Network Design with Coverage Costs

Transcription

1 1 / 27 Network Design with Coverage Costs Siddharth Barman 1 Shuchi Chawla 2 Seeun Umboh 2 1 Caltech 2 University of Wisconsin-Madison Workshop on Flexible Network Design 2013 Fields Institute

2 Physical Flow vs Data Flow 2 / 27 vs. Commodity Network Internet Unlike physical commodities, data can be easily duplicated, compressed, and combined.

3 Physical Flow vs Data Flow 2 / 27 Flow = 1 P = Flow = 2 vs. P = Flow = 1 P = Unlike physical commodities, data can be easily duplicated, compressed, and combined.

4 Overarching Goals 3 / 27 Design expressive and tractable models to represent information flow. In particular, focus on a cost structure that captures savings obtained by eliminating redundancy in data. Design (approximation) algorithms within this framework: ALG α OPT.

5 Outline 4 / 27 Network Design with Coverage Costs (single-source s for this talk). Related Work. Our results.

6 Load Function 5 / 27 We focus on redundant-data elimination: Load on an edge l e = # distinct data packets on it. P 1 : P 1 : P 2 : Load = 2

7 6 / 27 Problem Definition Input: Graph with edge costs, g terminal groups X 1,...,X g V, source s. A global set of packets Π. A demand set D j Π for each X j. = {p 1,p 2,p 3,p 4 s X 1 = {s, t 1 X 2 = {s, t 2 X 3 = {s, t 3 t 2 t 3 D 2 = {p 1, p 2 t 1 D 1 = {p 3, p 4 D 3 = {p 3, p 4

8 7 / 27 Problem Definition Goal: For each j, find a Steiner tree for X j and route packets in D j on it. Objective: Minimize total routing cost: Cost of edge Load on edge edges = {p 1,p 2,p 3,p 4 s X 1 = {s, t 1 l e =2 l e =2 t l e =2 2 t 3 X 2 = {s, t 2 X 3 = {s, t 3 D t 2 = {p 1, p 2 1 D 3 = {p 3, p 4 D 1 = {p 3, p 4

9 7 / 27 Problem Definition Goal: For each j, find a Steiner tree for X j and route packets in D j on it. Objective: Minimize total routing cost: Cost of edge # distinct data packets on edge edges = {p 1,p 2,p 3,p 4 s X 1 = {s, t 1 l e =2 l e =2 t l e =2 2 t 3 X 2 = {s, t 2 X 3 = {s, t 3 D t 2 = {p 1, p 2 1 D 3 = {p 3, p 4 D 1 = {p 3, p 4

10 Related Work 8 / 27 Submodular costs on edges [Hayrapetyan-Swamy-Tardos 05] O(log V ) via tree embeddings.

11 Related Work 8 / 27 Submodular costs on edges [Hayrapetyan-Swamy-Tardos 05] O(log V ) via tree embeddings. Buy-at-Bulk Network Design [Salman et al. 97] Single-source: O(1) [Guha et al. 01, Talwar 02, Gupta-Kumar-Roughgarden 03]. Multiple-source: polylog hardness [Andrews 04]

12 Outline 9 / 27 Network Design with Coverage Costs (single-source s for this talk). Related Work. Our results: Laminar Demands. Sunflower Demands.

13 Laminar Family of Demands 10 / 27 The family of demand sets is said to be laminar if i,j one of the following holds: D i D j = ϕ or D i D j or D j D i. D 1 D 2 D 5 D3 D 4

14 Results 11 / 27 Previous work: O(log Π ) [Barman and Chawla 12]. Our result: 2.

15 Intuition 12 / 27 D 1 D 2 D 3 s X 1 = {s, t 1 X 2 = {s, t 2 t 2 t 3 X 3 = {s, t 3 t 1

16 Intuition 12 / 27 D 1 D 2 D 3 s X 1 = {s, t 1 X 2 = {s, t 2 t 2 t 3 X 3 = {s, t 3 t 1 Bottom-up Route D 2 and D 3 on shortest paths for t 2 and t 3 first. t 1 can route via t 2 or t 3 at a cheaper cost.

