Approximating Source Location and Star Survivable Network Problems Zeev Nutov The Open University of Israel Joint work with Guy Kortsarz

The problems Source Location (SL) Given: A graph G=(V,E) with node-costs {c v :v ∈ V}, connectivity demands {d(v):v ∈ V}. Find: A min-cost set S ⊆ V of sources such that (S,v)-connectivity ≥ d(v) for every node v. Network Augmentation (NA) (a.k.a. SNDP) Given: A graph G=(V,E) and an edge set F on V, edge-costs {c e :e ∈ F} or node-costs {c v :v ∈ V}, and connectivity requirements {r(u,v) : (u,v) ∈ D ⊆ V × V}. Find: A minimum-cost edge set I ⊆ F such that in G+I the (u,v)-connectivity ≥ r(u,v) for every (u,v) ∈ D.

Network Augmentation (NA) Given: A graph G=(V,E) and an edge set F on V, edge-costs {c e :e ∈ F} or node-costs {c v :v ∈ V}, connectivity requirements {r(u,v):(u,v) ∈ D}. Find: A minimum-cost edge set I ⊆ F such that in G+I the (u,v)-connectivity ≥ r(u,v) for all (u,v) ∈ D. Some special cases of NA Survivable Network Design Problem (SNDP): E= ∅. Rooted NA: The demand set D is a star with center s. Star-NA: The edge set F is a star with center t. Connectivity Augmentation: Any edge has cost 1.

Connectivity functions (u,v)-q-connectivity is defined as the maximum (u,v)-flow value (every edge has capacity 1) Let {p v :v ∈ V} be “connectivity bonuses” (node v chosen to S gets connectivity bonus p v ). (S,v)-(p,q)-connectivity is defined by Let G=(V,E) be a network with node capacities {q v :v ∈ V}. Observation: The set function is submodular.

Example S v (s,v)-q-connectivity = max (s,v)-flow value (S,v)-(p,q)-connectivity = max (s,v)-flow value in the network obtained by adding a new node s and p u edges from s to every u ∈ S (p,q) ( ∞,1) (3,1) (2,1) (2,2) s

Example s S (s,v)-q-connectivity = max (s,v)-flow value (S,v)-(p,q)-connectivity = max (s,v)-flow value in the network obtained by adding a new node s and p u edges from s to every u ∈ S (p,q) ( ∞,1) (3,1) (2,1) (2,2) ∞ v

edge-connectivity λ G (S,v) p v =q v = ∞ for all v ∈ V (if v ∈ S then λ G (S,v)= ∞ ) node-connectivity p v = ∞ and q v =1 for all v ∈ V (if v ∈ S then ) node-connectivity κ’ G (S,v) p v =q v = 1 for all v ∈ V. node-connectivity κ G (S,v) v ∈ S : κ G (S,v) = ∞ v ∉ S: κ G (S,v) = maximum number of (S,v)-disjoint path. In digraphs, this version is equivalent to λ G (S,v). Connectivity functions in SL problems

Network Augmentation (NA) Given: A graph G=(V,E) and an edge set F on V, edge-costs {c e :e ∈ F} or node-costs {c v :v ∈ V}, node capacities {q v :v ∈ V}, and connectivity requirements {r(u,v):(u,v) ∈ D}. Find: A minimum-cost edge set I ⊆ F such that for all (u,v) ∈ D. Some special cases of NA Rooted NA: The demand set D is a star with center s. Star-NA: The edge set F is a star with center t.

Lemma: For both graphs and digraphs, SL is equivalent to Rooted Star-NA with node-costs and s=t. Relation between SL and NA problems Corollary: Directed SL is Set-Cover hard for unit demands and costs. For uniform demands and p=q=1, the problem is in P. s G c s =0 v r(s,v)=d(v) p v edges Add a new node s of cost 0. For every v ∈ V do:  set r(s,v)=d(v).  put p v sv-edges into F.

