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

ONLINE DEGREE-BOUNDED STEINER Sina Dehghani Saeed Seddighin NETWORK DESIGN Ali Shafahi Fall 2015

ONLINE STEINER FOREST PROBLEM  An initially given graph 𝐻. 𝑡 2 𝑡 1  A sequence of demands ( 𝑡 𝑗 , 𝑢 𝑗 ) arriving one-by-one. 𝑢 2 𝑢 1  Buy new edges to connect demands.

DEGREE-BOUNDED STEINER FOREST  There is a given bound 𝑐 𝑤 for every vertex 𝑤 . 𝑡 2 𝑡 1 𝑒𝑓𝑕 𝐼 (𝑤)  degree violation ≔ . 𝑐 𝑤  Find a Steiner forest 𝐼 minimizing the degree violations. 𝑢 2 𝑢 1

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

PREVIOUS OFFLINE WORK  Edge-weighted degree-bounded variant: Problem Paper Result EW DB Steiner forest MRSRRH. ’98 ⟨𝑃 log 𝑜), 𝑃(log 𝑜 ⟩ -approx. min weight, max deg ≤ 𝑐 ∗ + 2 EW DB Spanning tree G ’06 min weight, max deg ≤ 𝑐 ∗ + 1 EW DB Spanning tree LS ‘07

PREVIOUS ONLINE WORK  Online weighted Steiner network (no degree bound) 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

OUR CONTRIBUTION  Online degree-bounded Steiner network: 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.

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(log 2 𝑜) -competitive fractional solution, but rounding that is hard! min 𝛽 ∀𝑤 ∈ 𝑊 𝑦 𝑓 ≤ 𝛽. 𝑐 𝑤 limits degree violations. 𝑓∈𝜀 𝑤 ensures connectivity. ∀𝑇 ∈ 𝑻 𝑦 𝑓 ≥ 1 𝑓∈𝜀 𝑇 𝒚 𝑓 , 𝛽 ∈ ℝ +

REDUCTION TO UNIFORM DEGREE BOUNDS  Replace 𝑤 with 𝑤 1 … 𝑤 𝑐𝑤 . 𝑤 1  Connect each 𝑤 𝑗 to all neighbors of 𝑤 . 𝑤 2 𝑤 𝑂𝑓𝑗𝑕ℎ(𝑤)  Set all degree bounds to 1 . 𝑤 𝑐 𝑤  Uniformly distribute edges of 𝜀 𝐼 (𝑤) among 𝑤 𝑗 ’s .  The degree violation remains almost the same.

GREEDY ALGORITHM 𝑡 𝑗 𝑡 𝑗 𝑡 𝑗 𝑢 𝑗 𝑢 𝑗 𝑢 𝑗

GREEDY ALGORITHM 𝑡  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 𝑚 𝐼 𝑄 ∗ = max 𝑤∈𝑄 ∗ deg 𝐼 (𝑤) . 𝑄 ∗ 𝑄  Algorithm: Can be done 1. Initiate 𝐼 = 𝜚 . polynomially. 2. For every new demand ( 𝑡 𝑗 , 𝑢 𝑗 ): ∗ . 1. Find the path 𝑄 𝑗 with the minimum 𝑚 𝐼 𝑄 𝑗 ∗ . 2. 𝐼 = 𝐼 ∪ 𝑄 𝑗 𝑢

ANALYSIS Γ 𝑠  Let Γ 𝑠 be the set of vertices with deg 𝐼 𝑤 ≥ 𝑠 . ∗ is at least 𝑠 .  Let 𝐸 𝑠 be demands for which 𝑚 𝐼 𝑄 𝑗 Remark : 𝛥(𝑠) is a cut-set for 𝑡 𝑗 and 𝑢 𝑗 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 . 𝑡 𝑗 𝑢 𝑗 𝐷𝐷 𝑠 Remark: ∀r: 𝑃𝑄𝑈 ≥ |Γ 𝑠 | .

