arXiv subjects

Ansh Nagda

Publications and source records attributed to Ansh Nagda.

8 recordsLinked to original sources

arXiv2026

Reinforced Generation of Combinatorial Structures: Ramsey Numbers

We present improved lower bounds for nine classical Ramsey numbers: $\mathbf{R}(3, 13)$ is increased from $60$ to $61$, $\mathbf{R}(3, 18)$ from $99$ to $100$, $\mathbf{R}(4, 13)$ from $138$ to $139$, $\mathbf{R}(4, 14)$ from $147$ to $148$, $\mathbf{R}(4, 15)$ from $158$ to $159$, $\mathbf{R}(4, 16)$ from $170$ to $174$, $\mathbf{R}(4, 18)$ from $205$ to $209$, $\mathbf{R}(4, 19)$ from $213$ to $219$, and $\mathbf{R}(4, 20)$ from $234$ to $237$. These results were achieved using AlphaEvolve, an LLM-based code mutation agent. Beyond these new results, we successfully recovered lower bounds for all Ramsey numbers known to be exact, and matched the best known lower bounds across many other cases. These include bounds for which previous work does not detail the algorithms used. Virtually all known Ramsey lower bounds are derived computationally, with bespoke search algorithms each delivering a handful of results. AlphaEvolve is a single meta-algorithm yielding search algorithms for all of our results.

math.CO↗

arXiv2025

Reinforced Generation of Combinatorial Structures: Hardness of Approximation

Can AI based methods help us make advances in complexity theory? We provide evidence towards answering this in the affirmative, using AlphaEvolve (an LLM code mutation agent) to obtain new results in three settings: a) We improve a recent result of Kunisky and Yu to obtain near-optimal upper and (conditional) lower bounds on certification algorithms for MAX-CUT and MAX-Independent Set on random 3- and 4-regular graphs. Our improved lower bounds are obtained by constructing nearly extremal Ramanujan graphs on as many as $163$ vertices, and our upper bounds are obtained via analytical arguments. b) We obtain new inapproximability results for MAX-4-CUT and MAX-3-CUT, proving that it is NP-hard to approximate them within factors of $0.987$ and $0.9649$ respectively, using AlphaEvolve to discover new gadget reductions. Our MAX-4-CUT result improves upon the SOTA of $0.9883$, and our MAX-3-CUT result improves on the current best gadget-based inapproximability result of $0.9853$, but falls short of the SOTA of $16/17$ that relies on a custom PCP (rather than a reduction from ``standard'' H{\aa}stad-style PCPs). c) Inapproximability for the metric Traveling Salesman Problem (TSP): We show that it is NP-hard to approximate the minimum cost tour within a factor of $111/110$ using AlphaEvolve to discover a new gadget, thus improving the SOTA of $117/116$. Along the way, we provide new modular soundness and completeness arguments that can be of independent interest. A key technical challenge we faced: verifying a candidate construction produced by AlphaEvolve is costly (sometimes requiring time exponential in the size of the construction). We used AlphaEvolve itself to evolve the verification procedure to be faster (sometimes by $10,000\times$ for our gadgets). Our results suggest that gadget based proofs would benefit from a pass through AI-based tools to obtain stronger results.

cs.LG↗

arXiv2025

On optimal distinguishers for Planted Clique

In a distinguishing problem, the input is a sample drawn from one of two distributions and the algorithm is tasked with identifying the source distribution. The performance of a distinguishing algorithm is measured by its advantage, i.e., its incremental probability of success over a random guess. A classic example of a distinguishing problem is the Planted Clique problem, where the input is a graph sampled from either $G(n,1/2)$ -- the standard Erd\H{o}s-R\'{e}nyi model, or $G(n,1/2,k)$ -- the Erd\H{o}s-R\'{e}nyi model with a clique planted on a random subset of $k$ vertices. The Planted Clique Hypothesis asserts that efficient algorithms cannot achieve advantage better than some absolute constant, say $1/4$, whenever $k=n^{1/2-\Omega(1)}$. In this work, we aim to precisely understand the optimal distinguishing advantage achievable by efficient algorithms on Planted Clique. We show the following results under the Planted Clique hypothesis: 1. Optimality of low-degree polynomials: No efficient algorithm can beat the advantage the optimal low-degree polynomial. Concretely, this means that the advantage of any efficient algorithm is at most $(1+o(1))\cdot k^2/(\sqrt{\pi}n)$, which is optimal in light of a simple edge-counting algorithm achieving this bound. 2. Harder planted distributions: There is an efficiently sampleable distribution $\mathcal{P}^*$ supported on graphs containing $k$-cliques such that no efficient algorithm can distinguish $\mathcal{P}^*$ from $G(n,1/2)$ with advantage $n^{-d}$ for an arbitrarily large constant $d$. In other words, there exist alternate planted distributions that are much harder than $G(n,1/2,k)$. Along the way, we prove a constructive hard-core lemma for a broad class of distributions with respect to low-degree polynomials. This result is applicable much more widely beyond Planted Clique and might be of independent interest.

