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Covers, Dominations, Independent Sets and Matchings AmirHossein Bayegan Amirkabir University of Technology

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Basic Concepts Vertex Cover ◦ If G(V,E) is a graph, then a vertex of G is said to cover any edge of G that is incident with ◦ A set of vertices in G that covers every edge of G is called a vertex cover Edge Cover ◦ An edge covers its endpoints ◦ A set of edges in G that covers every vertex of G is called an edge cover

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Vertex Dominating Set ◦ A vertex is said to dominate both itself and any adjacent vertex ◦ A vertex dominating set is a set of vertices S such that every vertex in G is dominated by some vertex in S Edge Dominating Set ◦ An edge is said to dominate the edges incident to its endpoints as well as itself ◦ An edge dominating set is a set of vertices X such that every edge in G is dominated by some edge in X

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Independent Set of Vertices ◦ A set of vertices no two of which are adjacent Independent Set of Edges or Matching ◦ A set of edges no two of which are adjacent Minimum cardinality of a vertex cover of G: Min-VCov(G) Minimum cardinality of an edge cover of G: Min-ECov(G) Minimum cardinality of a vertex dominating set: Min-VDom(G) Minimum cardinality of a edge dominating set: Min-EDom(G) Maximum cardinality of a matching on G: Max-Match(G) Maximum cardinality of a vertex independent set: Max-Indep(G)

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Theorem : if G(V,E) is a graph with no isolated vertices then 1.Min-EDom(G) Max-Match(G) Min-VCov(G), and 2.Min-VDom(G) Max-Indep(G) Min-ECov(G) Theorem (Gallai’ s Complementarity Relations): if G(V,E) is a graph with no isolated vertices then 1.Max-Match(G) + Min-ECov(G) = |V(G)| 2.Max-Indep(G) + Min-VCov(G) = |V(G)|

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We present an O(n log n) time approximation algorithm with absolute worst-case ratio, due to Chiba, Nishizeki and Saito The algorithm is based on the following ideas: (i) we can reduce the problem on a general planar graph to one on a planar graph having no vertices of degree 4 or less; (ii) we can design an O(n log n) time “on-line” algorithm to execute any sequence of vertex-identifications and edge deletions of a given planar graph. Approximation algorithm for maximum independent set in planar graphs

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Lemma 7.1. Let a planar graph G = (V, E) contain a vertex v of degree 4 with N(v) = {v l, v 2, v 3, v 4 }, and let H be a subgraph of G induced by {v} U N(v). Then, renaming vertices in N(v) if necessary, we can assume that either (i) H is isomorphic with H 4 ; or (ii) H contains H’ 4 or H” 4 as a subgraph, and moreover (v 1,v 3 ) E Recall: Corollary 1.4. Every planar graph contains a vertex of degree at most 5

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Theorem 7.2. For any planar graph G = (V, E) with n vertices, the algorithm finds an independent set of at least vertices. Proof : We proceed by induction on the number n of vertices of G ◦ for any planar graph having n < 4 vertices, our algorithm always finds an independent vertex set of at least one vertex 1.Inductive hypothesis 2. algorithm always adds at least one vertex to I(G’) to form an independent set I(G) 3. the degree of v is 5 or less 1+2+3

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Theorem: If G is a planar connected graph with minimum degree 3, 4, or 5, a maximum independent set of G contains less than 2n/3, 2n/4, or 2n/5 vertices, respectively. Proof. Let I* V be a maximum independent set of G. Let G’ be the spanning subgraph obtained from G by deleting all the edges having both ends in V - I*, and assume that G’ has m’ edges. Since G‘ is a planar bipartite graph with n vertices, d(G, v) = d(G’, v) for every vertex v in so according to the minimum degree:, or So, or

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Theorem 7.1. For any planar graph G = (V, E), the algorithm finds an independent set of G with worst-case ratio Proof. ◦ If d m (G) 5 we use last two theorems ◦ If d m (G) 4 it suffice to show that We consider three cases depending on d m (G):

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The bound on the worst-case ratio is sharp

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The data structure of the algorithm:

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Lemma 7.2. Let G = (V, E) be a graph, and let s be any sequence of edge deletions on G. There exists an on-line algorithm to execute s in O(m) time. Lemma 7.3. Let G = (V, E) be a graph, and let s be any sequence of identifications on G. There exists an on-line algorithm to execute s in at most O(min {mlogn, n 2 }) time.

