Maximum-Life Routing Schedule Peng-Jun Wan wan@cs.iit.edu Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 1 / 42
Outline Problem Description Min-Cost Routing Ellipsoid Algorithm Price-Directive Algorithm Flow-Based Algorithm Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 2 / 42
A Motivating Example 250 s 16 90 u s u s 9 9 u 160 16 (a) t 250 (b) t u (c) t Figure: Consider the unicast from s to t in (a). If only one path in either (b) or (c) is used, the life is 10. On the other hand, we can use both paths for 10 time units each to achieve an overall life of 20. Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 3 / 42
Network model Communication topology: D = (V, A; c) Adjustable transmission power Power consumption: same as in the previous chapter Receiving power consumption is ignored Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 4 / 42
Communication Tasks Concurrent Unicasts Aggregation Broadcast Multicast Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 5 / 42
Communication Tasks Concurrent Unicasts Aggregation Broadcast Multicast R: a collection of routes for a given communication task Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 5 / 42
Routing Schedule b 2 R V +: energy budget function A routing schedule is a set of pairs (H i, x i ) 2 R R + for i = 1,, m satisfying that m p Hi (u) x i b (u), 8u 2 V. i=1 Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 6 / 42
Routing Schedule b 2 R V +: energy budget function A routing schedule is a set of pairs (H i, x i ) 2 R R + for i = 1,, m satisfying that m p Hi (u) x i b (u), 8u 2 V. i=1 The life (or length) of this schedule is m i=1 x i. Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 6 / 42
Max-Life Routing Schedule Max-Life Routing Schedule (MLRS): nding a routing schedule of maximum life. Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 7 / 42
Max-Life Routing Schedule Max-Life Routing Schedule (MLRS): nding a routing schedule of maximum life. max H 2R x H s.t. H 2R x H p H (u) b (u), 8u 2 V ; x H 0, 8H 2 R. Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 7 / 42
Max-Life Routing Schedule Max-Life Routing Schedule (MLRS): nding a routing schedule of maximum life. max H 2R x H s.t. H 2R x H p H (u) b (u), 8u 2 V ; x H 0, 8H 2 R. jv j = n constraints ) 9 an optimal solution using at most n routes. Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 7 / 42
Max-Life Routing Schedule Max-Life Routing Schedule (MLRS): nding a routing schedule of maximum life. max H 2R x H s.t. H 2R x H p H (u) b (u), 8u 2 V ; x H 0, 8H 2 R. jv j = n constraints ) 9 an optimal solution using at most n routes. # of variables jrj is exponential ) standard LP solvers are not practical. Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 7 / 42
Summary on Algorithms Ellipsoid Algorithm (EA) Price-Directive Algorithm (PDA) Flow-Based Algorithm (FBA) EA PDA FBA Conc. Unicasts exact 1 + ε exact Aggregation exact 1 + ε exact Broadcast 2H (n 1) 1 (1 + ε) (2H (n 1) 1) N/A Multicast O (k ε ) O (k ε ) N/A Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 8 / 42
Roadmap Problem Description Min-Cost Routing Ellipsoid Algorithm Price-Directive Algorithm Flow-Based Algorithm Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 9 / 42
Min-Cost Routing y 2 R V +: a price function H: a subgraph of D cost of H w.r.t. y = y (u) p H (u) u2v Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 10 / 42
Min-Cost Routing y 2 R V +: a price function H: a subgraph of D cost of H w.r.t. y = y (u) p H (u) u2v Min-Cost Routing (MCR): nd an H 2 R of minimum cost w.r.t y. Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 10 / 42
Min-Cost Routing y 2 R V +: a price function H: a subgraph of D cost of H w.r.t. y = y (u) p H (u) u2v Min-Cost Routing (MCR): nd an H 2 R of minimum cost w.r.t y. A generalization of Min-Power Routing Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 10 / 42
(Approximation) Algorithms for MCR By applying the algorithms developed in the previous chapter for MPR, we immediately have the following algorithmic results: 1 Concurrent Unicasts: polynomial 2 Aggregation: polynomial 3 Broadcast: (2H (n 1) 1)-approximation algorithm 4 Multicast: O (k ε )-approximation algorithm for any xed ε > 0 Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 11 / 42
MCR vs. Dual of MLRS Dual of MLRS: min s.t. u2v b (u) y (u) u2v p H (u) y (u) 1, 8H 2 R y (u) 0, 8u 2 V Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 12 / 42
MCR vs. Dual of MLRS Dual of MLRS: min s.t. u2v b (u) y (u) u2v p H (u) y (u) 1, 8H 2 R y (u) 0, 8u 2 V MCR: separation problem of the dual of MLRS Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 12 / 42
