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Online Node-weighted Steiner Connectivity Problems

Vahid LiaghatUniversity of Maryland

MohammadTaghi Hajiaghayi(UMD)

Debmalya Panigrahi(Duke)

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Node-Weighted Steiner Forest

• Given– An undirected graph .– A weight associated with each

vertex – A set of connectivity demands .

• Goal: Finding a subgraph that connects these demands.

• Objective: Minimize the total weight .

𝑠1

𝑠3

𝑠2

𝑡1

𝑡 2

𝑡 3

55

5330

30

4

103

4

Node-Weighted Steiner Forest

• Given– An undirected graph .– A weight associated with each

vertex – A set of connectivity demands .

• Goal: Finding a subgraph that connects these demands.

• Objective: Minimize the total weight .

𝑠1

𝑡1

55

5330

30

4

103

4

𝑠2𝑡 2

Node-Weighted Steiner Forest

• Given– An undirected graph .– A weight associated with each

vertex – A set of connectivity demands .

• Goal: Finding a subgraph that connects these demands.

• Objective: Minimize the total weight .

𝑠1

𝑠2

𝑡1

𝑡 2

55

5330

30

4

103

4

𝑠3

𝑡 3

5

Node-Weighted Steiner Forest

• Given– An undirected graph .– A weight associated with each

vertex – A set of connectivity demands .

• Goal: Finding a subgraph that connects these demands.

• Objective: Minimize the total weight .

𝑠1

𝑠2

𝑡1

𝑡 2

55

5330

30

4

103

4

𝑠3

𝑡 3

6

Online Steiner Forest

• Given– An undirected graph .– A weight associated with each

vertex – An online sequence of

demands .

• Goal: At iteration , finding a subgraph that satisfies the first demands.

• Objective: Minimize the competitive ratio

𝑠1

𝑠3

𝑠2

𝑡1

𝑡 2

𝑡 3

55

5330

30

4

103

4

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Hardness

• Node-weighted Steiner forest

• Node-weighted Steiner tree

• Set Cover: Given a collection of subsets of a universe, find the minimum number of subsets which cover all the universe.

Special Case

A lower bound of for any online algorithm where and denote the size of the universe and the number

of sets respectively. [AAABN’09]

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Known Results

Online Problem Lower Bound Upper Bound

Node-weighted Steiner Forest

Node-weighted Steiner Tree

Non-metric Facility Location

Set Cover

Special Case

Ω( log2𝑛log log𝑛 )

Ω( log2𝑛log log𝑛 )

Ω( log2𝑛log log𝑛 )

Ω( log2𝑛log log𝑛 )

[AAABN’03]

[AAABN’04]

[NPS’11, HLP’14]

[HLP’13]

One more log factor forprize-collecting variants

[HLP’14]

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Node-Weighted SF

A randomized - competitive algorithm

for the Steiner forest problem

A deterministc - competitive algorithm for SF when the

underlying graph excludes a fixed minor

Special Case

Results carry over to

network design problems

characterized by {0,1}-proper

functions

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Node-Weighted SF

A randomized - competitive algorithm

for the Steiner forest problem

A deterministc - competitive algorithm for SF when the

underlying graph excludes a fixed minor

Special Case

Results carry over to

network design problems

characterized by {0,1}-proper

functions

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Edge-Weighted Steiner Forest [Berman, Coulston]

A Greedy Candidate:

• Let be the current solution. Let • Let be the new terminal and let be the distance

between and (w.r.t. to )• Buy the shortest path!

• Try putting a disk centered at or at with radius (almost)

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Edge-Weighted Steiner Forest [Berman, Coulston]

𝑠 𝑡𝑫𝐷 /4

𝑢𝑠 𝑢t

Neighborhood ClearanceYes? We are good!

No? Bad!

Failure witness Failure witness

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Edge-Weighted Steiner Forest [Berman, Coulston]

𝑠 𝑡𝑫

¿𝑫 /𝟐¿𝑫 /𝟐

One layer for every possible radius, rounded up to powers of two.

𝑢𝑠 𝑢t

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Node-weighted

𝑫𝑡𝑠

For Planar Graphs:If the degree of the center of spider is large,

maybe this cannot happen too often?

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Node-weighted

𝑫𝑡𝑠

How about the general graphs?

Connect the terminals to the intersection vertices using a competitive facility location algorithm

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Node-Weighted SF

A randomized - competitive algorithm

for the Steiner forest problem

A deterministc - competitive algorithm for SF when the

underlying graph excludes a fixed minor

Special Case

Results carry over to

network design problems

characterized by {0,1}-proper

functions

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center

continent boundary

0

61

3

412

5

10

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Non-overlapping Disks & Binding Spiders

• A set of disks are non-overlapping if for every vertex the colored amount is less than the weight, i.e.,

• A tight vertex is an intersection vertex, if further growth of a disk over-colors

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H-Minor Free Graphs

• A graph is a minor of a graph if it can be derived from by repeatedly contracting an edge or removing an edge (or a vertex).

• For a graph , the family of -minor free graphs comprise all graphs which exclude as a minor.

• For example planar graphs are both -minor free and -minor free.

• Many interesting properties! (separators, treewidth, pathwidth, tree-depth, …)

• In particular, the average degree of a graph excluding as a minor, is at most where is the number of vertices of .

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SF in H-Minor-Free Graphs

• Let be the current solution. Let • Consider a large enough constant • Let be the new demand and let be the distance

between and (w.r.t. to )• First, buy the shortest path!

• Choose layer such that

• Try putting a disk centered at or in layer • Neighborhood Clearance? We’re good!• No? Buy both binding spiders

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2𝑖

𝑑~𝑤 (𝑠 ,𝑡 )≥𝜇2𝑖𝑠 𝑡

𝑐𝑠 𝑐𝑡

Failure witnesses

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Analysis

• If we charge the cost of our solution to the (radii) of disks, then we have an -competitive algorithm!

