Constant Factor Approximation of Vertex Cuts in Planar Graphs Eyal Amir, Robert Krauthgamer, Satish Rao Presented by Elif Kolotoglu

Outline  Introduction  Definitions  The Structural Theorem  The Algorithm  Conclusion

Outline  Introduction  Definitions  The Structural Theorem  The Algorithm  Conclusion

Introduction  Graph partitioning is used in many areas e.g. VLSI design, task scheduling etc.  An important graph partitioning problem is vertex-separator problem  Quotient vertex-cut problem is closely related to vertex-separator problem

Introduction  First constant approximation algorithm for minimum quotient vertex-cuts in planar graphs  Approximation ratio 1+4/3(1+ε) with running time O(W.n 3+2/ε )

Outline  Introduction  Definitions  The Structural Theorem  The Algorithm  Conclusion

Definitions  Vertex-separator problem (NP-hard) Let G=(V,E) be a graph with |V|=n, and vertex costs c:V N and vertex weights w:V N. A vertex-cut of G is a partition of V into three disjoint sets A, B, C s.t. no edge in G has one endpoint in A and the other in B. The cost of the vertex-cut is c(C)=∑ vєC c(v). A vertex cut is a vertex-separator if max{w(A),w(B)}≤2/3w(V) The vertex-separator problem is to find a minimum-cost vertex separator in an input graph G.

Definitions  Quotient vertex-cut problem (NP-hard) The quotient-cost of the vertex–cut is defined as: q(A,B,C)=c(C)/min{w(A),w(B)}+w(C) The minimum quotient vertex-cut problem is to find a vertex-cut with minimum quotient-cost in an input graph.

Notations for the Structural Theorem  G: a simple planar graph (input graph)  G 0 : plane graph of G  G T : face-vertex graph of G  Face-vertex graph: a triangular (every face has exactly three distinct edges in its boundary) graph obtained by adding a vertex to each face f of G 0 and connecting the new vertex to every vertex in the boundary of f.

Notations for the Structural Theorem  Vertex weights and vertex costs of G 0 =(V 0,E 0 ) are extended to G T =(V T, E T ) by w(v)=c(v)=0 for all vє V T /V 0 w(v) and c(v) are the same in G T as in G 0 for all the vertices in V 0  Any vertex-cut of G 0 corresponds to a vertex-cut in G T with the same weights and costs, and vice versa

Example G 0 G T

Definitions for the Structural Theorem  λ-quotient cost of a vertex-cut (A,B,C) is q λ (A,B,C)=c(C)/min{w(A),w(B)}+λ.w(C)

Outline  Introduction  Definitions  The Structural Theorem  The Algorithm  Conclusion

The Structural Theorem  In every plane graph there is a vertex-cut that is defined by a d-CAST (Cycles Arranged in a Shallow Tree, a collection of directed cycles that are arranged in a tree-like structure of constant depth) in G T and its 0-quotient cost is within a factor of 4(d+1)/3d from the minimum (over all vertex-cuts) in G 0.

The Structural Theorem  This theorem can be extended to any λ-quotient cost for arbitrary λ  Then the factor becomes [4(d+1)/3d]+1  Using this theorem they device an algorithm for finding a cut with near- optimal 1-quotient cost

Outline  Introduction  Definitions  The Structural Theorem  The Algorithm  Conclusion

The Algorithm  Outline of the algorithm  Given a planar input graph G, fix a planar embedding G 0 of it in the plane  Compute its face vertex graph G T  Assign weights to the faces of G T, s.t. the weight of face f of G T is ∑ vєf w(v)/deg(v)  Construct a search graph G S which is a directed version of G T with some edge weights (using this graph search for a d-CAST structure in G T )  Among a certain family of closed walks in G S, find a walk that has minimum cost to weight quotient

The Algorithm  The Search Graph  The search graph G S is defined to be the digraph obtained from G T by replacing every undirected edge (u,v) by a pair of directed edges (u,v) and (v,u).  The vertices of G S have the same cost they have in G T  Let T be any spanning tree of the graph G T. Choose a vertex r on the outer face as a root. Orient the tree, order the children of any vertex in counterclockwise direction

The Algorithm  Give a unique label to each region immediately surrounding each vertex v in the tree as follows:  Starting at the root, traverse the tree in a DFS manner  Each time a vertex is visited in this traversal, one immediate region around v, namely, the one between the edges used to enter and to exit v, is labeled with the traversal’s step number

The Algorithm  Assign the weight of each directed edge (u,v) in G S as follows:  If the underlying undirected edge belongs to the tree T, then w(u,v) = 0; otherwise, (u,v) is embedded in some region relative to each of its endpoints. Let t(u,i) and t(v,j) be the labels of these regions  Assume without loss of generality that t(u,i) < t(v,j), and let w(u,v) be the total weight of the faces enclosed by the simple cycle that (u,v) closes in the tree T, let w(v,u) = -w(u,v)

Example

The Algorithm  Walks with pebbles  For a walk Π, let Π e i denote the ith edge in the walk and let Π i denote the ith vertex in the walk  The total weight of a walk Π is w(Π)=∑w(Π e i ) and its total cost is c(Π)=∑c(Π i ).  The actual cost of a walk Π is the sum of costs of all the distinct vertices in Π.  d-pebble walks is a family of walks that allows, for a limited number of vertices, additional visits to these vertices at no additional cost

The Algorithm  Counted cost of a walk Π is the sum of the cost of all non-free visits to the vertices of the walk  Counted cost of a walk is an upper bound on its actual cost  Use a dynamic programming algorithm to find a special walk called d-pebble walk of minimum counted cost  A vertex-cut of small quotient can be extracted from the walk found in the dynamic program

Outline  Introduction  Definitions  The Structural Theorem  The Algorithm  Conclusion

Conclusion  Vertex separators and vertex-cuts are closely related with minimum width tree decompositions and branch decompositions  A method for solving these problems can be obtained by recursively using a balanced vertex-cut algorithm

Conclusion  Using the technique proposed in this paper, they could find a tree decomposition (for planar graphs) whose width is within a constant factor of the optimal, but it is worse than the best known approximation ratio so far  Authors claim that the running time can be improved using this technique, though not the approximation ratio

Homework  What is the main idea given in the structural theorem?  What is the contribution of the paper to the graph partitioning field and specifically to the tree decompositions?

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