ONLINE DEGREE-BOUNDED STEINER Sina Dehghani Saeed Seddighin - - PowerPoint PPT Presentation

ONLINE DEGREE-BOUNDED STEINER NETWORK DESIGN

Sina Dehghani Saeed Seddighin Ali Shafahi Fall 2015

ONLINE STEINER FOREST PROBLEM

• An initially given graph 𝐻.
• A sequence of demands (𝑡𝑗, 𝑢𝑗) arriving one-by-one.
• Buy new edges to connect demands.

𝑡1 𝑢1 𝑡2 𝑢2

DEGREE-BOUNDED STEINER FOREST

• There is a given bound 𝑐𝑤 for every vertex 𝑤.
• degree violation ≔

𝑒𝑓𝑕𝐼(𝑤) 𝑐𝑤

.

• Find a Steiner forest 𝐼 minimizing the degree violations.

𝑡1 𝑢1 𝑡2 𝑢2

PREVIOUS OFFLINE WORK

Problem Paper Result Degree-bounded Spanning tree FR ’90 𝑃(log 𝑜)-approximation Degree-bounded Steiner tree AKR ’91 𝑃(log 𝑜)-approximation Degree-bounded Steiner forest FR ’94 maximum degree ≤ 𝑐∗ + 1

• Degree-bounded network design:

PREVIOUS OFFLINE WORK

Problem Paper Result EW DB Steiner forest

• MRSRRH. ’98

⟨𝑃 log 𝑜), 𝑃(log 𝑜 ⟩-approx. EW DB Spanning tree G ’06 min weight, max deg ≤ 𝑐∗ + 2 EW DB Spanning tree LS ‘07 min weight, max deg ≤ 𝑐∗ + 1

• Edge-weighted degree-bounded variant:

PREVIOUS ONLINE WORK

Problem Paper Result Online edge-weighted Steiner tree IW ‘91 𝑃 log 𝑜 -competitive Online edge-weighted Steiner forest AAB ‘96 𝑃(log 𝑜)-competitive

• Online weighted Steiner network (no degree bound)

OUR CONTRIBUTION

Problem Result Online degree-bounded Steiner forest 𝑃(log 𝑜)-competitive greedy algorithm Online degree-bounded Steiner tree Ω(log 𝑜) lower bound Online edge-weighted degree-bounded Steiner tree Ω 𝑜 lower bound Online degree-bounded group Steiner tree Ω(𝑜) lower bound for det. algorithms.

• Online degree-bounded Steiner network:

LINEAR PROGRAM

∀𝑓 ∈ 𝐹: 𝑦 𝑓 = 1 if and only if 𝑓 is selected. 𝑻 be the collection of separating sets of demands. OMPC has an O(log2 𝑜)-competitive fractional solution, but rounding that is hard!

min 𝛽 ∀𝑤 ∈ 𝑊

𝑓∈𝜀 𝑤

𝑦 𝑓 ≤ 𝛽. 𝑐𝑤 ∀𝑇 ∈ 𝑻

𝑓∈𝜀 𝑇

𝑦 𝑓 ≥ 1 𝒚 𝑓 , 𝛽 ∈ ℝ+ ensures connectivity. limits degree violations.

REDUCTION TO UNIFORM DEGREE BOUNDS

• Replace 𝑤 with 𝑤1 … 𝑤𝑐𝑤.
• Connect each 𝑤𝑗 to all neighbors of 𝑤.
• Set all degree bounds to 1.
• Uniformly distribute edges of 𝜀𝐼(𝑤) among 𝑤𝑗’s.
• The degree violation remains almost the same.

𝑤 𝑂𝑓𝑗𝑕ℎ(𝑤) 𝑤1 𝑤2 𝑤𝑐𝑤

GREEDY ALGORITHM

𝑡𝑗 𝑢𝑗 𝑡𝑗 𝑢𝑗 𝑡𝑗 𝑢𝑗

GREEDY ALGORITHM

• Definitions:
• Let 𝐼 denote the online output of the previous step.
• For an (s, 𝑢)-path 𝑄 the extension part is P∗ = {𝑓|𝑓 ∈ 𝑄, 𝑓 ∉ 𝐼}.
• The load of 𝑄∗ is 𝑚𝐼 𝑄∗ = max

𝑤∈𝑄∗ deg𝐼(𝑤).

• Algorithm:

1. Initiate 𝐼 = 𝜚. 2. For every new demand (𝑡𝑗, 𝑢𝑗): 1. Find the path 𝑄

𝑗 with the minimum 𝑚𝐼 𝑄 𝑗 ∗ .

2. 𝐼 = 𝐼 ∪ 𝑄

𝑗 ∗.

𝑡 𝑢 𝑄 𝑄∗ Can be done polynomially.

ANALYSIS

• Let Γ 𝑠 be the set of vertices with deg𝐼 𝑤 ≥ 𝑠.
• Let 𝐸 𝑠 be demands for which 𝑚𝐼 𝑄

𝑗 ∗ is at least 𝑠.

Remark: 𝛥(𝑠) is a cut-set for 𝑡𝑗 and 𝑢𝑗 for every 𝑗 ∈ 𝐸 𝑠 .

