Network Design with Coverage Costs

Transcription

1 Network Design with Coverage Costs Siddharth Barman 1 Shuchi Chawla 2 Seeun William Umboh 2 1 Caltech 2 University of Wisconsin-Madison APPROX-RANDOM 2014

2 Motivation

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

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

5 Why Coverage Costs? Want a cost structure that captures bandwidth savings obtained by eliminating redundancy in data. Packet deduplication capability exists in networks today.

6 Problem Statement

7 Load Function 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

8 Input: Graph with edge costs, g terminal pairs (s 1,t 1 ),...,(s g,t g ) s 1 t 1 s 2 t 2

9 Input: Graph with edge costs, g terminal pairs (s 1,t 1 ),...,(s g,t g ) A global set of packets Π. s 1 t 1 s 2 t 2

10 Input: Graph with edge costs, g terminal pairs (s 1,t 1 ),...,(s g,t g ) A global set of packets Π. A demand set D j Π for each j. s 1 t 1 s 2 t 2 D 1 D 2

11 Input: Graph with edge costs, g terminal pairs (s 1,t 1 ),...,(s g,t g ) A global set of packets Π. A demand set D j Π for each j. Goal: Find (s j,t j ) path on which to route D j for each j minimizing Cost of edge Load on edge edges s 1 t 1 s 2 t 2 D 1 D 2

12 Input: Graph with edge costs, g terminal pairs (s 1,t 1 ),...,(s g,t g ) A global set of packets Π. A demand set D j Π for each j. Goal: Find (s j,t j ) path on which to route D j for each j minimizing Cost of edge # distinct packets on edge edges s 1 t 1 s 2 t 2 D 1 D 2

13 Input: Graph with edge costs, g terminal pairs (s 1,t 1 ),...,(s g,t g ) A global set of packets Π. A demand set D j Π for each j. Goal: Find (s j,t j ) path on which to route D j for each j minimizing Cost of edge # distinct packets on edge edges s 1 t 1 s 2 t 2 D 1 D 2

14 Input: Graph with edge costs, g terminal pairs (s 1,t 1 ),...,(s g,t g ) A global set of packets Π. A demand set D j Π for each j. Goal: Find (s j,t j ) path on which to route D j for each j minimizing Cost of edge # distinct packets on edge edges s 1 t 1 D1 D1 D2 D1 U D2 D2 D 1 D 2 s 2 t 2

15 More generally, we consider terminal groups. Input: Graph with edge costs, g terminal groups X 1,...,X g V. A global set of packets Π. A demand set D j Π for each j. Goal: Find a Steiner tree for X j on which to broadcast D j for each j minimizing Cost of edge # distinct packets on edge edges

16 Special cases (for terminal pairs): Pairwise disjoint demands = shortest-paths s 1 t 1 D1 D2 D 1 D 2 s 2 t 2

17 Special cases (for terminal pairs): Pairwise disjoint demands = shortest-paths s 1 t 1 D1 D2 D 1 D 2 s 2 t 2 Identical demands = Steiner forest s 1 t 1 D D D D D s 2 t 2 D 1, D 2 = D

18 Challenges Intuitively, difficulty depends on complexity of demand sets

19 Challenges Intuitively, difficulty depends on complexity of demand sets Desire an approximation in terms of g (number of groups), e.g. O(log g) (tree embeddings only yield O(log n)).

20 Our results

21 Laminar Demands 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 1 D 3 D 5 D 5 D3 D 4

22 Laminar Demands 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 1 D 3 D 5 D 5 D3 D 4 Theorem NDCC with laminar demands admits a 2-approximation. Previous work: O(log Π ) [Barman-Chawla 12].

23 Sunflower Demands The family of demand sets is said to be a sunflower family if there exists a core C Π such that i,j, D i D j = C. D i \ C C D j \ C

24 Sunflower Demands The family of demand sets is said to be a sunflower family if there exists a core C Π such that i,j, D i D j = C. D i \ C C D j \ C No O(log 1/4 γ g)-approximation for any γ > 0 (reduction from single-cable buy-at-bulk [Andrews-Zhang 02]).

25 Sunflower Demands The family of demand sets is said to be a sunflower family if there exists a core C Π such that i,j, D i D j = C. D i \ C C D j \ C No O(log 1/4 γ g)-approximation for any γ > 0 (reduction from single-cable buy-at-bulk [Andrews-Zhang 02]). Theorem NDCC with sunflower demands admits an O(log g) approximation over unweighted graphs with V = j X j.

26 Related Work Network design with economies of scale. Cost of edge Load on edge edges

27 Related Work Network design with economies of scale. Cost of edge Load on edge edges Submodular costs on edges [Hayrapetyan-Swamy-Tardos 05] O(log n) via tree embeddings.

28 Related Work Network design with economies of scale. Cost of edge Load on edge edges Submodular costs on edges [Hayrapetyan-Swamy-Tardos 05] O(log n) 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: Ω(log 1 4 n) hardness [Andrews 04]

29 Approach: Laminar

30 Laminar D 1 D 2 D 1 D 3 D 5 D 5 D3 D 4 Naive LP does not have much structure Key idea: exploit laminarity to write a different LP Run an extension of AKR-GW primal-dual algorithm to get 2-approx

31 Approach: Sunflower (single-source for this talk)

32 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. 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.

33 Structure of OPT Define: E j = Steiner tree for X j on which OPT broadcasts D j C E 0 = j E j

34 Structure of OPT Define: E j = Steiner tree for X j on which OPT broadcasts D j C E 0 = j E j Note: E 0 spans V because V = j X j and single-source (each X j contains s)

35 Structure of OPT Define: E j = Steiner tree for X j on which OPT broadcasts D j C E 0 = j E j Note: E 0 spans V because V = j X j and single-source (each X j contains s) c(opt) = C c(e 0 ) + D j \ C c(e j ) j C c(mst) + D j \ C c(opt Steiner for X j ) j

36 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.

37 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. 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.

38 Using Group Spanners 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,2β}opt.

39 Group Spanners Result 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.

40 Summary Coverage cost model for designing information networks. P1 : P1 : P2 : Load = 2

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

42 Summary Coverage cost model for designing information networks. P1 : P1 : P2 : Load = 2 Laminar demands: 2-approximation via primal-dual. D1 D2 D1 D2 D4 D3 D3 D5 D5 D4 Sunflower demands: O(log g)-approximation (unweighted graph, no Steiner vertices). D i \ C C D j \ C Group spanners: (O(1), O(log g)) group spanners (unweighted graph, no Steiner vertices).

43 Open Problems Group spanners: (O(log g), O(log g)) in general?

44 Open Problems Group spanners: (O(log g), O(log g)) in general? Sunflower demands: O(log g) approximation in general?

45 Open Problems Group spanners: (O(log g), O(log g)) in general? Sunflower demands: O(log g) approximation in general? Terminal pairs with single-source: O(1)?

46 Open Problems Group spanners: (O(log g), O(log g)) in general? Sunflower demands: O(log g) approximation in general? Terminal pairs with single-source: O(1)? Thanks!