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- W4385469930 abstract "We revisit the algorithmic problem of finding a triangle in a graph (textsc{Triangle Detection}), and examine its relation to other problems such as textsc{3Sum} and textsc{Independent set}. (I) Given a graph $G=(V,E)$ with $n$ vertices and $m$ edges, we present a randomized algorithm that computes an approximation of the number of triangles in $G$ and finds a witness with high probability. As an application, we limit the range of a conjecture of Pv{a}trac{s}cu (2010) regarding the triangle detection problem. (II) We present an algorithm which given a graph $G=(V,E)$ performs one of the following tasks in $O(m+n)$ (ie, linear) time: (i)~compute a $Omega(1/sqrt{n})$-approximation of a maximum independent set in $G$, or (ii)~find a triangle in $G$. The run-time is faster than that for any previous method for each of these tasks. (III) Let $G=(V,E)$ be a graph with $n$ vertices and $m$ edges. We consider data structures for answering independent set queries of the form: Given a subset $U subset V$, is $U$ independent? We give a simple implementation of a candidate data structure. Assuming a lower bound hypothesis for triangle detection as implied by state-of-the-art algorithms for this task, we obtain a conditional lower bound for answering independent set queries. (IV) The second result suggests the following broader research direction: if it is difficult to find (A) or (B) separately, can one find one of the two efficiently? This motivates the emph{dual pair} concept we introduce. We discuss and provide several instances of dual-pair approximation." @default.
- W4385469930 created "2023-08-02" @default.
- W4385469930 creator A5091313409 @default.
- W4385469930 date "2021-05-03" @default.
- W4385469930 modified "2023-09-23" @default.
- W4385469930 title "Finding Triangles or Independent Sets" @default.
- W4385469930 doi "https://doi.org/10.48550/arxiv.2105.01265" @default.
- W4385469930 hasPublicationYear "2021" @default.
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