Approximating Minimum Power Degree and Connectivity Problems

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

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

5. undirected directed c(H) = n-1 p(H) = 1 c(H) = n-1 p(H) = n Power vs Cost Definition: Let H=(V,F) be a graph with edge-costs {c(e):eF} power of v in H: pF(v) = max{c(e):eF(v)} = maximum cost of an edge leaving v The power of H: p(H) = pF(V)= ∑vVpF(v) 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 −−−

6. Defining the Problems Minimum Power Edge-Multi-Cover (MPEMC) Definition: Given a degree requirement function r on V, an edge set F on V is an r-edge-cover if degF(v) ≥ r(v) for all vV 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 Hof G so that H is an r-edge-cover. 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 Hof G.

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

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 • Theorem: • ≤ 2 + [HKMN05, JKMWY05] •  ≤ 2+1 [HKMN05] •  ≤  [LN07] Corollary: =Θ() provided =O(ρ). Currently, ρ= O(log k log n/(n-k))=O(log2k) [FL08,N08] Previous best value of (and of ): O(log4n) [HKMN05]

9. 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)). Theorem 2 MPEC admits a 3/2-approximation algorithm. Our Result Previous ratio for MPEMC, MPkCS: O(log4n) [HKMN05]. What about MPEC, when we have 0,1 requirements? Previous ratio for MPEC: 2.

10. Proof of Theorem 1 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 • 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

11. Reduction to bipartite graphs  - approximation algorithm for bipartite MPEMC implies 2-approximation algorithm for general MPEMC. • 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 auav , avauwith cost c(uv) each; • r '(bv)=r(v)for bv B and r '(av)=0for avin A. av au A u v B bv bu

12. Algorithm for bipartite graphs Definition: For an edge set I, the residual requirement of bB is: rI(b)=max{r(b)-degI(b),0}; let rI(B)=ΣbB rI(b). • 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) pI(V) ≤(1+γ) τ • (2) rI(B) ≤ (1-) r(B)=(1-1/e)(1-1/γ) • The Algorithm: • Initialization: F ← , γ← 1/2 • Whiler(B) > 0 do: • Find the smallest τ so that A returns I E satisfying (1),(2). • F ← F +I, E ← E–I, r←rI . • EndWhile The approximation ratio:O(log r(B))=O(log n2)=O(log n).

13. Proof of The Main Lemma Definition: Let R=r(B). An edge abE is dangerous if c(ab) ≥ γτ · r(b)/R. • Lemma 1: If τ ≥ opt then rJ(B) ≥ R(1-1/γ) for any set J of • dangerous edges with pJ(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. Proof:Let D={bB :degJ(b) ≥ 1}. Then Thus r(D) ≤ R/γ, which implies rJ(B) ≥ R-r(D) ≥ R(1-1/γ) Lemma 2: pF(B) ≤ γτfor any set F of non-dangerous edges. Proof:

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 γτat B. Thus after the dangerous edges are ignored, we obtain the problem: (*) max{r(B)-rI(B) : I E,pI(A) ≤ τ} 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: • Delete all dangerous edges. • Let I be the edge set returned by the (1-1/e)-approximation algorithm for Problem (*). • If rI(B) ≤ (1-)R then return I; • Else declare “τ≤ opt”.

15. A 3/2-Approximation Algorithm for MPEC Minimum Power Edge-Cover (MPEC) Instance: A graph G=(V,E), edge-costs {c(e):eE}, and SV. Objective: Find a minimum power S-cover I E. The idea behind the algorithm: Reduction to Min-Cost Edge-Cover (solvable in polynomial time) with loss of 3/2in the approximation ratio. Algorithm: • 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. • 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)). • Find a minimum-cost edge-cover I′ in G′,c′. • Return I = {I(uv) : uvI′}. U

16. Approximation Ratio The Main Lemma: • If I′ is an edge cover in G′ then I covers S in G and p(I) ≤ c′(I′). • opt′ ≤3/2 ·opt (opt′ = minimum-cost of an edge-cover in G′,c′) The ratio 3/2 follows since: p(I) ≤ c′(I′) = opt′ ≤ 3/2 ·opt. 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)

17. Definition: 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. Lemma: For general costs, any star I admits a 2-decomposition D so that: p(D) ≤ 3/2 · p(I) 2-Decompositions of Stars p(I) = 7 p(D)=10 p(I) = 6 p(D)=8 For unit costs, p(I)=l+1 and p(D) ≤3·l/2+1, so p(D)/p(I) ≤ 3/2.

18. Summary and Open Questions Results: • O(log n)-approximation for MPEMC. • O(log n log n/(n-k))-approximation for MPkCS. • 3/2-approximation for MPEC. Open Questions: • Constant ratio for MPEMC? • (log n)-hardness for MPEMC? • Approximation hardness of MPkCS/MCkCS… • 4/3-approximation for MPEC?

19. Questions?