• How can we do that?• The total cost := the shortest path + the binding

spider

• If we put a new disk, we’re good: • Otherwise, we buy two spiders.• We have two different cases:

– We are buying an expensive spider with at least legs!

– Both spiders are cheap (at most legs)

2𝑖

𝑑~𝑤 (𝑠 ,𝑡 )≥𝜇2𝑖𝑠 𝑡

𝑐𝑠 𝑐𝑡

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Analysis• Recall that cost of a spider (#legs)

• If both spiders are cheap, charge to the number of connected components.

• Otherwise, we show # legs in expensive spiders = O(# disks)

2𝑖

𝑑~𝑤 (𝑠 ,𝑡 )≥𝜇2𝑖𝑠 𝑡

𝑐𝑠 𝑐𝑡

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Disks may intersect only on the boundaries.

Average degree at most

Minimum degree of aBlack vertex is at least

Average degree ofBlue vertices is

at most

Cost of Expensive Spiders #legs #edges (#blue vertices) O(2^i) O(total radii in layer )

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Summary• We use Disk Painting as a framework for solving node-weighted network

design problems• A randomized -competitive algorithm for online network design problems

characterized by proper functions• A deterministic -competitive algorithm for online network design problems

characterized by proper functions when the underlying graph excludes a fixed minor

• All the results can be extended to prize-collecting counterparts (tomorrow morning)

• Primal-Dual techniques for Group Steiner Tree? Higher Connectivity?

• Stochastic settings?• Streaming or parallel models?

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Thank You!

Questions?

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Hardness

• Node-weighted Steiner forest

• Node-weighted Steiner tree

• Set Cover: Given a collection of subsets of a universe, find the minimum number of subsets which cover all the universe.

Special Case

𝒆𝟏 𝒆𝟐 𝒆𝟑 𝒆𝟒

𝑺𝟏 𝑺𝟐

covered

𝟏 𝟏

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Disks and Paintings

• Let denote the length of shortest path connecting and , including the weight of endpoints.

• Disk of radius centered at

• Continent: vertex is inside if .• Boundary: not inside, but has a neighbor inside.

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center

continent boundary

0

61

3

412

5

10

31

Non-overlapping Disks & Binding Spiders

• A set of disks are non-overlapping if for every vertex the colored amount is less than the weight, i.e.,

• A tight vertex is an intersection vertex, if further growth of a disk over-colors

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A Few Observations

• We consider non-overlapping disks.• Disks may intersect only on the boundaries.• The radii of all disks are the same, denoted by .

If there are disks centered at terminals, then

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1)

• The arriving clients are at least far from each other.• Thus an overlap may acquire only at the boundaries,

i.e, the possible facilities.

2𝑖

𝐎𝐏𝐓𝑶𝑷𝑻 𝒊≤

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2) O(cost of )

• The total cost := the shortest path + paths to witnesses

+ the simulation cost

• Simulation cost cost of • At each Type iteration:

The shortest path + paths to witnesses .

# Type iterations O(# clients demanded in layer )

incurs at least for every arriving client.

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2) O(cost of )

• The neighborhood of a new client is clear!• So we need to open a new facility in the boundary of a

disk of radius .

• If we successfully add a client, then we are good!• If not, we will reduce #connected components having

a client of layer .

# Type iterations O(# clients demanded in layer )

incurs at least for every arriving client.

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References[1] U. Feige. A threshold of ln n for approximating set cover. JACM’98.[2] P. Klein and R. Ravi. A nearly best-possible approximation algorithm for node-weighted Steiner trees. Journal of Algorithms’95.[3] A. Moss and Y. Rabani. Approximation algorithms for constrained node weighted Steiner tree problems. STOC’01, SICOMP’07.[4] D. Johnson, M. Minkoff, and S. Philips. The prize collecting Steiner tree problem: theory and practice. Soda’00.[5] Sudipto Guha, Anna Moss, Joseph (Seffi) Naor, and Baruch Schieber. Efficient recovery from power outage. STOC’99.[6] M.H. Bateni, M.T. Hajiaghayi, V. Liaghat. Improved Approximation Algorithms for (Budgeted) Node-weighted Steiner Problems. Submitted to ICALP’13.[7] Alon, N., Awerbuch, B., Azar, Y., Buchbinder, N., Naor, J.: The online set cover problem. SIAM J’09.[8] Naor, J., Panigrahi, D., Singh, M. Online node-weighted steiner tree and related problems. FOCS’11.[9] Alon, N., Moshkovitz, D., Safra, S. Algorithmic construction of sets for k-restrictions. ACM Trans. Algorithms’06.

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Our Results [Hajiaghayi, Panigrahi, L ’13]

• A randomized algorithm for the Steiner forest problem with the competitive ratio . The competitive ratio is tight to a logarithmic factor.

• A deterministic primal-dual algorithm for theSteiner Forest problem with an (asymptotically)tight competitive ratio when theunderlying graph excludes a fixed minor.– Also implies a simple algorithm for Edge-Weighted variant.

• The same guarantees carry over to a general family of network design problems characterized by proper functions.

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Our Results [Hajiaghayi, Panigrahi, L ’13]

• A randomized algorithm for the Steiner forest problem with the competitive ratio . The competitive ratio is tight to a logarithmic factor.

• A deterministic primal-dual algorithm for theSteiner Forest problem with an (asymptotically)tight competitive ratio when theunderlying graph excludes a fixed minor.– Also implies a simple algorithm for Edge-Weighted variant.

• The same guarantees carry over to a general family of network design problems characterized by proper functions.