Max-Life vs. Min-Cost opt: life of a max-life routing schedule. For any price function y 2 R V +, let α (y) = min p H (u) y (u) : min-cost of routes in R w.r.t.y, H 2R u2v β (y) = b (u) y (u) : total energy cost w.r.t. y. u2v Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 13 / 42
Max-Life vs. Min-Cost opt: life of a max-life routing schedule. For any price function y 2 R V +, let α (y) = min p H (u) y (u) : min-cost of routes in R w.r.t.y, H 2R u2v β (y) = b (u) y (u) : total energy cost w.r.t. y. u2v Lemma For any y 2 R V +, α (y) β(y ) opt. In addition, there exists some y 2 RV + such that α (y) = β(y ) opt. Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 13 / 42
Max-Life vs. Min-Cost First Part: Trivial if α (y) = 0. So, we assume that α (y) > 0. Then, is a feasible solution of the dual LP. Hence y opt β = β (y) β (y) ) α (y) α (y) α (y) opt. y α(y ) Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 14 / 42
Max-Life vs. Min-Cost First Part: Trivial if α (y) = 0. So, we assume that α (y) > 0. Then, is a feasible solution of the dual LP. Hence y opt β = β (y) β (y) ) α (y) α (y) α (y) opt. Second Part: Suppose y is an optimal solution to dual LP. Then, opt = β (y) and α (y) = 1 ) α (y) = β (y) opt. y α(y ) Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 14 / 42
Roadmap Problem Description Min-Cost Routing Ellipsoid Algorithm Price-Directive Algorithm Flow-Based Algorithm Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 15 / 42
Ellipsoid Method with (Approximate) Separation Oracle N : a network class Theorem Suppose that there is a polynomial (respectively, a polynomial µ-approximation) algorithm for MCR for a communication task restricted to N. Then, there is a polynomial (respectively, a polynomial µ-approximation) algorithm for MLRS for the same communication task restricted to N. Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 16 / 42
Ellipsoid Algorithm for MLRS Ellipsoid Algorithm Conc. Unicasts exact Aggregation exact Broadcast 2H (n 1) 1 Multicast O (k ε ) Drawback: very slow practically Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 17 / 42
Roadmap Problem Description Min-Cost Routing Ellipsoid Algorithm Price-Directive Algorithm Flow-Based Algorithm Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 18 / 42
Basic Idea An iterative algorithm, in each iteration: Set the prices of the nodes with low residue energy relatively higher Nodes with low residue energy are protected from getting drained of energy quickly Nodes with high residue energy are enforced to contribute more energy Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 19 / 42
Basic Idea An iterative algorithm, in each iteration: Set the prices of the nodes with low residue energy relatively higher Nodes with low residue energy are protected from getting drained of energy quickly Nodes with high residue energy are enforced to contribute more energy Challenge: how to choose the prices properly? Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 19 / 42
Parameters A: a µ-approximation algorithm for MCR if µ = 1, the algorithm A is optimal for MCR ε: a constant parameter 2 (0, 1) output: an (1 + ε) µ-approximation. Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 20 / 42
Variables H: the set of chosen routes; x H for each H 2 H: the duration of H; Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 21 / 42
Variables H: the set of chosen routes; x H for each H 2 H: the duration of H; z 2 R V +: the energy consumption percentage vector de ned by z (u) = H 2H x H p H (u), 8u 2 V ; b (u) φ: the maximum energy consumption percentage max u2v z (u); Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 21 / 42
Variables H: the set of chosen routes; x H for each H 2 H: the duration of H; z 2 R V +: the energy consumption percentage vector de ned by z (u) = H 2H x H p H (u), 8u 2 V ; b (u) φ: the maximum energy consumption percentage max u2v z (u); y 2 R V +: the price vector; β: the total energy cost u2v b (u) y (u). Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 21 / 42
Outline of PDA H ; 8u 2 V, z (u) 0; φ 0; 1 8u 2 V, y (u) b(u) ; β n; repeat compute an H 2 R using A on (D, y) ; t min v 2V b (v) /p H (v); if H 2 H then x H x H + t, else H H [ fhg and x H t; 8u 2 V, z (u) φ max u2v z (u) ; 8u 2 V, y (u) y (u) z (u) + t p H (u) b(u) ; β u2v b (u) y (u) ; until 0 < φ 1+ε ε ln β n ; Output f(h, x H /φ) : H 2 Hg. 1 + εt p H (u) b(u) ; Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 22 / 42
Running Time And Approximation Bound Theorem The algorithm l PDA produces m an (1 + ε) µ-approximation in at most K = n iterations. (1+ε) ln n (1+ε) ln(1+ε) ε Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 23 / 42
