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Fall 2008 CMPT 881. Algorithms for Maximum Induced Matching Problem. Somsubhra Sharangi. Outline. Maximum Induced matching problem An Application: Multi-Hop Wireless Networks with Constraints Complexity Results Approximation schemes for Induced Matching on Regular Graphs
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Fall 2008 CMPT 881 Algorithms for Maximum Induced Matching Problem Somsubhra Sharangi
Outline • Maximum Induced matching problem • An Application: Multi-Hop Wireless Networks with Constraints • Complexity Results • Approximation schemes for Induced Matching on Regular Graphs • Approximation schemes for Induced Matching on Geometric Graphs
u1 v1 u2 v2 u3 v3 u4 v4 Maximum Induced Matching An induced matching M is a matching such that no pair of edges of M are joined by an edge in graph G. u1 u1 v1 v1 u2 u2 v2 v2 u3 u3 v3 v3 u4 u4 v4 v4 Graph G Induced Matching M Ordinary Matching
An Application: Multi-Hop Wireless Network with Constraints • Static, Shared Channel • No collision detection mechanism • Example: Mesh Networks, Sensor Networks etc. • Protocol: IEEE 802.11 RTS/CTS model What is the capacity of the network?
Complexity Results • Maximum Induced Matching is NP-Hard (from 3 SAT). (x1 ν x2 v x3)Λ(~x1 v x2 v x3) Λ(x1 v ~x2 v ~x3) • k-separated matching • k-separated matching for k≥2 is not approximable within |V|(1/2 - Є) unless P=NP and within |V|(1 - Є) unless NP = ZPP for any Є>0 (from max Independent Set). • k-separated matching can be approximated within a factor of Θ(|E|/(log|E|)2) (from max Independent Set)
Maximum Induced Matching on Regular Graphs • For any d-regular graph, where d 3: • An approximation algorithm with asymptotic performance guarantee d – 1 let M be the empty matching; select an edge {u,v} from E; add {u,v} to M; delete each edge at distance ≤ 2 from {u,v}; delete each vertex adjacent to u or v; while there is some edge in G loop choose a vertex u of minimum degree; choose a vertex v of minimum degree adjacent to u; add {u,v} to M; delete each edge at distance ≤ 2 from {u,v}; delete each vertex adjacent to u or v; end loop
u u v v u v u v u u v v Maximum Induced Matching on Regular Graphs
Maximum Induced Matching on Regular Graphs • Let G=(V,E) be a d-regular graph, where n=|V| • The algorithm produces an induced matching M where • Any induced matching M* satisfies • let M be an induced matching returned by A and let M* be a maximum induced matching in G • The algorithm has asymptotic performance guarantee d - 1
Maximum Induced Matching on Geometric Graphs • The k-hop interference set of an edge eЄ E, denoted by Ik(e), is the set of edges uЄ E such that d(e, u) ≤ k • A subset S of Ik(e) is called k-maximal if no other edge u Є Ik(e) can be added to S such that we have d(u, v) > k, for all v ЄIk(e) • The k-hop interference degreeof an edge e Є E, denoted by dk(e), is defined as • The K-hop interference degree of the graph G = (V,E), denoted by dK(G), is defined as
Maximum Induced Matching on Geometric Graphs let M be the empty matching and i = 1; arrange edges of E in descending order of weight starting with e1,e2,... repeat for all edges in E if M U ei is k-separated matching then M:= M U ei and increment i end loop • The weight of the matching returned by the greedy algorithm is always within a factor dk(G) of the weight of an optimal matching.
