Using Homogeneous Weights for Approximating the Partial Cover Problem Reuven Bar-Yehuda

The Partial Set Cover problem Given: Find: min s.t. The partial set cover problem

The partial set cover problem Minimal t-cover C is a t-cover (a feasible solution) if and C is a minimal t-cover if , is not a t-cover The partial set cover problem

Example (a simple graph) 2 7 5 1 1 1 1 4 7 6 6 1 2 4 3 3 The partial set cover problem

The partial set cover problem Example: 13-cover 2 7 5 1 1 1 1 4 7 6 6 1 2 4 3 3 The partial set cover problem

Example: minimal 13-cover 2 7 5 1 1 1 1 4 7 6 6 1 2 4 3 3 The partial set cover problem

Example: optimal 13-cover 2 7 5 1 1 1 1 4 7 6 6 1 2 4 3 3 The partial set cover problem

Example (a hyper-graph) V 1 6 2 4 a 7 5 10 2 b c e={a,b,c,d} d The partial set cover problem

Example: minimal 11-cover 6 2 4 a 7 b 5 c 10 d 2 The partial set cover problem

Example: optimal 11-cover 6 2 4 a 7 b 5 c 10 d 2 The partial set cover problem

Homogeneous Weight function Definition: Property: any minimal cover is a “good” approximation The partial set cover problem

Homogeneous Weight function Given: Claim: C is a minimal t-cover (define: ) The partial set cover problem

The partial set cover problem Proof: Lemma 1: If C is a t-cover then Lemma 2: If C is a minimal t-cover then The partial set cover problem

The partial set cover problem Proof: Lemma 1: If C is a t-cover then If √ Else The partial set cover problem

The partial set cover problem Proof: Lemma 2 (a simple graph): If C is a minimal t-cover then Case 1: Case 2: (case 1) The partial set cover problem

The partial set cover problem Proof: Lemma 2 (a simple graph): If C is a minimal t-cover then Case 3: The partial set cover problem

L2: If C is a minimal t-cover then Case 3: Define: The partial set cover problem

The partial set cover problem To complete the proof: Let , and C is a minimal t-cover  But  The partial set cover problem

Example: hyper-graph, C is a minimal t-cover, δ(C) > 2t 4 1 a 1,9 b 1,9 c 2,5 d 3,5 e 5,4 The partial set cover problem

The partial set cover problem Proof: Lemma 2 (hyper-graph): If C is a minimal t-cover then Case 1: Case 2: (case 1) The partial set cover problem

L2: If C is a minimal t-cover then Case 3: Define: The partial set cover problem

The partial set cover problem To complete the proof: Let , and C is a minimal t-cover  But  The partial set cover problem

Homogeneous Weight function Claim: C is a minimal t-cover, C* optimal Proof: By Lemma 2: By Lemma 1: The partial set cover problem

The partial set cover problem Weight Decomposition Local-Ratio Theorem: C is an approximation w.r.t. C is an approximation w.r.t. ( ) The partial set cover problem

The partial set cover problem Algorithm Cover ( ) If return If return “no solution” If return Cover( ) Cover( ) The partial set cover problem

The partial set cover problem For each C = Cover( ) If is a t-cover Return C The partial set cover problem

The partial set cover problem Example 2 4 19 1 1 1 1 10 1 3 2 23 1 The partial set cover problem

The partial set cover problem Example 2 4,4 19,9 1 1 1 1,6 10 ω,δ 1 3 2,3 23,9 1 The partial set cover problem

The partial set cover problem Example 2 10/3 35/2 1 1 1 10 ω2 1 3 3/2 43/2 1 The partial set cover problem

The partial set cover problem Example 10/3,3 35/2,3 2 1 10 ω,δ 1 3/2,2 43/2,3 The partial set cover problem

The partial set cover problem Example 2 13/12 61/4 1 10 ω2 77/4 1 The partial set cover problem

The partial set cover problem Example 13/12,1 61/4,1 2 10 ω,δ 77/4,1 The partial set cover problem

The partial set cover problem Example 2 4 19 1 1 1 1 10 1 3 2 23 1 The partial set cover problem

The partial set cover problem Example 2 4 19 1 1 1 1 10 1 3 2 23 1 The partial set cover problem

The partial set cover problem Algorithm Cover Claim: Algorithm Cover is a approximation for the t-cover problem Proof: (L3: Algorithm Cover has at most iterations) By induction on the number of iterations. Base: for the empty set is optimal Step: by the Local-Ratio Theorem The partial set cover problem

The partial set cover problem Time complexity Lemma 4: Algorithm Cover can be implemented in time Proof: vertex deletion  adjacent edges deletion For each such edge, l(e) is subtracted from its |e| vertices  At most iterations, each iteration The partial set cover problem

The partial set cover problem Related work t-VC: (simple graph) Bshouty and Burroughs, 1998, 2-approximation Hochbaum, 1998 Here: t-VC with edge lengths: (simple graph) Hochbaum, 1998, 3-approximation Here: 2-approximation The partial set cover problem

The partial set cover problem Related work t-SC: Burroughs, 1998, ( )-approximation t-SC with edge lengths: Here: approximation time The partial set cover problem