Running Time And Approximation Bound Theorem The algorithm l PDA produces m an (1 + ε) µ-approximation in at most K = n iterations. (1+ε) ln n (1+ε) ln(1+ε) ε PDA Conc. Unicasts 1 + ε Aggregation 1 + ε Broadcast (1 + ε) (2H (n 1) 1) Multicast O (k ε ) Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 23 / 42
Notations opt: life of an optimal solution Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 24 / 42
Notations opt: life of an optimal solution H 0, z 0, φ 0, y 0 and β 0 : initial values of H, z, φ, y and β resp. Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 24 / 42
Notations opt: life of an optimal solution H 0, z 0, φ 0, y 0 and β 0 : initial values of H, z, φ, y and β resp. H j, z j, φ j, y j and β j : values of H, z, φ, y and β resp. at the end of the j-th iteration for each j 1 Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 24 / 42
Notations opt: life of an optimal solution H 0, z 0, φ 0, y 0 and β 0 : initial values of H, z, φ, y and β resp. H j, z j, φ j, y j and β j : values of H, z, φ, y and β resp. at the end of the j-th iteration for each j 1 H j : route selected in the j-th iteration Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 24 / 42
Notations opt: life of an optimal solution H 0, z 0, φ 0, y 0 and β 0 : initial values of H, z, φ, y and β resp. H j, z j, φ j, y j and β j : values of H, z, φ, y and β resp. at the end of the j-th iteration for each j 1 H j : route selected in the j-th iteration t j : value of t computed in the j-th iteration Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 24 / 42
Notations opt: life of an optimal solution H 0, z 0, φ 0, y 0 and β 0 : initial values of H, z, φ, y and β resp. H j, z j, φ j, y j and β j : values of H, z, φ, y and β resp. at the end of the j-th iteration for each j 1 H j : route selected in the j-th iteration t j : value of t computed in the j-th iteration τ j = max u2v y j (u) b (u): maximum energy cost of all nodes at the end of j-th iteration Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 24 / 42
Upper Bound on φ j Claim: φ j log 1+ε τ j for each j 1. Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 25 / 42
Upper Bound on φ j Claim: φ j log 1+ε τ j for each j 1. Lemma For any ε > 0 and 0 t 1, t log 1+ε (1 + εt). Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 25 / 42
Upper Bound on φ j Claim: φ j log 1+ε τ j for each j 1. Lemma For any ε > 0 and 0 t 1, t log 1+ε (1 + εt). z j (u) z j 1 (u) log 1+ε (1 + ε (z j (u) z j 1 (u))) = log 1+ε y j (u) y j 1 (u), )z j (u) log 1+ε y j (u) y 0 (u) = log 1+ε (y j (u) b (u)) log 1+ε τ j. Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 25 / 42
Running Time Prove by contradiction: assume > K iterations. Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 26 / 42
Running Time Prove by contradiction: assume > K iterations. Among the rst K iterations some node v appears as a bottleneck node in at least K /n iterations. Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 26 / 42
Running Time Prove by contradiction: assume > K iterations. Among the rst K iterations some node v appears as a bottleneck node in at least K /n iterations. In each of K /n iterations, y (v) is increased by a factor of 1 + ε. Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 26 / 42
Running Time Prove by contradiction: assume > K iterations. Among the rst K iterations some node v appears as a bottleneck node in at least K /n iterations. In each of K /n iterations, y (v) is increased by a factor of 1 + ε. y K (v) y 0 (v) (1 + ε) K /n /n (1 + ε)k = b (v) ) τ K y k (v) b (v) (1 + ε) K /n ) φ K ln β K n log 1+ε τ K ln τ K n = 1 ln (1 + ε) ln n log 1+ε τ K 1 + ε ε Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 26 / 42
Running Time Prove by contradiction: assume > K iterations. Among the rst K iterations some node v appears as a bottleneck node in at least K /n iterations. In each of K /n iterations, y (v) is increased by a factor of 1 + ε. y K (v) y 0 (v) (1 + ε) K /n /n (1 + ε)k = b (v) ) τ K y k (v) b (v) (1 + ε) K /n ) φ K ln β K n log 1+ε τ K ln τ K n = 1 ln (1 + ε) ln n log 1+ε τ K 1 + ε ε By the stopping rule, the number of iterations K, which is a contradiction. Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 26 / 42
Correctness of The output Solution k: number of iterations. Claim: By the end of the j-th iteration for 1 j k, the energy consumption percentage of each node u is z j (u), i.e., z j (u) = H 2H j x H p H (u), 8u 2 V. b (u) Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 27 / 42
