arXiv subjects

Rishi Patel

Publications and source records attributed to Rishi Patel.

5 recordsLinked to original sources

arXiv2026

Knapsack Secretary is not $1/e$-Competitive

We prove that no algorithm for the knapsack secretary problem can be $1/e$-competitive. The knapsack secretary problem was first introduced by Babaioff, Immorlica, Kempe, and Kleinberg (2007). There have been many improvements to the achievable competitive ratio since then, but the $1/e$ impossibility barrier has remained unchanged. Many combinatorial variants of the secretary problem, including knapsack secretary, inherit the $1/e$ impossibility by embedding the single-choice problem as a special case. We construct a family of hard instances for the $1$-$B$ knapsack secretary problem, which is a special case of the general knapsack secretary problem, to improve the existing impossibility result. We show in this special case that the competitive ratio is at most $0.36437 < \frac{1}{e} - 0.0035$. Our construction is similar to the one used by Abels, Ladewig, Schewior, and Stinzend\"orfer (2022), for which they show an impossibility of $1/(1+e)$ for ordinal algorithms, where only the relative ranks of the items are known. Our work resolves an open question of theirs by showing that $1/e$ cannot be achieved even in the cardinal case of the $1$-$B$ knapsack secretary problem. We complement our impossibility result with a simple algorithm for $1$-$B$ knapsack secretary that is $(1/5.10-o(1))$-competitive for every fixed $B \geq 2$. This improves the guarantee obtained by applying general-purpose random-order knapsack algorithms to this special case.

cs.DS↗

arXiv2026

LLM vs. Human Unit Tests: Fault Detection on Real Python Bugs

Large language models (LLMs) have shown considerable promise for automated unit test generation, yet their practical effectiveness relative to human-written tests remains poorly understood. Existing evaluations commonly rely on coverage-oriented benchmarks that do not assess fault-detection capability directly. We present an empirical comparison of LLM-generated and human-written unit tests across three complementary Python benchmarks: 29 real historical bugs from BugsInPy, a function-level benchmark drawn from python-slugify and packaging, and a controlled paired benchmark. Our generation pipeline couples Gemini 2.5 Flash with a lightweight lexical retrieval mechanism that supplies bug-relevant context at generation time. Across eight quality dimensions, LLM-generated tests with retrieval-augmented context detect faults in 69% of cases compared to 17.2% for general-purpose human-written tests (Fisher's exact, $p < 0.001$, Cohen's $h = 1.10$). Critically, line and branch coverage are nearly identical between the two approaches (84.8% vs. 88.5% and 75.2% vs. 82.1%), confirming that coverage is an insufficient proxy for fault-detection capability. We discuss the conditions under which each approach excels, characterize their complementary strengths, and identify the critical role of retrieval context and reproducible benchmark construction in meaningful test-quality evaluation.

cs.SE↗

arXiv2025

Repeated Sales with Heterogeneous Buyer Sophistication

This paper considers behavior-based price discrimination in the repeated sale of a non-durable good to a single long-lived buyer, by a seller without commitment power. We assume that there is a mixed population of forward-looking ``sophisticated'' buyers and myopic ``naive'' buyers. We investigate the impact of these dynamics on the seller's ability to learn about the buyer and exploit this learning for revenue. We obtain conclusions that differ dramatically with the time horizon of the interactions. To understand short time horizons, we analyze a two-period model, and find that the strategic demand reduction observed with fully sophisticated buyers is robust to the introduction of naive types. In fact, despite the inability of naive buyers to game the pricing algorithm, their introduction can further harm the seller's revenue, due to more intense demand reduction overall. For long horizons, we consider an infinite-horizon model with time discounting. We find that the extreme demand reduction predicted by previous work does not survive the introduction of naive buyers. Instead, we observe equilibria where the seller learns meaningfully despite the sophisticated buyers' demand reduction. We prove that for a natural family of such equilibria, the seller's revenue is not just high, but approximates the revenue attainable with commitment power, even when the fraction of naive types is vanishingly small.

cs.GT↗

arXiv2023

Modelling of COVID-19 Using Fractional Differential Equations

In this work, we have described the mathematical modeling of COVID-19 transmission using fractional differential equations. The mathematical modeling of infectious disease goes back to the 1760s when the famous mathematician Daniel Bernoulli used an elementary version of compartmental modeling to find the effectiveness of deliberate smallpox inoculation on life expectancy. We have used the well-known SIR (Susceptible, Infected and Recovered) model of Kermack & McKendrick to extend the analysis further by including exposure, quarantining, insusceptibility and deaths in a SEIQRDP model. Further, we have generalized this model by using the solutions of Fractional Differential Equations to test the accuracy and validity of the mathematical modeling techniques against Canadian COVID-19 trends and spread of real-world disease. Our work also emphasizes the importance of Personal Protection Equipment (PPE) and impact of social distancing on controlling the spread of COVID-19.

physics.soc-ph↗

arXiv2021

Prior-Free Clock Auctions for Bidders with Interdependent Values

We study the problem of selling a good to a group of bidders with interdependent values in a prior-free setting. Each bidder has a signal that can take one of $k$ different values, and her value for the good is a weakly increasing function of all the bidders' signals. The bidders are partitioned into $\ell$ expertise-groups, based on how their signal can impact the values for the good, and we prove upper and lower bounds regarding the approximability of social welfare and revenue for a variety of settings, parameterized by $k$ and $\ell$. Our lower bounds apply to all ex-post incentive compatible mechanisms and our upper bounds are all within a small constant of the lower bounds. Our main results take the appealing form of ascending clock auctions and provide strong incentives by admitting the desired outcomes as obvious ex-post equilibria.

cs.GT↗