On the complexity of fixed parameter clique and dominating set Friedrich Eisenbrand, Fabrizio Grandoni(2004) Present by Xiaoyan Zhao
Paper Outline Fixed parameter clique problem clique in dense graphs(*) clique in sparse graphs Fixed parameter dominating sets(*) Detection of the induced diamonds The first two are believed not to be fixed parameter tractable 1/10
Definition and notation overview Clique A graph such that each pair of distinct nodes is adjacent, e.g., triangle is a 3 nodes clique. Dominating set A subset V' dominates a node u in V if either u belongs to V' or u is adjacent to at least one node in V'. The set V' is called a dominating set of G if all the nodes of G are dominated by V '. Induced subgraph G' =(V ' , E ' ) of G Two nodes are adjacent in G' if and only if they are adjacent in G. G' is denoted by G[V' ]. W(r,s,t) denotes the running time of the × multiplication of an matrix by an × s t r s n n n n matrix. 2/10
Fixed parameter clique problem Definition: determine whether a graph G of n nodes contains a clique of l nodes, where l is the parameter. Two facts about the clique problem: ω ω < 2 3 7 6 detection of a triangle: , where is the O ( n ) . exponent of fast square matrix multiplication. a node v is contained in an l clique if and only if the graph G[N(v)] induced on G contains an l-1 clique. Major improvement on this paper: previous result: new result: 3/10
Fixed parameter clique problem (2) Previous best algorithm outline Clique(G. l): if l == 3h , then creates an auxiliary graph G' in the following way: creates a node for each h -node clique Creates an edge between two nodes if and only if them form a 2h -node clique in G . G contains a 3h -node clique if and only if G' contains a triangle, and the time bound to find a ω triangle in G' is . h O ( n ) else for each node v in V apply the above algorithm on the induced graph G[N(v)] to detect l-1 node clique. α ( l ) α = ω + The time bound is where . O ( n ) , ( l ) l / 3 l ( m o d 3 ) 4/10
Fixed parameter clique problem (3) New algorithm outline Clique(G,l): = = − = Let , , . Create a 3-partite l l / 3 l ( l 1 ) / 3 l l / 3 1 2 3 auxiliary graph G' in the following way: Partition the nodes into sets , where the nodes in are the V V i i l i cliques of order of G, for i ={1,2,3}. Create an edge between a node u in and a node v in , if V V i j + and only if these two nodes induce an clique in G. l l i j Detect a triangle of G' in the following way: for each pair of nodes {u, v} where u in and v in , compute V V 1 3 the number P(u,v) of 2-length paths between u and v through a V node of . The graph G' contains a triangle if and only if there 2 V V is a pair of adjacent nodes {u, v} where u in and v in such 1 3 that P(u,v) >0. The time is bounded by adjacency matrices multiplication. 5/10
Fixed parameter clique problem(4) Result comparisons: 6/10
Fixed parameter dominating set Definition: determine whether an undirected graph G of n nodes contains a dominating set of l nodes. A previous fastest known algorithm: enumerates all the subsets of l nodes of G and test whether one of these subset forms + 1 a dominating set, in time . l O ( n ) A new improved algorithm based on fast ω matrix multiplication : in time . ( l , 1 , l ) O ( n ) 1 2 7/10
Fixed parameter dominating set(2) The new algorithm outline: : the set of subsets of V of cardinality h V h : a 0-1 matrix whose rows are indexed by the D h elements of and whose columns are indexed by V h the elements of V . Given w in and v in V , V h [ ] = 0 if and only if w dominates v . D w , v h Compute the matrix , where , . ' = ⋅ T = = D D D l l / 2 l l / 2 l l 1 2 1 2 Clearly that D' contains a zero entry if and only if G admits a dominating set of size l . The time is bounded by the matrix multiplication cost. 8/10
Fixed parameter dominating set(3) Result comparisons : 9/10
Questions? Thank you!! Present by Xiaoyan Zhao
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