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Proof: The total amount of time T(s) required for s satisfies: we shall show that T(s) is at most O(m log n): Consider multigraphs G* i corresponding to G i, in which the multiple edges and loops produced by identifications are not deleted. We have The surrogate of u i, in G* i+1 has degree at least twice as large as u i, in G* i. The degree of a vertex is nondecreasing, and the degree of a vertex in G* i+1 is clearly at most 2m. Therefore the total cost is at most 2m log2m = O(m log n)

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Theorem 7.3. The algorithm spends at most O(n logn) time on a planar graph G =(V,E). we can find a vertex v of minimum degree in a constant time IdentificationO(m logn) edge-deletion O(m) deletion of an isolated vertexO(n) TotalO(n logn)

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Baker’s algorithm Approximation algorithm for the maximum independent set problem on planar graphs Attains the worst-case ratio k/(k + 1) and runs in O(8 k kn) time for any positive integer k K Time worst-case ratio 1 linear ½ loglog n O(n(1og n) 3 log log n) log log n/(1 + log log n)

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1. choose k as an appropriate integer 2. consider a plane embedding of a given graph G and define the level of a vertex as follows: the vertices on the outer boundary have level 1 ; the vertices on the outer boundary of the subgraph obtained by deleting the level i - 1 vertices have level i; 3. for each i, 0 i k, do the following: construct graph G i = G - V i, where V i is the set of all the vertices of level i(mod k+ 1 ) (all the components of G, are “k-outerplanar”, that is, it has an embedding having no vertex of level > k); find a maximum independent set I*(G i ) of G, by taking union of those of the components; 4. output the largest I*(G i ) as the solution of G.

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A maximum independent set I*(G) of G is partitioned into k + 1 subsets {I*(G) ∩ V i | i = 0, 1, , k} Clearly the smallest one contains at most 1/(k + 1) of I*(G) Therefore the largest I*(G i ) satisfies |I*(G i )|≥k|I*(G)|/(k + 1) worst-case ratio k/(k + 1)

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Matching and processor scheduling Suppose we have set of tasks each takes a unit time The order the tasks can be scheduled is restricted by precedence constraints We will show how to schedule these tasks in parallel on a pair of processors in such a way as to minimize the earliest completion of the system We shall bound the completion time in terms of the size of matching

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The optimal scheduling algorithm is as follows: 1. Identify tasks T i1 to T ik having no successors and assign them priorities 1 to k, respectively. Say next to k+1 2. repeat (2a and b) until every task has been assigned a priority a.For each task T all of whose successor tasks s 1,…,s m have been assigned a priority, let v(T) be an m-vector components as the priority labels of tasks in ascending order b.Select lexicographically smallest vector-labelled task T, and set priority(T) to next. Then set next to next+1 3. Whenever a processor becomes free assign it to the highest priority task remaining to complete

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31 (1,3) (3) 2 (2,6) (2,4) (5,8) (7)

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Theorem(Scheduling Lower Bound). Let a system of tasks constrained by a set of precedence constraints represented by a digraph G(V,E) be defined as shown. Let t min denote the minimum time in which the task system can be completed. Then t min M() returns a maximum matching C() returns the complement U() returns undirected graph T() returns transitive closure

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T1T1 T2T2 T5T5 T3T3 T4T4 Task precedence graph T1T1 T4T4 T3T3 T2T2 T5T5 Associated graph C(U(T(G))) T1T1 T4T4 T3T3 T2T2 T5T5 M M

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