17 Intuition 12 / 27 D 1 D 2 D 3 s X 1 = {s, t 1 X 2 = {s, t 2 t 2 t 3 X 3 = {s, t 3 t 1 Bottom-up Route D 2 and D 3 on shortest paths for t 2 and t 3 first. t 1 can route via t 2 or t 3 at a cheaper cost. BUT: slightly longer paths for t 2 or t 3 can lead to greater sharing of edges with t 1.

18 Intuition 13 / 27 Idea Route D Π on a subgraph connecting groups whose demand sets include D. D 1 D 2 D 3 s X 1 = {s, t 1 X 2 = {s, t 2 t 2 t 3 X 3 = {s, t 3 t 1

19 13 / 27 Intuition Idea Route D Π on a subgraph connecting groups whose demand sets include D. D 1 D 2 D 3 s X 1 = {s, t 1 X 2 = {s, t 2 t 2 t 3 X 3 = {s, t 3 t 1 1. Connect s,t 1,t 2 and route D 2.

20 13 / 27 Intuition Idea Route D Π on a subgraph connecting groups whose demand sets include D. D 1 D 2 D 3 s X 1 = {s, t 1 X 2 = {s, t 2 t 2 t 3 X 3 = {s, t 3 t 1 2. Connect s,t 1,t 3 and route D 3.

21 13 / 27 Intuition Idea Route D Π on a subgraph connecting groups whose demand sets include D. D 1 D 2 D 3 s X 1 = {s, t 1 X 2 = {s, t 2 t 2 t 3 X 3 = {s, t 3 t 1 3. Connect s,t 1 and route D 1.

22 Primal Dual Generalization of Goemans-Williamson s Steiner forest primal dual algorithm. 14 / 27 D 1 s X 1 = {s, t 1 D 2 D 3 X 2 = {s, t 2 t 2 t 3 X 3 = {s, t 3 t 1 Theorem Network Design with Coverage Costs in the laminar demands setting admits a 2-approximation.

23 Outline 15 / 27 Network Design with Coverage Costs (single-source s for this talk). Related Work. Our results: Laminar Demands. Sunflower Demands.

24 Sunflower Family of Demands The family of demand sets is said to be a sunflower family if there exists C Π such that i,j, D i D j = C. 16 / 27 D i \ C C D j \ C

25 Sunflower Family of Demands The family of demand sets is said to be a sunflower family if there exists C Π such that i,j, D i D j = C. 16 / 27 D i \ C C D j \ C Theorem Network Design with Coverage Costs in the sunflower demands setting admits an O(log g) approximation over unweighted graphs with V = j X j.

26 Related Work 17 / 27 Shmoys-Swamy-Levi 04: O(1) approximation algorithm for facility location with service installation costs. Svitkina-Tardos 06: O(1) approximation algorithm for facility location with hierarchical costs.

27 Primal Dual? 18 / 27 Which order to consider demand sets?

28 Lower Bound on OPT 1. OPT routes D j C on a Steiner tree for X j. 2. V = j X j and single-source. 3. Edges carrying C spanning tree. 4. Edges carrying D j \ C Steiner tree for X j. 19 / 27 s s s D 1 D 2 X 1 X 2 OPT X 1 X 2 C X 1 X 2 D 1 \ C

29 Lower Bound on OPT 1. OPT routes D j C on a Steiner tree for X j. 2. V = j X j and single-source. 3. Edges carrying C spanning tree. 4. Edges carrying D j \ C Steiner tree for X j. 19 / 27 s s s D 1 D 2 X 1 X 2 OPT X 1 X 2 C X 1 X 2 D 1 \ C c(opt) C c(mst) + D j \ C c(opt Steiner for X j ) j

30 Group Spanners Definition For a graph G and g groups X 1,...,X g V, a subgraph H is a (α,β) group spanner if c(h) αc(mst), c(opt Steiner for X j in H) βc(opt Steiner for X j in G) for all j. G H =1 = / 27

31 Group Spanners Definition For a graph G and g groups X 1,...,X g V, a subgraph H is a (α,β) group spanner if c(h) αc(mst), c(opt Steiner for X j in H) βc(opt Steiner for X j in G) for all j. G H =1 =2 3 6 Usual spanners: X j s are all vertex pairs. c(shortest u v path in H) βc(shortest u v path in G) for all u,v V. 20 / 27

32 Using Group Spanners 21 / 27 Given (α,β) group spanner H, route D j along 2-approximate Steiner tree for X j in H. Cost C c(h) + D j \ C 2c(opt Steiner for X j in H) j C αc(mst) + D j \ C 2βc(opt Steiner for X j ) j max{α,2βopt.

