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

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

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

3 DEGREE-BOUNDED STEINER FOREST There is a given bound b v for every vertex v. s 1 s 2 degree violation deg H(v) b v. Find a Steiner forest H minimizing the degree violations. t 2 t 1

4 PREVIOUS OFFLINE WORK Degree-bounded network design: Problem Paper Result Degree-bounded Spanning tree FR 90 O(log n)-approximation Degree-bounded Steiner tree AKR 91 O(log n)-approximation Degree-bounded Steiner forest FR 94 maximum degree b + 1

5 PREVIOUS OFFLINE WORK Edge-weighted degree-bounded variant: Problem Paper Result EW DB Steiner forest MRSRRH. 98 O log n), O(log n -approx. EW DB Spanning tree G 06 min weight, max deg b + 2 EW DB Spanning tree LS 07 min weight, max deg b + 1

6 PREVIOUS ONLINE WORK Online weighted Steiner network (no degree bound) Problem Paper Result Online edge-weighted Steiner tree IW 91 O log n -competitive Online edge-weighted Steiner forest AAB 96 O(log n)-competitive

7 OUR CONTRIBUTION Online degree-bounded Steiner network: Problem Online degree-bounded Steiner forest Online degree-bounded Steiner tree Online edge-weighted degree-bounded Steiner tree Online degree-bounded group Steiner tree Result O(log n)-competitive greedy algorithm Ω(log n) lower bound Ω n lower bound Ω(n) lower bound for det. algorithms.

8 LINEAR PROGRAM e E: x e = 1 if and only if e is selected. S be the collection of separating sets of demands. OMPC has an O(log 2 n)-competitive fractional solution, but rounding that is hard! min α v V x e α. b v e δ v S S x e 1 e δ S x e, α R + limits degree violations. ensures connectivity.

9 REDUCTION TO UNIFORM DEGREE BOUNDS Replace v with v 1 v b v. Connect each v i to all neighbors of v. Set all degree bounds to 1. v 1 v 2 v Neigh(v) Uniformly distribute edges of δ H (v) among v i s. v bv The degree violation remains almost the same.

10 GREEDY ALGORITHM s i s i s i t i t i t i

11 GREEDY ALGORITHM Definitions: Let H denote the online output of the previous step. For an (s, t)-path P the extension part is P = {e e P, e H}. The load of P is l H P = max deg H(v). v P Algorithm: 1. Initiate H = φ. 2. For every new demand (s i, t i ): 1. Find the path P i with the minimum l H P i. 2. H = H P i. Can be done polynomially. s t P

12 ANALYSIS Let Γ r be the set of vertices with deg H v r. Γ r Let D r be demands for which l H P i is at least r. Remark: Γ(r) is a cut-set for s i and t i for every i D r. Let CC(r) denote the number of connected components of G\Γ r that have at least one endpoint of demand i D(r). Lemma: r: CC r D r + 1. s i t i Remark: r: OPT CC r Γ r.

13 ANALYSIS Γ(r) s have a hierarchical order, i.e. Γ r + 1 Γ(r). s i s j Every demand i D(r) copies some vertices to upper level. Γ(Δ) Out of all copies, at most 2(Γ r 1) are for internal edges. t i Γ(r) Γ(2) t j Lemma: r: D r Δ t=r+1 Γ t 2(Γ r 1). Γ(1)

14 ANALYSIS Lemma: For every sequence of integers a 1 a 2 a Δ > 0 max i { j=i Δ aj } Δ a i 2 log a 1. Partition to log a 1 groups. One group has at least Δ log a 1 numbers. a 1 a 2 2 log a 1 a i 2 k 2 k 1 a Δ 2 0

15 ANALYSIS Putting all together: r: OPT CC r Γ r OPT max r CC r Γ r. CC r D r + 1. Δ D r t=r+1 Γ t 2(Γ r 1). Setting a i = Γ i and using the lemma: OPT max r t=r Γ r O Γ r +1 Γ r Δ 2 log Γ 1 O 1 Ω( Δ log n )

16 LOWER BOUND Theorem: Every (randomized) algorithm for online degree-bounded Steiner tree is Ω(log n)-competitive. root z 1 z i z j z 2 l x 1 x i,j n O 2 (2l) b v = n if v = root 2 O. W. x 2 l 2

17 LOWER BOUND Theorem: Let OPT b denote the minimum weight of a Steiner tree with maximum degree b. Then for every (randomized) algorithm A for online edge-weighted degree-bounded Steiner tree either E max deg A v Ω n. b or E weight A Ω n. OPT b. n = 2k + 1 b = 3 weight A = n i+1 deg A root = i v 1 v 2 root v i v i+1 v k n n 2 n i n i+1 n k OPT 3 = i j=1 n j O(n i ) v k+1 v k+2 v k+i v k+i+1 v 2k

18 LOWER BOUND Theorem: Every deterministic algorithm A for online degree-bounded group Steiner tree is Ω(n)-competitive. All degree bounds are 1. v 1 v 2 v 3 v n 2 v n 1 deg A root = n 1. root

19 OPEN PROBLEMS The main open problem: Online edge-weighted degree-bounded Steiner forest, when the weights are polynomial to n. Other degree-bounded variants (with or without weights): Online group Steiner tree. Online survivable network design.

20 Thank you