arXiv subjects

Aranya Kumar Bal

Publications and source records attributed to Aranya Kumar Bal.

3 recordsLinked to original sources

arXiv2026

Estimating the Volume of a Union of Convex Bodies

The \textit{Union Volume Estimation problem}, i.e. yielding an $(\varepsilon, δ)$-approximation of the volume of the union of convex bodies, is a classical problem in computational geometry and formal methods. In this paper, we first detect where and why the efficient union volume estimators with approximate sampling and PAC volume estimation oracles fail. Specifically, we show that having only TV-distance guarantees on the approximate sampler does not suffice, and demonstrate this bottleneck by constructing the sunflower families of convex bodies. In this context, we propose the concept of \textit{tail-controlled samplers} that can yield with high-probability almost-$\ell_{\infty}$ closeness. Secondly, we leverage this concept to propose a generic algorithmic framework \texttt{Alpona} that properly combines a PAC volume computation and approximate point sampling oracles with tail-controls to yield an $(\varepsilon, δ)$-estimate of the union volume for any collection of convex bodies. We show that \texttt{Alpona} needs only linear (in number of convex bodies) oracle calls to be $(\varepsilon, δ)$-correct. Finally, we provide a tail-controlled analysis of the Hit-and-Run sampler~\citep{lovasz2006fast}. We further provide a novel geometric analysis to improve the required walk length of Hit-and-Run. We further instantiate \texttt{Alpona}'s correctness for the sparse frequency streams of convex bodies: a special case of the union volume estimation problem where no convex body intersects more than $τ$ many other bodies.

cs.DS↗

arXiv2026

Unlocking Fractional Moments in Delphic Set Streams

We consider estimation of non-integer frequency moments $F_k$ and related Bernstein-type statistics in the Delphic set stream model under a bounded-frequency assumption: every universe element appears at most $τ$ times. The main challenge of this model is to keep space low while also keeping update time low, which is not trivial because the sets can be exponential in size compared to their representations. Our core insight is that by sampling the stream at different rates and observing the resulting distinct-counts, we can 'probe' the frequency distribution and numerically integrate these probes to reconstruct a broad class of statistics. Building on that, we crucially observe that the distinct-count of a randomly sampled substream, viewed as a function of the sampling rate, is a single analytic object whose evaluations determine a broad class of statistics via a complementary Laplace-type integral. Algorithmically we exploit this by: 1. estimating those evaluations using only standard $F_0$ (distinct-count) algorithms on sampled substreams and 2. recovering target statistics by controlled numerical integration on a judiciously chosen grid. For $F_k$ with $k\in (0,1)$ we obtain the first one-pass streaming algorithms for Delphic set streams whose space and per-set update time are $\mathrm{poly}(\log|Ω|,\log m,\varepsilon^{-1},\log(1/δ))$ in the practically relevant regime $τ=\mathrm{polylog}(|Ω|,m)$; in general the bounds are polynomial in $τ$ and $\varepsilon^{-1}$ and logarithmic in $δ^{-1}$. We also give a complexity-theoretic barrier explaining why lower bounds for removing the bounded-frequency assumption appear difficult: ruling out polylogarithmic algorithms for unrestricted Delphic $F_k$ would imply a linear-space threshold-counting separation.

cs.DS↗

arXiv2026

A 64-Rectangle Counterexample to Wegner's Conjecture and LP Gaps up to $5/2$

Wegner conjectured that every finite family $\mathcal R$ of axis-parallel rectangles satisfies $τ(\mathcal R)\le 2ν(\mathcal R)-1$, where $ν$ is the packing number and $τ$ is the piercing number. Ajwani, Gajjala, Raman, and Ray recently disproved this by constructing a triangle-free counterexample on $2196\cdot 8^9$ rectangles and, using a computer-assisted package-and-port recursion, obtained a standard LP gap of $17891/8064$ for Maximum Independent Set of Rectangles. We give a simpler and hand-checkable counterexample with $64$ rectangles. It is built from an eight-rectangle gadget whose independent sets inject into four ordered slots; we then use four horizontal and four vertical copies of this gadget to form a triangle-free family with $ν=16$ and $τ\ge 32$. We use the same horizontal-vertical step to define recursive families of rectangles $P_r$ with $ν(P_r)=4^{2^r}$. For the standard clique, equivalently point, relaxation we obtain a finite gap $73/32$ at $P_3$, improving the previous benchmark of $17891/8064$. We then construct recursive fractional solutions and matching piercing sets showing $\lim_r α^*(P_r)/ν(P_r)=\lim_r τ(P_r)/ν(P_r)=5/2$. Finally, by disjoint union with isolated rectangles, we show that every rational $t\in[1,5/2)$ occurs as a standard LP gap and also as a packing-piercing ratio for suitable rectangle families.

math.CO↗