cs.CC↗

arXiv2025

Improved approximation algorithms for the EPR Hamiltonian

The EPR Hamiltonian is a family of 2-local quantum Hamiltonians introduced by King (arXiv:2209.02589). We introduce a polynomial time $\frac{1+\sqrt{5}}{4}\approx 0.809$-approximation algorithm for the problem of computing the ground energy of the EPR Hamiltonian, improving upon the previous state of the art of $0.72$ (arXiv:2410.15544). As a special case, this also implies a $\frac{1+\sqrt{5}}{4}$-approximation for Quantum Max Cut on bipartite instances, improving upon the approximation ratio of $3/4$ that one can infer in a relatively straightforward manner from the work of Lee and Parekh (arXiv:2401.03616).

quant-ph↗

arXiv2024

On approximability of the Permanent of PSD matrices

We study the complexity of approximating the permanent of a positive semidefinite matrix $A\in \mathbb{C}^{n\times n}$. 1. We design a new approximation algorithm for $\mathrm{per}(A)$ with approximation ratio $e^{(0.9999 + \gamma)n}$, exponentially improving upon the current best bound of $e^{(1+\gamma-o(1))n}$ [AGOS17,YP22]. Here, $\gamma \approx 0.577$ is Euler's constant. 2. We prove that it is NP-hard to approximate $\mathrm{per}(A)$ within a factor $e^{(\gamma-\epsilon)n}$ for any $\epsilon>0$. This is the first exponential hardness of approximation for this problem. Along the way, we prove optimal hardness of approximation results for the $\|\cdot\|_{2\to q}$ ``norm'' problem of a matrix for all $-1 < q < 2$.

cs.DS↗

arXiv2021

Sublinear Time Hypergraph Sparsification via Cut and Edge Sampling Queries

The problem of sparsifying a graph or a hypergraph while approximately preserving its cut structure has been extensively studied and has many applications. In a seminal work, Benczúr and Karger (1996) showed that given any $n$-vertex undirected weighted graph $G$ and a parameter $\varepsilon \in (0,1)$, there is a near-linear time algorithm that outputs a weighted subgraph $G'$ of $G$ of size $\tilde{O}(n/\varepsilon^2)$ such that the weight of every cut in $G$ is preserved to within a $(1 \pm \varepsilon)$-factor in $G'$. The graph $G'$ is referred to as a {\em $(1 \pm \varepsilon)$-approximate cut sparsifier} of $G$. Subsequent recent work has obtained a similar result for the more general problem of hypergraph cut sparsifiers. However, all known sparsification algorithms require $Ω(n + m)$ time where $n$ denotes the number of vertices and $m$ denotes the number of hyperedges in the hypergraph. Since $m$ can be exponentially large in $n$, a natural question is if it is possible to create a hypergraph cut sparsifier in time polynomial in $n$, {\em independent of the number of edges}. We resolve this question in the affirmative, giving the first sublinear time algorithm for this problem, given appropriate query access to the hypergraph.

cs.DS↗

arXiv2021

Counting and Sampling Perfect Matchings in Regular Expanding Non-Bipartite Graphs

We show that the ratio of the number of near perfect matchings to the number of perfect matchings in $d$-regular strong expander (non-bipartite) graphs, with $2n$ vertices, is a polynomial in $n$, thus the Jerrum and Sinclair Markov chain [JS89] mixes in polynomial time and generates an (almost) uniformly random perfect matching. Furthermore, we prove that such graphs have at least $Ω(d)^n$ any perfect matchings, thus proving the Lovasz-Plummer conjecture [LP86] for this family of graphs.

cs.DS↗

arXiv2020

Near-linear Size Hypergraph Cut Sparsifiers

Cuts in graphs are a fundamental object of study, and play a central role in the study of graph algorithms. The problem of sparsifying a graph while approximately preserving its cut structure has been extensively studied and has many applications. In a seminal work, Benczúr and Karger (1996) showed that given any $n$-vertex undirected weighted graph $G$ and a parameter $\varepsilon \in (0,1)$, there is a near-linear time algorithm that outputs a weighted subgraph $G'$ of $G$ of size $\tilde{O}(n/\varepsilon^2)$ such that the weight of every cut in $G$ is preserved to within a $(1 \pm \varepsilon)$-factor in $G'$. The graph $G'$ is referred to as a {\em $(1 \pm \varepsilon)$-approximate cut sparsifier} of $G$. A natural question is if such cut-preserving sparsifiers also exist for hypergraphs. Kogan and Krauthgamer (2015) initiated a study of this question and showed that given any weighted hypergraph $H$ where the cardinality of each hyperedge is bounded by $r$, there is a polynomial-time algorithm to find a $(1 \pm \varepsilon)$-approximate cut sparsifier of $H$ of size $\tilde{O}(\frac{nr}{\varepsilon^2})$. Since $r$ can be as large as $n$, in general, this gives a hypergraph cut sparsifier of size $\tilde{O}(n^2/\varepsilon^2)$, which is a factor $n$ larger than the Benczúr-Karger bound for graphs. It has been an open question whether or not Benczúr-Karger bound is achievable on hypergraphs. In this work, we resolve this question in the affirmative by giving a new polynomial-time algorithm for creating hypergraph sparsifiers of size $\tilde{O}(n/\varepsilon^2)$.

cs.DS↗