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تسجيل مستخدم جديدThe Landscape of Minimum Label Cut (Hedge Connectivity) Problem
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Minimum Label Cut (or Hedge Connectivity) problem is defined as follows: given an undirected graph $G=(V, E)$ with $n$ vertices and $m$ edges, in which, each edge is labeled (with one or multiple labels) from a label set $L={ell_1,ell_2, ..., ell_{|L|}}$, the edges may be weighted with weight set $W ={w_1, w_2, ..., w_m}$, the label cut problem(hedge connectivity) problem asks for the minimum number of edge sets(each edge set (or hedge) is the edges with the same label) whose removal disconnects the source-sink pair of vertices or the whole graph with minimum total weights(minimum cardinality for unweighted version). This problem is more general than edge connectivity and hypergraph edge connectivity problem and has a lot of applications in MPLS, IP networks, synchronous optical networks, image segmentation, and other areas. However, due to limited communications between different communities, this problem was studied in different names, with some important existing literature citations missing, or sometimes the results are misleading with some errors. In this paper, we make a further investigation of this problem, give uniform definitions, fix existing errors, provide new insights and show some new results. Specifically, we show the relationship between non-overlapping version(each edge only has one label) and overlapping version(each edge has multiple labels), by fixing the error in the existing literature; hardness and approximation performance between weighted version and unweighted version and some useful properties for further research.
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Connectivity is a central notion of graph theory and plays an important role in graph algorithm design and applications. With emerging new applications in networks, a new type of graph connectivity problem has been getting more attention--hedge conne
ctivity. In this paper, we consider the model of hedge graphs without hedge overlaps, where edges are partitioned into subsets called hedges that fail together. The hedge connectivity of a graph is the minimum number of hedges whose removal disconnects the graph. This model is more general than the hypergraph, which brings new computational challenges. It has been a long open problem whether this problem is solvable in polynomial time. In this paper, we study the combinatorial properties of hedge graph connectivity without hedge overlaps, based on its extremal conditions as well as hedge contraction operations, which provide new insights into its algorithmic progress.
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{sc Neighbor}, and {sc Adjacency} queries. Given $epsilon in (0,1)$, the algorithm with high probability outputs an estimate $hat{t}$ satisfying the following $(1-epsilon) t leq hat{t} leq (1+epsilon) t$, where $m$ is the number of edges in the graph and $t$ is the size of minimum cut in the graph. The expected number of local queries used by our algorithm is $minleft{m+n,frac{m}{t}right}mbox{poly}left(log n,frac{1}{epsilon}right)$ where $n$ is the number of vertices in the graph. Eden and Rosenbaum showed that $Omega(m/t)$ many local queries are required for approximating the size of minimum cut in graphs. These two results together resolve the query complexity of the problem of estimating the size of minimum cut in graphs using local queries. Building on the lower bound of Eden and Rosenbaum, we show that, for all $t in mathbb{N}$, $Omega(m)$ local queries are required to decide if the size of the minimum cut in the graph is $t$ or $t-2$. Also, we show that, for any $t in mathbb{N}$, $Omega(m)$ local queries are required to find all the minimum cut edges even if it is promised that the input graph has a minimum cut of size $t$. Both of our lower bound results are randomized, and hold even if we can make {sc Random Edge} query apart from local queries.
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