The Sparsest Cut problem is NP-hard and therefore cannot be solved exactly in polynomial- time unless P = NP. We can hope only to find an approximate solution in polynomial-time. An approximation algorithm is an efficient algorithm that finds an approximate solution with a provable approximation guarantee.

What is Vertex cut?

In other words, a vertex cut is a subset of vertices of a graph which, if removed (or "cut")--together with any incident edges--disconnects the graph (i.e., forms a disconnected graph). A vertex cut set of size 1 corresponds to an articulation vertex.

What is an S T cut?

In a flow network, an s–t cut is a cut that requires the source and the sink to be in different subsets, and its cut-set only consists of edges going from the source's side to the sink's side. ... The capacity of an s–t cut is defined as the sum of the capacity of each edge in the cut-set.

What is minimum cut in a graph?

In graph theory, a minimum cut or min-cut of a graph is a cut (a partition of the vertices of a graph into two disjoint subsets) that is minimal in some metric. Variations of the minimum cut problem consider weighted graphs, directed graphs, terminals, and partitioning the vertices into more than two sets.

What is minimum vertex cut?

(definition) Definition: The smallest set of vertices in an undirected graph which separate two distinct vertices. That is, every path between them passes through some member of the cut.

SOS Seminar lecture 4 - Sparsest cut problem and ARV algorithm (updated)

32 related questions found

Is Min cut NP-complete?

We show that the Min Cut Linear Arrangement Problem (Min Cut) is NP-complete for trees with polynomial size edge weights and derive from this the NP-completeness of Min Cut for planar graphs with maximum vertex degree 3.

How do you calculate minimum cuts?

The minimum cut is a partition of the nodes into two groups. Once you find the max flow, the minimum cut can be found by creating the residual graph, and when traversing this residual network from the source to all reachable nodes, these nodes define one part of the partition.

What is minimum cut algorithm?

Min-Cut of a weighted graph is defined as the minimum sum of weights of (at least one)edges that when removed from the graph divides the graph into two groups. Mechthild Stoer and Frank Wagner proposed an algorithm in 1995 to find minimum cut in an undirected weighted graphs.

What is Mincut problem?

The minimum cut problem (abbreviated as “min cut”), is defined as follows: Input: Undirected graph G = (V,E) Output: A minimum cut S, that is, a partition of the nodes of G into S and V \ S that minimizes the number of edges going across the partition. ... Let n be the number of vertices and m be the number of edges.

What is cut edge with example?

Example. By removing the edge (c, e) from the graph, it becomes a disconnected graph. In the above graph, removing the edge (c, e) breaks the graph into two which is nothing but a disconnected graph. Hence, the edge (c, e) is a cut edge of the graph.

What is the capacity of an ST cut?

The minimum capacity of an s-t cut is 250 and the sum of the revenue of each project is 450; therefore the maximum profit g is 450 − 250 = 200, by selecting projects p₂ and p₃.

What is DFS graph?

Depth-first search (DFS) is an algorithm for traversing or searching tree or graph data structures. The algorithm starts at the root node (selecting some arbitrary node as the root node in the case of a graph) and explores as far as possible along each branch before backtracking.

What is cut vertices and cut edges?

A cut vertex is a vertex that when removed (with its boundary edges) from a graph creates more components than previously in the graph. A cut edge is an edge that when removed (the vertices stay in place) from a graph creates more components than previously in the graph. 31) The cut vertex is c. There are no cut edges.

What is 2 connected?

A graph is connected if for any two vertices x, y ∈ V (G), there is a path whose endpoints are x and y. A connected graph G is called 2-connected, if for every vertex x ∈ V (G), G − x is connected.

What is a cut set matrix?

A Cut Set Matrix is a minimal set of branches of a connected graph such that the removal of these branches causes the graph to be cut into exactly two parts. ... A Cut Set Matrix consists of one and only one branch of the network tree, together with any links which must be cut to divide the network into two parts.

What is graph cut algorithm?

In the graph-cut method, each image is represented as a graph of nodes. Each node corresponds to an image pixel and links connecting the nodes are called edges. A pathway is constructed connecting all the edges to travel across the graph.

What is bipartite matching?

The bipartite matching is a set of edges in a graph is chosen in such a way, that no two edges in that set will share an endpoint. The maximum matching is matching the maximum number of edges. ... For a bipartite graph, there can be more than one maximum matching is possible.

Are minimum cuts unique?

If all edge capacities are distinct, the max flow is unique. If all edge capacities are distinct, the min cut is unique. If all edge capacities are increased by an additive constant, the min cut remains unchanged. If all edge capacities are multiplied by a positive integer, the min cut remains unchanged.

Which algorithm is used to solve a minimum cut algorithm?

Which algorithm is used to solve a minimum cut algorithm? Explanation: Minimum cut algorithm is solved using Stoer-Wagner algorithm. Maximum flow problem is solved using Ford-Fulkerson algorithm.

Can there be multiple minimum cuts?

8 (final answer): If that cut is identical to your original cut, then there is only one. Otherwise, you just found 2 cuts, so the original one can't possibly be unique.

Is min cut problem NP hard?

(2) the weighted Min Cut problem is NP-complete even when restricted to trees with polynomial size weights. We also solve an open problem about the complexity of the search number problem described in a recent paper by Megiddo, Hakimi, Garey, Johnson, and Papadimitriou [IS] and several related problems.

Is Max flow NP-complete?

The maximum flow problem with minimum quantities was introduced in [4], where the problem was shown to be weakly NP-complete even on series-parallel graphs and Lagrangean relaxation techniques and heuristics for solving the problem were studied.