1. Approximating Minimum Power Degree and Connectivity Problems Zeev Nutov The Open University of Israel Joint Work with: Guy Kortsarz Vahab Mirrokni Elena Tsanko

2. 2 Talk Outline Min-Power Problems - Motivation Defining the Problems Relations Between the Problems Our Results O(log n)-Approximation Algorithm for Min-Power Edge-Multi-Cover (MPEMC) 3/2-Approximation Algorithm for Min-Power Edge-Cover (MPEC)

3. 3 The Cost Measure-Wired Networks: connecting every two nodes incurs a cost. Nodes in the network correspond to transmitters. More power  larger transmission range. Transmission range = usually (but not always) disk centered at the node. The Power Measure-Wireless Networks: every node connects to all nodes in its “range”. The Power Measure-Motivation Goal: Find min-power range assignment so that the resulting communication network satisfies some prescribed property.

4. 4 b a c d g f e a b d g f e c Range assignment Communication network Example

5. 5 Relating powers and costs: Directed: c(H)/Δ(H) ≤ p(H) ≤ c(H) (Δ(H)=max-outdegree) Undirected: c(H)/√|F|/2 ≤ p(H) ≤ 2c(H) c(H) ≤ p(H) ≤ 2c(H) if H is a forest c(H) = n-1 p(H) = n c(H) = n-1 p(H) = 1 directed undirected Power vs Cost Definition: Let H=(V,F) be a graph with edge-costs {c(e):e  F} power of v in H: p F (v) = max{c(e):e  F (v)} = maximum cost of an edge leaving v The power of H: p(H) = p F (V)= ∑ v  V p F (v) −−−

6. 6 Minimum Power Edge-Multi-Cover (MPEMC) Instance: A graph G = (V,E) with edge costs {c(e):e  E}, and degree requirements {r(v):v  V}. Objective: Find a minimum power subgraph H of G so that H is an r-edge-cover. Defining the Problems Definition: Given a degree requirement function r on V, an edge set F on V is an r-edge-cover if deg F (v) ≥ r(v) for all v  V Minimum Power k-Connected Subgraph (MPkCS) Instance: A graph G = (V,E) with edge costs {c(e):e  E}, and an integer k. Objective: Find a minimum power k-connected spanning subgraph H of G.

7. 7 k-1 k-clique (r max +1)- Approximation for MPEMC Algorithm: For every v  V pick a set F(v) of r(v) cheapest edges incident to v. Tight example costs = 1 requirements: r(v)=k-1 for clique nodes. opt = k (the clique edges) Algorithm : k·k (edges of the stars) Claim: The approximation ratio is (r max +1) and this is tight. Proof: Let π(v) = max{c(e):e  F(v)}. Clearly, Σ v  V π(v) ≤ opt.

8. 8 Relating Approximation Ratios  = approximation ratio for MPkCS  = approximation for MPEMC with r(v)=k-1 for all v  V ρ = approximation ratio for MCkCS Currently, ρ = O ( log k log n/(n-k) ) =O(log 2 k) [FL08,N08] Corollary:  =Θ(  ) provided  =O(ρ). Theorem:  ≤ 2  +  [HKMN05, JKMWY05]  ≤ 2  +1 [HKMN05]  ≤  [LN07] Previous best value of  (and of  ): O(log 4 n) [HKMN05]

9. 9 Our Result Theorem 1 MPEMC admits an O(log n)-approximation algorithm. Thus MPkCS admits an approximation algorithm with ratio O ( log n + log k log n/(n-k) ) = O ( log n log n/(n-k) ). Previous ratio for MPEMC, MPkCS: O(log 4 n) [HKMN05]. What about MPEC, when we have 0,1 requirements? Previous ratio for MPEC: 2. Theorem 2 MPEC admits a 3/2-approximation algorithm.

10. 10 Proof of Theorem 1 Reduction to bipartite graphs Algorithm: iteratively covers a constant fraction of the total requirement with edge set of power ~ opt Ignoring dangerous edges: Reduction to a special case of Budgeted Multi-Coverage with Group Constraints problem Remark: Standard greedy methods do not work, because: Claim: The “budgeted” version of MPEMC is harder than the Densest k-Subgraph problem. Proof: Given an instance G,k of DkS set: {c(e)=1: e  E}, {r(v)=k-1: v  V}, and budget P=k. In the budgeted MPEMC we seek a k-subgraph with maximum number of edges; this is exactly DkS. Proof Outline

11. 11 Reduction to bipartite graphs  - approximation algorithm for bipartite MPEMC implies 2  -approximation algorithm for general MPEMC. auau avav bvbv bubu A B u v The Reduction: Given an instance G=(V,E),c,r of MPEMC, construct an instance G'=(A+B,E′),c ',r ' of bipartite MPEMC: -each of A,B is a copy of V; -for every uv  E there are edges a u a v, a v a u with cost c(uv) each; -r '(b v )=r(v) for b v  B and r '(a v )=0 for a v in A.