• Let 𝐷𝐷(𝑠) denote the number of connected components of

𝐻\𝛥 𝑠 that have at least one endpoint of demand i ∈ 𝐸(𝑠). Lemma: ∀𝑠: 𝐷𝐷 𝑠 ≥ 𝐸 𝑠 + 1. Remark: ∀r: 𝑃𝑄𝑈 ≥

𝐷𝐷 𝑠 |Γ 𝑠 | .

𝑡𝑗 𝑢𝑗 Γ 𝑠

ANALYSIS

• 𝛥(𝑠)’s have a hierarchical order, i.e. 𝛥 𝑠 + 1 ⊆ 𝛥(𝑠).
• Every demand 𝑗 ∈ 𝐸(𝑠) copies some vertices to upper level.
• Out of all copies, at most 2(𝛥 𝑠 − 1) are for internal edges.

Lemma: ∀𝑠: 𝐸 𝑠 ≥ 𝑢=𝑠+1

𝛦

𝛥 𝑢 − 2(𝛥 𝑠 − 1).

Γ(1) Γ(2) Γ(Δ) Γ(𝑠) 𝑡𝑗 𝑢𝑗 𝑡

𝑘

𝑢𝑘

ANALYSIS

Lemma: For every sequence of integers 𝑏1 ≥ 𝑏2 ≥ ⋯ ≥ 𝑏Δ > 0 max

𝑗 { 𝑘=𝑗

Δ

𝑏𝑘 𝑏𝑗

} ≥

Δ 2 log 𝑏1 .

• Partition to log 𝑏1 groups.
• One group has at least

Δ log 𝑏1 numbers.

𝑏1 𝑏2 2 log 𝑏1 𝑏Δ … 20 … 𝑏𝑗 2𝑙 2𝑙−1 … ≥ ≥ ≥ ≥ ≥ …

ANALYSIS

• Putting all together:

∀𝑠: 𝑃𝑄𝑈 ≥ 𝐷𝐷 𝑠 Γ 𝑠 𝐷𝐷 𝑠 ≥ 𝐸 𝑠 + 1. 𝐸 𝑠 ≥ 𝑢=𝑠+1

Δ

Γ 𝑢 − 2(Γ 𝑠 − 1).

• Setting 𝑏𝑗 = Γ 𝑗

and using the lemma:

𝑃𝑄𝑈 ≥ max

𝑠 𝑢=𝑠

Δ

Γ 𝑠 −𝑃 Γ 𝑠 +1 Γ 𝑠

≥

Δ 2 log Γ 1 − 𝑃 1 ∈ Ω( Δ log 𝑜)

⇒ 𝑃𝑄𝑈 ≥ max

𝑠 𝐷𝐷 𝑠 Γ 𝑠 .

LOWER BOUND

𝑠𝑝𝑝𝑢 𝑨1 𝑨2𝑚 𝑦1 𝑦 2𝑚

2

𝑨𝑗 𝑨

𝑘

𝑦𝑗,𝑘 … … … … …

Theorem: Every (randomized) algorithm for online degree-bounded Steiner tree is 𝛻(𝑚𝑝𝑕 𝑜)-competitive.

𝑜 ∈ 𝑃 2(2𝑚) 𝑐𝑤 = 𝑜 𝑗𝑔 𝑤 = 𝑠𝑝𝑝𝑢 2 𝑃. 𝑋.

LOWER BOUND

• Theorem: Let 𝑃𝑄𝑈

𝑐 denote the minimum weight of a Steiner tree with maximum degree 𝑐. Then

for every (randomized) algorithm 𝐵 for online edge-weighted degree-bounded Steiner tree either

• 𝐹 max deg𝐵 𝑤

≥ Ω 𝑜 . 𝑐

• r
• 𝐹 𝑥𝑓𝑗𝑕ℎ𝑢 𝐵

≥ Ω 𝑜 . 𝑃𝑄𝑈

𝑐.

… 𝑤1 𝑤𝑙+1 𝑤2 𝑤𝑙+2 𝑤𝑗 𝑤𝑙+𝑗 𝑤𝑗+1 𝑤𝑙+𝑗+1 𝑠𝑝𝑝𝑢 𝑤𝑙 𝑤2𝑙 𝑜 𝑜2 𝑜𝑗 𝑜𝑗+1 𝑜𝑙 … … … 𝑜 = 2𝑙 + 1 𝑐 = 3 𝑥𝑓𝑗𝑕ℎ𝑢 𝐵 = 𝑜𝑗+1 deg𝐵 𝑠𝑝𝑝𝑢 = 𝑗 𝑃𝑄𝑈3 =

𝑘=1 𝑗

𝑜𝑘 ∈ 𝑃(𝑜𝑗)

LOWER BOUND

Theorem: Every deterministic algorithm 𝐵 for online degree-bounded group Steiner tree is 𝛻(𝑜)-competitive.

𝑠𝑝𝑝𝑢 𝑤1 𝑤2 𝑤3 𝑤𝑜−2 𝑤𝑜−1 All degree bounds are 1. deg𝐵 𝑠𝑝𝑝𝑢 = 𝑜 − 1.

OPEN PROBLEMS

• The main open problem:
• Online edge-weighted degree-bounded Steiner forest,

when the weights are polynomial to 𝑜.

• Other degree-bounded variants (with or without weights):
• Online group Steiner tree.
• Online survivable network design.

Thank you