Algorithms for Maximum Induced Matching Problem Somsubhra Sharangi Fall 2008 CMPT 881

2 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

3 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. u1u1 u2u2 u3u3 u4u4 v1v1 v2v2 v3v3 v4v4 u1u1 u2u2 u3u3 u4u4 v1v1 v2v2 v3v3 v4v4 u1u1 u2u2 u3u3 u4u4 v1v1 v2v2 v3v3 v4v4 Graph G Ordinary Matching Induced Matching M

4 An Application: Multi-Hop Wireless Network with Constraints  Static, Shared Channel  No collision detection mechanism  Example: Mesh Networks, Sensor Networks etc.  Protocol: IEEE RTS/CTS model What is the capacity of the network?

5 Complexity Results  Maximum Induced Matching is NP-Hard (from 3 SAT). (x 1 ν x 2 v x 3 ) Λ (~x 1 v x 2 v x 3 ) Λ (x 1 v ~x 2 v ~x 3 )  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)

6 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

7 Maximum Induced Matching on Regular Graphs u v uv uv uv uv u v

8  Let G=(V,E) be a d-regular graph, where n=|V|  The algorithm produces an induced matching M where  let M be an induced matching returned by A and let M * be a maximum induced matching in G  Any induced matching M* satisfies  The algorithm has asymptotic performance guarantee d - 1

9 Maximum Induced Matching on Geometric Graphs  The k-hop interference set of an edge e Є E, denoted by I k (e), is the set of edges u Є E such that d(e, u) ≤ k  A subset S of I k (e) is called k-maximal if no other edge u Є I k (e) can be added to S such that we have d(u, v) > k, for all v Є I k (e)  The k-hop interference degree of an edge e Є E, denoted by d k (e), is defined as  The K-hop interference degree of the graph G = (V,E), denoted by dK(G), is defined as

10 Maximum Induced Matching on Geometric Graphs  The weight of the matching returned by the greedy algorithm is always within a factor d k (G) of the weight of an optimal matching. let M be the empty matching and i = 1; arrange edges of E in descending order of weight starting with e 1,e 2,... repeat for all edges in E if M U e i is k-separated matching then M:= M U e i and increment i end loop

11 Maximum Induced Matching on Geometric Graphs

12 PTAS for Geometric Graph