33 Group Spanners Result 22 / 27 Lemma Given an unweighted graph G and g groups X 1,...,X g V with V = j X j, we can construct in polynomial time a (O(1), O(log g)) group spanner. Theorem Network Design with Coverage Costs in the sunflower demands setting admits an O(log g) approximation over unweighted graphs with V = j X j.

34 Related Work on Spanners 23 / 27 X j s are all vertex pairs (usual spanners): (O(log n),o(log n)) [Chandra et al. 92, ] X j s are all vertex pairs containing s: (2, 3) [Khuller, Raghavachari, Young 93]

35 Constructing Group Spanners Assume optimal Steiner tree T j for X j has no Steiner vertices (at a loss of 2 in β). 24 / 27 T j

36 24 / 27 Constructing Group Spanners Assume optimal Steiner tree T j for X j has no Steiner vertices (at a loss of 2 in β). Algorithm (sketch) 1. H MST. 2. While e T j such that d H (S e,x j \ S e ) > log g: add e to H. T j H e T j \ H

37 24 / 27 Constructing Group Spanners Assume optimal Steiner tree T j for X j has no Steiner vertices (at a loss of 2 in β). Algorithm (sketch) 1. H MST. 2. While e T j such that d H (S e,x j \ S e ) > log g: add e to H. T j H e T j \ H Se X j \ S e

38 24 / 27 Constructing Group Spanners Assume optimal Steiner tree T j for X j has no Steiner vertices (at a loss of 2 in β). Algorithm (sketch) 1. H MST. 2. While e T j such that d H (S e,x j \ S e ) > log g: add e to H. T j H e T j \ H S e X j \ S e > log g

39 Constructing Group Spanners Assume optimal Steiner tree T j for X j has no Steiner vertices (at a loss of 2 in β). Algorithm (sketch) 1. H MST. 2. While e T j such that d H (S e,x j \ S e ) > log g: add e to H. T j H e T j \ H S e X j \ S e > log g girth(h) log g = c(h) O(1)c(MST) 24 / 27

40 Group Spanners Result 25 / 27 Lemma Given an unweighted graph G and g groups X 1,...,X g V with V = j X j, we can construct in polynomial time a (O(1), O(log g)) group spanner. Theorem Network Design with Coverage Costs in the sunflower demands setting admits an O(log g) approximation over unweighted graphs with V = j X j.

41 26 / 27 Summary Coverage cost model for designing information networks. P1 : P1 : P2 : Load = 2

42 26 / 27 Summary Coverage cost model for designing information networks. P1 : P1 : P2 : Load = 2 Laminar demands: 2-approximation via primal-dual. D1 D2 D3 D5 D4

43 26 / 27 Summary Coverage cost model for designing information networks. P1 : P1 : P2 : Load = 2 Laminar demands: 2-approximation via primal-dual. D1 D2 D3 D5 D4 Sunflower demands: O(log g)-approximation. D i \ C C D j \ C Group spanners: unweighted graphs have (O(1), O(log g)) group spanners.

44 Open Problems 27 / 27 Group spanners: (O(log g), O(log g)) for general graphs?

45 Open Problems 27 / 27 Group spanners: (O(log g), O(log g)) for general graphs? Sunflower demands: O(1) approximation?

46 Open Problems 27 / 27 Group spanners: (O(log g), O(log g)) for general graphs? Sunflower demands: O(1) approximation? General demand families. Submodular costs.

47 Open Problems 27 / 27 Group spanners: (O(log g), O(log g)) for general graphs? Sunflower demands: O(1) approximation? General demand families. Submodular costs. Thanks!