arXiv subjects

Arka Ray

Publications and source records attributed to Arka Ray.

6 recordsLinked to original sources

arXiv2026

A Simple Sub-Polynomial Degree Coboundary Expander

High dimensional expanders simultaneously satisfying spectral and combinatorial (coboundary) expansion have recently played a major role in breakthroughs in PCP and coding theory, but the only known construction of such complexes is extremely involved, requiring deep algebraic number theory. In this work, we give an extremely simple combinatorial construction of a sub-polynomial degree complex based on projections of the flags complex (subspace chains) that is (i) a local spectral expander, (ii) a coboundary expander, and (iii) a swap coboundary expander. As a corollary, we also give the first near-linear size combinatorial hypergraphs with good agreement tests in the '1%' regime, and a simple PCP construction with near-linear size.

math.CO↗

arXiv2026

Phoenix-VL 1.5 Medium Technical Report

We introduce Phoenix-VL 1.5 Medium, a 123B-parameter natively multimodal and multilingual foundation model, adapted to regional languages and the Singapore context. Developed as a sovereign AI asset, it demonstrates that deep domain adaptation can be achieved with minimal degradation to broad-spectrum intelligence and alignment. Continued pretraining was performed on Mistral Medium 3.1 using a localized 1-trillion tokens multimodal corpus, followed by a 250-billion tokens long-context extension phase. Subsequent post-training incorporated a novel human-annotated Singapore multimodal dataset and curated textual corpus on Singapore culture, knowledge, and legislation, totaling 22-billion tokens. An additional 5 billion tokens of model alignment was performed through Online Direct Preference Optimization. Phoenix-VL 1.5 Medium achieves state-of-the-art performance for its size on Singapore multimodal, legal, and government policy benchmarks while remaining globally competitive on general multimodal intelligence, multilingual, and STEM benchmarks. We also introduce a novel evaluation suite encompassing localized knowledge benchmarks and an institutionally aligned model behavior and safety framework. We report the data curation principles, training methodology, and highlight benchmark and inference performance.

cs.CL↗

arXiv2024

Improved linearly ordered colorings of hypergraphs via SDP rounding

We consider the problem of linearly ordered (LO) coloring of hypergraphs. A hypergraph has an LO coloring if there is a vertex coloring, using a set of ordered colors, so that (i) no edge is monochromatic, and (ii) each edge has a unique maximum color. It is an open question as to whether or not a 2-LO colorable 3-uniform hypergraph can be LO colored with 3 colors in polynomial time. Nakajima and \v{Z}ivn\'{y} recently gave a polynomial-time algorithm to color such hypergraphs with $\widetilde{O}(n^{1/3})$ colors and asked if SDP methods can be used directly to obtain improved bounds. Our main result is to show how to use SDP-based rounding methods to produce an LO coloring with $\widetilde{O}(n^{1/5})$ colors for such hypergraphs. We show how to reduce the problem to cases with highly structured SDP solutions, which we call balanced hypergraphs. Then, we discuss how to apply classic SDP-rounding tools to obtain improved bounds.

cs.DS↗

arXiv2023

Improved Hardness of Approximation for Geometric Bin Packing

The Geometric Bin Packing (GBP) problem is a generalization of Bin Packing where the input is a set of $d$-dimensional rectangles, and the goal is to pack them into unit $d$-dimensional cubes efficiently. It is NP-Hard to obtain a PTAS for the problem, even when $d=2$. For general $d$, the best-known approximation algorithm has an approximation guarantee exponential in $d$, while the best hardness of approximation is still a small constant inapproximability from the case when $d=2$. In this paper, we show that the problem cannot be approximated within $d^{1-\epsilon}$ factor unless NP=P. Recently, $d$-dimensional Vector Bin Packing, a closely related problem to the GBP, was shown to be hard to approximate within $\Omega(\log d)$ when $d$ is a fixed constant, using a notion of Packing Dimension of set families. In this paper, we introduce a geometric analog of it, the Geometric Packing Dimension of set families. While we fall short of obtaining similar inapproximability results for the Geometric Bin Packing problem when $d$ is fixed, we prove a couple of key properties of the Geometric Packing Dimension which highlight fundamental differences between Geometric Bin Packing and Vector Bin Packing.

cs.DS↗

arXiv2022

Sparse Cuts in Hypergraphs from Random Walks on Simplicial Complexes

There are a lot of recent works on generalizing the spectral theory of graphs and graph partitioning to hypergraphs. There have been two broad directions toward this goal. One generalizes the notion of graph conductance to hypergraph conductance [LM16, CLTZ18]. In the second approach one can view a hypergraph as a simplicial complex and study its various topological properties [LM06, MW09, DKW16, PR17] and spectral properties [KM17, DK17, KO18a, KO18b, Opp20]. In this work, we attempt to bridge these two directions of study by relating the spectrum of {\em up-down walks} and {\em swap-walks} on the simplicial complex to hypergraph expansion. In surprising contrast to random-walks on graphs, we show that the spectral gap of swap-walks and up-down walks between level $m$ and $l$ with $1 < m \leq l$ can not be used to infer any bounds on hypergraph conductance. Moreover, we show that the spectral gap of swap-walks between $X(1)$ and $X(k-1)$ can not be used to infer any bounds on hypergraph conductance, whereas we give a Cheeger-like inequality relating the spectral of walks between level $1$ and $l$ for any $l \leq k$ to hypergraph expansion. This is a surprising difference between swaps-walks and up-down walks! Finally, we also give a construction to show that the well-studied notion of {\em link expansion} in simplicial complexes can not be used to bound hypergraph expansion in a Cheeger-like manner.

cs.DS↗

arXiv2021

There is no APTAS for 2-dimensional vector bin packing: Revisited

We study the Vector Bin Packing and the Vector Bin Covering problems, multidimensional generalizations of the Bin Packing and the Bin Covering problems, respectively. In the Vector Bin Packing, we are given a set of $d$-dimensional vectors from $[0,1]^d$ and the aim is to partition the set into the minimum number of bins such that for each bin $B$, each component of the sum of the vectors in $B$ is at most 1. Woeginger [Woe97] claimed that the problem has no APTAS for dimensions greater than or equal to 2. We note that there was a slight oversight in the original proof. In this work, we give a revised proof using some additional ideas from [BCKS06,CC09]. In fact, we show that it is NP-hard to get an asymptotic approximation ratio better than $\frac{600}{599}$. An instance of Vector Bin Packing is called $\delta$-skewed if every item has at most one dimension greater than $\delta$. As a natural extension of our general $d$-Dimensional Vector Bin Packing result we show that for $\varepsilon\in (0,\frac{1}{2500})$ it is NP-hard to obtain a $(1+\varepsilon)$-approximation for $\delta$-Skewed Vector Bin Packing if $\delta>20\sqrt \varepsilon$. In the Vector Bin Covering problem given a set of $d$-dimensional vectors from $[0,1]^d$, the aim is to obtain a family of disjoint subsets (called bins) with the maximum cardinality such that for each bin $B$, each component of the sum of the vectors in $B$ is at least 1. Using ideas similar to our Vector Bin Packing result, we show that for Vector Bin Covering there is no APTAS for dimensions greater than or equal to 2. In fact, we show that it is NP-hard to get an asymptotic approximation ratio better than $\frac{998}{997}$.

cs.DS↗