new lower bound on the size of depth-3 circuits with bottom fan-in at most 3 computing Majority. In this paper, we give an optimal algorithm that solves local enumeration on monotone formulas for $k = 3$ and all $t \le n/2$. In particular, we obtain an optimal lower bound on the size of monotone depth-3 circuits with bottom fan-in at most 3 computing Majority.

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arXiv2025

Condensing and Extracting Against Online Adversaries

We study the tasks of deterministically condensing and extracting from Online Non-Oblivious Symbol Fixing (oNOSF) sources, a natural model of defective randomness where extraction is impossible in many parameter regimes [AORSV, EUROCRYPT'20]. A $(g,\ell)$-oNOSF source is a sequence of $\ell$ blocks where at least $g$ blocks are good (independent, with min-entropy) and the remaining bad blocks are controlled by an online adversary and can be arbitrarily correlated with prior blocks. Previously, [CGR, FOCS'24] proved impossibility of condensing beyond rate $1/2$ when $g\le 0.5 \ell$ and showed existence of condensers for when $g \ge 0.51\ell$ and $n$ is exponential in $\ell$. In this work, not only do we construct the first explicit condensers matching the existential results of [CGR, FOCS'24], but we make a doubly exponential improvement by handling the case when $g\ge 0.51\ell$ and $n$ is only polylogarithmic in $\ell$. We also obtain a much improved explicit construction for transforming low-entropy oNOSF sources into uniform oNOSF sources. Next, we essentially resolve the question of the existence of condensers for oNOSF sources by showing the existence of condensers even when $n$ is a large enough constant and $\ell$ is growing (provided $g \ge 0.51\ell$). We apply our condensers to collective coin flipping and collective sampling, widely studied problems in fault-tolerant distributed computing, and provide very simple protocols for them. Finally, we study the possibility of extraction from oNOSF sources. For lower bounds, we introduce the notion of online influence - extending the notion of influence of boolean functions - and establish tight bounds that imply extraction lower bounds. We also construct explicit extractors via leader election protocols that beat standard resilient functions [AL, Combinatorica'93].

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arXiv2025

Two-Sided Lossless Expanders in the Unbalanced Setting

We present the first explicit construction of two-sided lossless expanders in the unbalanced setting (bipartite graphs that have polynomially many more nodes on the left than on the right). Prior to our work, all known explicit constructions in the unbalanced setting achieved only one-sided lossless expansion. Specifically, we show that the one-sided lossless expanders constructed by Kalev and Ta-Shma (RANDOM'22) -- that are based on multiplicity codes introduced by Kopparty, Saraf, and Yekhanin (STOC'11) -- are, in fact, two-sided lossless expanders. Moreover, we show that our result is tight, thus completely characterizing the graph of Kalev and Ta-Shma. Using our unbalanced bipartite expander, we easily obtain lossless (non-bipartite) expander graphs on $N$ vertices with polynomial degree $\ll N$ and expanding sets of size $N^{0.49}$.

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arXiv2025

Local Enumeration: The Not-All-Equal Case

Gurumukhani et al. (CCC'24) proposed the local enumeration problem Enum(k, t) as an approach to break the Super Strong Exponential Time Hypothesis (SSETH): for a natural number $k$ and a parameter $t$, given an $n$-variate $k$-CNF with no satisfying assignment of Hamming weight less than $t(n)$, enumerate all satisfying assignments of Hamming weight exactly $t(n)$. Furthermore, they gave a randomized algorithm for Enum(k, t) and employed new ideas to analyze the first non-trivial case, namely $k = 3$. In particular, they solved Enum(3, n/2) in expected $1.598^n$ time. A simple construction shows a lower bound of $6^{\frac{n}{4}} \approx 1.565^n$. In this paper, we show that to break SSETH, it is sufficient to consider a simpler local enumeration problem NAE-Enum(k, t): for a natural number $k$ and a parameter $t$, given an $n$-variate $k$-CNF with no satisfying assignment of Hamming weight less than $t(n)$, enumerate all Not-All-Equal (NAE) solutions of Hamming weight exactly $t(n)$, i.e., those that satisfy and falsify some literal in every clause. We refine the algorithm of Gurumukhani et al. and show that it optimally solves NAE-Enum(3, n/2), namely, in expected time $poly(n) \cdot 6^{\frac{n}{4}}$.

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arXiv2024

On Extremal Properties of k-CNF: Capturing Threshold Functions

We consider a basic question on the expressiveness of $k$-CNF formulas: How well can $k$-CNF formulas capture threshold functions? Specifically, what is the largest number of assignments (of Hamming weight $t$) accepted by a $k$-CNF formula that only accepts assignments of weight at least $t$? Among others, we provide the following results: - While an optimal solution is known for $t \leq n/k$, the problem remains open for $t > n/k$. We formulate a (monotone) version of the problem as an extremal hypergraph problem and show that for $t = n-k$, the problem is exactly the Turán problem. - For $t = αn$ with constant $α$, we provide a construction and show its optimality for $2$-CNF. Optimality of the construction for $k>2$ would give improved lower bounds for depth-$3$ circuits.