12. 12 Algorithm for bipartite graphs The Main Lemma: There exists a polynomial algorithm A that given an integer τ and γ > 1 either establishes that τ ≤ opt, or returns an edge set I so that: (1) p I (V) ≤ (1+γ) τ (2) r I (B) ≤ (1-  ) r(B)  =(1-1/e)(1-1/γ) Definition: For an edge set I, the residual requirement of b  B is: r I (b)=max{r(b)-deg I (b),0}; let r I (B)= Σ b  B r I (b). The Algorithm: Initialization: F ← , γ ← 1/2 While r(B) > 0 do: Find the smallest τ so that A returns I  E satisfying (1),(2). F ← F +I, E ← E–I, r←r I. EndWhile The approximation ratio: O ( log r(B) ) =O(log n 2 )=O(log n).

13. 13 Proof of The Main Lemma Lemma 1: If τ ≥ opt then r J (B) ≥ R(1-1/γ) for any set J of dangerous edges with p J (B) ≤ τ. Thus: - The dangerous edges in OPT cover at most R/γ of the demand; - The non-dangerous edges cover at least (1-1/γ)R of the demand. Definition: Let R=r(B). An edge ab  E is dangerous if c(ab) ≥ γτ · r(b)/R. Proof: Let D={b  B :deg J (b) ≥ 1}. Then Lemma 2: p F (B) ≤ γτ for any set F of non-dangerous edges. Proof: Thus r(D) ≤ R/γ, which implies r J (B) ≥ R-r(D) ≥ R(1-1/γ)

14. 14 Finishing the Proof of The Main Lemma Corollary: If τ ≥ opt then the non-dangerous edges: - cover at least (1-1/γ)R of the demand; - incur power at most γτ at B. Thus after the dangerous edges are ignored, we obtain the problem: Problem (*) admits a (1-1/e)-approximation algorithm: The proof is slightly more complicated than the proof of [KMN99] that Budgeted Max-Coverage admits a (1-1/e)-approximation algorithm. Algorithm A : 1.Delete all dangerous edges. 2.Let I be the edge set returned by the (1-1/e)- approximation algorithm for Problem (*). 3.If r I (B) ≤ (1-  )R then return I; Else declare “τ ≤ opt”. (*) max{r(B)-r I (B) : I  E, p I (A) ≤ τ}

15. 15 A 3/2-Approximation Algorithm for MPEC Minimum Power Edge-Cover (MPEC) The idea behind the algorithm: Reduction to Min- Cost Edge-Cover (solvable in polynomial time) with loss of 3/2 in the approximation ratio. 1.For every u,v  S compute a minimum {u,v}-cover I(uv) that consists of the edge uv or of two adjacent edges su,sv. 2.Construct an instance G′=(S,E′),c′ of Min- Cost Edge-Cover: G′ is a complete graph on S and c′(uv)=p ( I(uv) ). 3.Find a minimum- cost edge-cover I′ in G′,c′. 4.Return I = {I(uv) : uv  I′}. Instance: A graph G=(V,E), edge-costs {c(e):e  E}, and S  V. Objective: Find a minimum power S-cover I  E. U Algorithm:

16. 16 Proof-Sketch: Any inclusion minimal S-cover is a collection of stars. Thus it is enough to consider the case when OPT is a star. Recall Step 1 in the algorithm: For every u,v  S compute a minimum {u,v}-cover I(uv) that consists of one edge or of two adjacent edges. We prove: any star I with costs can be decompose into 2-stars and single edges (with at least one edge) so that: The sum of the powers of 2-stars and edges ≤ 3/2·p(I) (i)If I′ is an edge cover in G′ then I covers S in G and p(I) ≤ c′(I′). (ii)opt′ ≤ 3/2 · opt (opt′ = minimum-cost of an edge-cover in G′,c′) The Main Lemma: Approximation Ratio The ratio 3/2 follows since: p(I) ≤ c′(I′) = opt′ ≤ 3/2 ·opt.

17. 17 For unit costs, p(I)= l +1 and p( D ) ≤ 3· l /2+1, so p( D )/p(I) ≤ 3/2. 2-Decompositions of Stars A 2-decomposition of a star I is a partition D of I into 2-stars and edges (with at least one edge) that covers the nodes of I. The power of D is the sum of the powers of is parts. Definition: p(I) = 6 p( D )=8 p(I) = 7 p( D )=10 Lemma: For general costs, any star I admits a 2-decomposition D so that: p( D ) ≤ 3/2 · p(I)

18. 18 Summary and Open Questions 1. O(log n)-approximation for MPEMC. 2. O ( log n log n/(n-k) ) -approximation for MPkCS. 3. 3/2-approximation for MPEC. 1.Constant ratio for MPEMC? 2.  (log n)-hardness for MPEMC? 3.Approximation hardness of MPkCS/MCkCS… 4.4/3-approximation for MPEC? Results: Open Questions: