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1
Online Node-weighted Steiner Connectivity Problems
Vahid LiaghatUniversity of Maryland
MohammadTaghi Hajiaghayi(UMD)
Debmalya Panigrahi(Duke)
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
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
7
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]
8
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]
9
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
10
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
11
12
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)
13
Edge-Weighted Steiner Forest [Berman, Coulston]
𝑠 𝑡𝑫𝐷 /4
𝑢𝑠 𝑢t
Neighborhood ClearanceYes? We are good!
No? Bad!
Failure witness Failure witness
14
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
15
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
17
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
18
center
continent boundary
0
61
3
412
5
10
19
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
20
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 .
21
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
22
2𝑖
𝑑~𝑤 (𝑠 ,𝑡 )≥𝜇2𝑖𝑠 𝑡
𝑐𝑠 𝑐𝑡
Failure witnesses
23
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
24
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
25
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 )
26
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?
27
Thank You!
Questions?
28
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
29
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.
30
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
32
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
33
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≤
34
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.
35
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.
36
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.
37
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.
38
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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