[Bar Ilan, Kortsarz, Peleg 96] Edge-Connectivity SL admits ratio 1+ln d(V). [Sakashita, Makino, Fujishige, LATIN 06] SL with connectivity functions admits ratio 1+ln d(V), and this is tight. [Kortzars, N, ICALP 06] Directed Edge-Connectivity Augmentation admits ratio 1+ln r(D) (reduction to a special case of Star-NA). Some previous work on digraphs NA: r(D) = sum of the requirements, p max = max # of parallel edges in F SL : d(V) = sum of the demands, p max = max connectivity bonus Common technique: Greedy algorithm for Submodular Cover.

[N, Approx 03] Undirected Rooted Connectivity Augmentation admits ratio O(ln 2 k) and is Set-Cover hard, where k= max r(s,v) = maximum requirement. [Chuzhoy, Khanna FOCS 09] Undirected SNDP with edge-costs admits ratio O(k 3 ln n). [N, FOCS 09] Undirected Rooted NA with edge-costs admits ratio O(k ln k). [Fukunaga, TAMC 11] Undirected κ ‘-SL is equivalent to Rooted Star-NA with edge-costs. Thus undirected κ ‘-SL admits ratio O(k ln k). Some previous work on graphs

Theorem 1: Directed Star-NA admits ratios 1+ln n for edge costs; 1+ln r(D) for node-costs. Thus directed SL admits ratio 1+ ln r(D), and this is so for any submodular connectivity function. Main Results Theorem 2: Undirected Star-NA admits ratios O(ln 2 k) for edge-costs; p max ∙ O(ln 2 k) for node-costs. Thus undirected SL admits ratio p max ∙ O(ln 2 k), and κ ‘-SL (the case p max =1) admits ratio O(ln 2 k). This improves the ratio O(k ln k) of [Fukunaga, TAMC 11].

Theorem 2: Directed Star-NA admits ratios 1+ ln n for edge costs; 1+ ln r(D) for node-costs. Proof Sketch of Theorem 1 Submodular Covering Problem Given: A groundset U with costs {c u :u ∈ U} and a submodular function g on 2 U with g( ∅ )=0. Find: A minimum-cost set A ⊆ U with g(A)=g(U). Greedy Algorithm Repeatedly adds a ∈ A\U to A with maximum. Approximation ratio: 1+ln max u ∈ U g({u}) [Wolsey 82]. Reduction: U=F, and for I ⊆ F let. Properties of g: g is submodular, g( ∅ )=0, and max u ∈ U g({u}) ≤|T|. Node costs: The proof is slightly more complicated.

Theorem 2: Undirected Star-NA admits ratio O(ln 2 k) for edge-costs. Proof Sketch of Theorem 2 (the edge-costs case) Consider Star-NA instances where we seek to increase the connectivity between pairs in D only by 1; namely for all (u,v) ∈ D If this problem admits ratio ρ (k) then (by “Reverse Augmentation”) the general problem admits ratio ρ (k) ∙ ln k. The increasing connectivity by 1 problem admits ratio ρ (k)=O(ln k): Reducible to finding a Min-Cost Hitting-Set of a hypergraph. This hypergraph has maximum degree O(k 2 ). Thus the Hitting-Set problem admits ratio O(ln k 2 ) = O(ln k).

Reducing Star-NA to Hitting Set U ⊆ V is a tight set if there is (u,v) ∈ D and a partition U,C,U* of V such that one of u,v is in U and the other in U*, and An edge set I ⊆ F is a feasible solution to Star-NA iff the endnodes of I cover all tight sets. Thus the problem is equivalent to finding a Min-Cost Hitting Set in the hypergraph of minimal tight sets. t U u U* v C

Theorem 1: Directed Star-NA admits a logarithmic ratio, and so is directed SL. Summary Theorem 2: Undirected Star-NA admits ratio O(p max ∙ ln 2 k) and so is undirected SL. In particular κ ‘-SL (the case p max =1) admits ratio O(ln 2 k). Lemma 1: SL is equivalent to Rooted Star-NA with s=t.

Questions?