ANALYSIS 𝑡 𝑘  𝛥(𝑠) ’s have a hierarchical order, i.e. 𝛥 𝑠 + 1 ⊆ 𝛥(𝑠) . 𝑡 𝑗 Γ(Δ)  Every demand 𝑗 ∈ 𝐸(𝑠) copies some vertices to upper level. 𝑢 𝑘 𝑢 𝑗  Out of all copies, at most 2(𝛥 𝑠 − 1) are for internal edges . Γ(𝑠) Γ(2) 𝛦 Lemma: ∀𝑠: 𝐸 𝑠 − 2(𝛥 𝑠 − 1) . ≥ 𝑢=𝑠+1 Γ(1) 𝛥 𝑢

ANALYSIS Lemma: For every sequence of integers 𝑏 1 ≥ 𝑏 2 ≥ ⋯ ≥ 𝑏 Δ > 0 Δ 𝑘=𝑗 𝑏 𝑘 Δ 2 log 𝑏 1 . max 𝑗 { } ≥ 𝑏 𝑗  Partition to log 𝑏 1 groups. 𝑏 1 𝑏 2 … ≥ 𝑏 𝑗 … … ≥ … 𝑏 Δ ≥ ≥ ≥ Δ  One group has at least log 𝑏 1 numbers. 2 𝑙 2 log 𝑏 1 2 𝑙−1 2 0

ANALYSIS  Putting all together : ∀𝑠: 𝑃𝑄𝑈 ≥ 𝐷𝐷 𝑠 𝐷𝐷 𝑠 Γ 𝑠 . ⇒ 𝑃𝑄𝑈 ≥ max Γ 𝑠 𝑠 + 1 . 𝐷𝐷 𝑠 ≥ 𝐸 𝑠 Δ − 2(Γ 𝑠 − 1) . 𝐸 𝑠 ≥ 𝑢=𝑠+1 Γ 𝑢  Setting 𝑏 𝑗 = Γ 𝑗 and using the lemma: Δ 𝑢=𝑠 Γ 𝑠 −𝑃 Γ 𝑠 +1 Δ Δ 𝑃𝑄𝑈 ≥ max ≥ 2 log Γ 1 − 𝑃 1 ∈ Ω( log 𝑜 ) Γ 𝑠 𝑠

LOWER BOUND Theorem: Every (randomized) algorithm for online degree-bounded Steiner tree is 𝛻(𝑚𝑝𝑕 𝑜) -competitive. 𝑠𝑝𝑝𝑢 𝑨 𝑗 𝑨 𝑨 2 𝑚 𝑨 1 𝑘 … … … … … 𝑦 1 𝑦 𝑗,𝑘 𝑦 2𝑚 2 𝑜 ∈ 𝑃 2 (2𝑚) 𝑐 𝑤 = 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 deg 𝐵 𝑤 ≥ Ω 𝑜 . 𝑐 or  𝐹 𝑥𝑓𝑗𝑕ℎ𝑢 𝐵 𝑐 . ≥ Ω 𝑜 . 𝑃𝑄𝑈 𝑠𝑝𝑝𝑢 𝑜 = 2𝑙 + 1 𝑐 = 3 𝑤 1 𝑤 2 𝑤 𝑗 𝑤 𝑗+1 𝑤 𝑙 … … 𝑥𝑓𝑗𝑕ℎ𝑢 𝐵 = 𝑜 𝑗+1 deg 𝐵 𝑠𝑝𝑝𝑢 = 𝑗 𝑜 2 𝑜 𝑙 𝑜 𝑗 𝑜 𝑗+1 𝑜 𝑗 … … 𝑜 𝑘 ∈ 𝑃(𝑜 𝑗 ) 𝑃𝑄𝑈 3 = 𝑤 𝑙+1 𝑤 2𝑙 𝑤 𝑙+2 𝑤 𝑙+𝑗 𝑤 𝑙+𝑗+1 𝑘=1

LOWER BOUND Theorem: Every deterministic algorithm 𝐵 for online degree-bounded group Steiner tree is 𝛻(𝑜) -competitive. All degree bounds are 1. 𝑤 1 𝑤 3 𝑤 2 𝑤 𝑜−2 𝑤 𝑜−1 deg 𝐵 𝑠𝑝𝑝𝑢 = 𝑜 − 1 . 𝑠𝑝𝑝𝑢

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.

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