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arXiv2024

On the Existence of Seedless Condensers: Exploring the Terrain

We prove several new results for seedless condensers in the context of three related classes of sources: Non-Oblivious Symbol Fixing (NOSF) sources, online NOSF (oNOSF) sources [AORSV, EUROCRYPT'20], and adversarial Chor-Goldreich (aCG) source [DMOZ, STOC'23]. We think of these sources as a sequence of random variables $\mathbf{X}=\mathbf{X}_1,\dots,\mathbf{X}_\ell$ on $\ell$ symbols where at least $g$ out of these $\ell$ symbols are "good" (i.e., have some min-entropy requirement), denoted as a $(g,\ell)$-source, and the remaining "bad" $\ell-g$ symbols may adversarially depend on these $g$ good blocks. The difference between each of these sources is realized by restrictions on the power of the adversary. Prior to our work, the only known seedless condenser upper or lower bound in these settings is due to [DMOZ, STOC'23], where they explicitly construct a seedless condenser for a restricted subset of $(g,\ell)$-aCG sources. We show: 1) oNOSF sources a) When $g\leq\ell/2$, we prove that condensing with error 0.99 above rate $\frac{1}{\lfloor \ell/g \rfloor}$ is impossible. In fact, we show that this is tight. b) For $g> \ell/2$, we show the existence of excellent condensers for uniform oNOSF sources. In addition, we show the existence of similar condensers for oNOSF sources with only logarithmic min-entropy. 2) aCG sources a) We observe that uniform aCG sources are equivalent to uniform oNOSF sources and consequently inherit the same results. b) We show that one cannot condense beyond the min-entropy gap of each block or condense low min-entropy CG sources above rate $1/2$. 3) NOSF sources a) We show that condensing with constant error above rate $\frac{g}{\ell}$ is impossible for uniform NOSF sources for any $g$ and $\ell$, thus ruling out the possibility of any non-trivial condensing. This shows a distinction between NOSF sources and oNOSF sources.

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arXiv2024

Local Enumeration and Majority Lower Bounds

Depth-3 circuit lower bounds and $k$-SAT algorithms are intimately related; the state-of-the-art $Σ^k_3$-circuit lower bound and the $k$-SAT algorithm are based on the same combinatorial theorem. In this paper we define a problem which reveals new interactions between the two. Define Enum($k$, $t$) problem as: given an $n$-variable $k$-CNF and an initial assignment $α$, output all satisfying assignments at Hamming distance $t$ from $α$, assuming that there are no satisfying assignments of Hamming distance less than $t$ from $α$. Observe that: an upper bound $b(n, k, t)$ on the complexity of Enum($k$, $t$) implies: - Depth-3 circuits: Any $Σ^k_3$ circuit computing the Majority function has size at least $\binom{n}{\frac{n}{2}}/b(n, k, \frac{n}{2})$. - $k$-SAT: There exists an algorithm solving $k$-SAT in time $O(\sum_{t = 1}^{n/2}b(n, k, t))$. A simple construction shows that $b(n, k, \frac{n}{2}) \ge 2^{(1 - O(\log(k)/k))n}$. Thus, matching upper bounds would imply a $Σ^k_3$-circuit lower bound of $2^{Ω(\log(k)n/k)}$ and a $k$-SAT upper bound of $2^{(1 - Ω(\log(k)/k))n}$. The former yields an unrestricted depth-3 lower bound of $2^{ω(\sqrt{n})}$ solving a long standing open problem, and the latter breaks the Super Strong Exponential Time Hypothesis. In this paper, we propose a randomized algorithm for Enum($k$, $t$) and introduce new ideas to analyze it. We demonstrate the power of our ideas by considering the first non-trivial instance of the problem, i.e., Enum($3$, $\frac{n}{2}$). We show that the expected running time of our algorithm is $1.598^n$, substantially improving on the trivial bound of $3^{n/2} \simeq 1.732^n$. This already improves $Σ^3_3$ lower bounds for Majority function to $1.251^n$. The previous bound was $1.154^n$ which follows from the work of Håstad, Jukna, and Pudlák (Comput. Complex.'95).

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arXiv2024

Extractors for Polynomial Sources over $\mathbb{F}_2$

We explicitly construct the first nontrivial extractors for degree $d \ge 2$ polynomial sources over $\mathbb{F}_2^n$. Our extractor requires min-entropy $k\geq n - \tildeΩ(\sqrt{\log n})$. Previously, no constructions were known, even for min-entropy $k\geq n-1$. A key ingredient in our construction is an input reduction lemma, which allows us to assume that any polynomial source with min-entropy $k$ can be generated by $O(k)$ uniformly random bits. We also provide strong formal evidence that polynomial sources are unusually challenging to extract from, by showing that even our most powerful general purpose extractors cannot handle polynomial sources with min-entropy below $k\geq n-o(n)$. In more detail, we show that sumset extractors cannot even disperse from degree $2$ polynomial sources with min-entropy $k\geq n-O(n/\log\log n)$. In fact, this impossibility result even holds for a more specialized family of sources that we introduce, called polynomial non-oblivious bit-fixing (NOBF) sources. Polynomial NOBF sources are a natural new family of algebraic sources that lie at the intersection of polynomial and variety sources, and thus our impossibility result applies to both of these classical settings. This is especially surprising, since we do have variety extractors that slightly beat this barrier - implying that sumset extractors are not a panacea in the world of seedless extraction.

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