Approximating Source Location and Star Survivable Network Problems

Zeev NutovThe Open University of Israel

Joint work with Guy Kortsarz

The problemsSource Location (SL)Given: A graph G=(V,E) with node-costs {cv: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 {ce:e∈F} or node-costs {cv: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 {ce:e∈F} or node-costs {cv: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 themaximum (u,v)-flow value (every edge has capacity 1)

Let {pv:v∈V} be “connectivity bonuses”(node v chosen to S gets connectivity bonus pv ). (S,v)-(p,q)-connectivity is defined by

qG u,v

Let G=(V,E) be a network with node capacities {qv:v∈V}.

p,qG S,v

vp v S

v S

qGp,q

G qG

S,vS,v

S,v

Observation: The set function is submodular. v qGf (S) S,v

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 pu edges from s to every u∈S

(p,q)

(∞,1)

(3,1)

(2,1)(2,2) p,qG S,v = 3

s

Example

sS

(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 pu edges from s to every u∈S

(p,q)

(∞,1)

(3,1)

(2,1)(2,2) p,qG S,v =

∞v

edge-connectivity λG(S,v)pv=qv=∞ for all v∈V (if v∈S then λG(S,v)=∞)

node-connectivity pv=∞ and qv=1 for all v∈V (if v∈S then )

node-connectivity κ’G(S,v) pv=qv=1 for all v∈V.

node-connectivity κG(S,v) v∈S :κG(S,v) =∞v∉S: κG(S,v) = maximum number of (S,v)-disjoint path. In digraphs, this version is equivalent to λG(S,v).

Connectivity functions in SL problems

G S,v G S,v =

Network Augmentation (NA)Given: A graph G=(V,E) and an edge set F on V, edge-costs {ce:e∈F} or node-costs {cv:v∈V}, node capacities {qv: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.

qG+I u,v r u,v

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

cs=0

vr(s,v)=d(v)

pv edges

• Add a new node s of cost 0.• For every v∈V do:

set r(s,v)=d(v). put pv 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 digraphsNA: r(D) = sum of the requirements, pmax = max # of parallel edges in FSL : d(V) = sum of the demands, pmax = max connectivity bonus

ˆ, ,

Common technique: Greedy algorithm for Submodular Cover.

[N, Approx 03]Undirected Rooted Connectivity Augmentation admits ratio O(ln2 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(k3 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(ln2 k) for edge-costs; pmax∙O(ln2 k) for node-costs.Thus undirected SL admits ratio pmax∙O(ln2 k),and κ‘-SL (the case pmax=1) admits ratio O(ln2 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 {cu:u∈U} and a submodular function g on 2U 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 maxu∈Ug({u}) [Wolsey 82].

• Reduction: U=F, and for I⊆F let .

• Properties of g: g is submodular, g(∅)=0, and maxu∈Ug({u}) ≤|T|.

• Node costs: The proof is slightly more complicated.

g A + u - g Ac a

q qG sv G+I

sv T

g I = r T - λ T max r - ,0λ

Theorem 2: Undirected Star-NA admits ratio O(ln2 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(k2). Thus the Hitting-Set problem admits ratio O(ln k2) = O(ln k).

1 qGr u,v u,v

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.

tU

u

U*

v

C

qG|q(C)|+| (U,U*)| u,v

Theorem 1: Directed Star-NA admits a logarithmic ratio, and so is directed SL.

Summary

Theorem 2: Undirected Star-NA admits ratio O(pmax∙ln2 k) and so is undirected SL. In particular κ‘-SL (the case pmax=1) admits ratio O(ln2 k).

Lemma 1: SL is equivalent to Rooted Star-NA with s=t.

Questions?