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Consider the following 2-respecting min-cut problem. Given a weighted graph $G$ and its spanning tree $T$, find the minimum cut among the cuts that contain at most two edges in $T$. This problem is an important subroutine in Kargers celebrated randomized near-linear-time min-cut algorithm [STOC96]. We present a new approach for this problem which can be easily implemented in many settings, leading to the following randomized min-cut algorithms for weighted graphs. * An $O(mfrac{log^2 n}{loglog n} + nlog^6 n)$-time sequential algorithm: This improves Kargers $O(m log^3 n)$ and $O(mfrac{(log^2 n)log (n^2/m)}{loglog n} + nlog^6 n)$ bounds when the input graph is not extremely sparse or dense. Improvements over Kargers bounds were previously known only under a rather strong assumption that the input graph is simple [Henzinger et al. SODA17; Ghaffari et al. SODA20]. For unweighted graphs with parallel edges, our bound can be improved to $O(mfrac{log^{1.5} n}{loglog n} + nlog^6 n)$. * An algorithm requiring $tilde O(n)$ cut queries to compute the min-cut of a weighted graph: This answers an open problem by Rubinstein et al. ITCS18, who obtained a similar bound for simple graphs. * A streaming algorithm that requires $tilde O(n)$ space and $O(log n)$ passes to compute the min-cut: The only previous non-trivial exact min-cut algorithm in this setting is the 2-pass $tilde O(n)$-space algorithm on simple graphs [Rubinstein et al., ITCS18] (observed by Assadi et al. STOC19). In contrast to Kargers 2-respecting min-cut algorithm which deploys sophisticated dynamic programming techniques, our approach exploits some cute structural properties so that it only needs to compute the values of $tilde O(n)$ cuts corresponding to removing $tilde O(n)$ pairs of tree edges, an operation that can be done quickly in many settings.
Minimum-weight cut (min-cut) is a basic measure of a networks connectivity strength. While the min-cut can be computed efficiently in the sequential setting [Karger STOC96], there was no efficient way for a distributed network to compute its own min-
In this work, we resolve the query complexity of global minimum cut problem for a graph by designing a randomized algorithm for approximating the size of minimum cut in a graph, where the graph can be accessed through local queries like {sc Degree},
We present a practically efficient algorithm for maintaining a global minimum cut in large dynamic graphs under both edge insertions and deletions. While there has been theoretical work on this problem, our algorithm is the first implementation of a
Given a graph $G=(V,E)$ with two distinguished vertices $s,tin V$ and an integer parameter $L>0$, an {em $L$-bounded cut} is a subset $F$ of edges (vertices) such that the every path between $s$ and $t$ in $Gsetminus F$ has length more than $L$. The
We study two variants of textsc{Maximum Cut}, which we call textsc{Connected Maximum Cut} and textsc{Maximum Minimal Cut}, in this paper. In these problems, given an unweighted graph, the goal is to compute a maximum cut satisfying some connectivity