Correctness of The output Solution k: number of iterations. Claim: By the end of the j-th iteration for 1 j k, the energy consumption percentage of each node u is z j (u), i.e., z j (u) = H 2H j x H p H (u), 8u 2 V. b (u) Therefore, the nal scaling by a factor φ k results in a feasible solution. Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 27 / 42
Lower Bound on t j Claim: t j 1 β j β j 1 εµ β j 1 opt for each 1 j k, Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 28 / 42
Lower Bound on t j Claim: t j 1 β j β j 1 εµ β j 1 opt for each 1 j k, β j = b (u) y j (u) u2v = b (u) y j u2v 1 (u) 1 + εt j p Hj (u) b (u) = b (u) y j 1 (u) + εt j p Hj (u) y j u2v u2v = β j 1 + εt j p Hj (u) y j u2v β j 1 + εt j µ β j 1 opt. 1 (u)! 1 (u)! Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 28 / 42
Approximation Bound k t j opt k β j β j 1 j=1 εµ opt j=1 β j 1 εµ ln β k = opt β 0 εµ ln β k n, Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 29 / 42
Approximation Bound So, k t j opt k β j β j 1 j=1 εµ opt j=1 β j 1 εµ ln β k = opt β 0 εµ ln β k n, k j=1 t j 1 ln β k n opt 1 ε opt opt = φ k εµ φ k εµ 1 + ε (1 + ε) µ. Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 29 / 42
Roadmap Problem Description Min-Cost Routing Ellipsoid Algorithm Price-Directive Algorithm Flow-Based Algorithm Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 30 / 42
Single Flow D = (V, A): a digraph with two distinct nodes s and t f 2 R A + is an s t ow in D if f δ out (v) = f δ in (v), 8v 2 V n fs, tg ( ow conservation law) s v1 2 v 3 6 4 0 6 0 4 0 5 5 v 2 5 v 4 t Figure: An an s t ow of value 11. Value of f : val (f ) = f (δ out (s)) f δ in (s). Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 31 / 42
Single Flow D = (V, A): a digraph with two distinct nodes s and t f 2 R A + is an s t ow in D if f δ out (v) = f δ in (v), 8v 2 V n fs, tg ( ow conservation law) s v1 2 v 3 6 4 0 6 0 4 0 5 5 v 2 5 v 4 t Figure: An an s t ow of value 11. Value of f : val (f ) = f (δ out (s)) f δ in (s). f is subject to an arc-capacity z 2 R A + if f z. Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 31 / 42
Flow Decomposition s 6 v 1 2 v 3 4 4 5 5 v 2 5 v 4 (a) 6 t s 5 v 2 5 (b) v 4 5 t s 6 v 2 1 v 3 4 4 6 t s v 1 2 v 3 2 2 t (c) v 4 (d) s v 1 v 3 4 4 4 4 t (e) v 4 Figure: Any s t ow of value L can be decomposed into at most jaj s t paths of total value L and possibly some circuits. Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 32 / 42
Maximum Flow Maximum Flow: nding an s t ow f subject to a given arc-capacity z 2 R A + such that val (f ) is maximized. Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 33 / 42
Maximum Flow Maximum Flow: nding an s t ow f subject to a given arc-capacity z 2 R A + such that val (f ) is maximized. Solvable in polynomial time by ow-augmentation algorithms. Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 33 / 42
Multi ow Given k commodities with s i, t i being the source and sink, resp., for commodity i. F i : the set of s i t i ows. A k- ow is a sequence hf 1, f 2,, f k i with f i 2 F i 81 i k. Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 34 / 42
Multi ow Given k commodities with s i, t i being the source and sink, resp., for commodity i. F i : the set of s i t i ows. A k- ow is a sequence hf 1, f 2,, f k i with f i 2 F i 81 i k. A k- ow hf 1, f 2,, f k i is subject to an arc-capacity z 2 R A + if k i=1 f i z. Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 34 / 42
Maximum Concurrent Multi ow max s.t. L f i 2 F i, 81 i k val (f i ) = L, 81 i k k i=1 f i z Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 35 / 42
Concurrent Unicasts Given k unicasts are treated as k commodities. A k- ow hf 1, f 2,, f k i is subject to an energy budget b 2 R V + if c (e) e2δ out (v )! k f i (e) b (v), 8v 2 V. i=1 Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 36 / 42
Concurrent Unicasts Given k unicasts are treated as k commodities. A k- ow hf 1, f 2,, f k i is subject to an energy budget b 2 R V + if c (e) e2δ out (v )! k f i (e) b (v), 8v 2 V. i=1 MLRS corresponds to maximum concurrent multi ow subject to energy budget b Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 36 / 42
MLRS for Concurrent Unicasts Step 1: Solve the LP max s.t. L f i 2 F i, 81 i k val (f i ) = L, 81 i k e2δ out (v ) c (e) k i=1 f i (e) b (v), 8v 2 V Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 37 / 42
MLRS for Concurrent Unicasts Step 1: Solve the LP max s.t. L f i 2 F i, 81 i k val (f i ) = L, 81 i k e2δ out (v ) c (e) k i=1 f i (e) b (v), 8v 2 V Step 2: Decompose each f i into at most jaj s i t i paths of total value L and discarding the rest circuits if there is any. Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 37 / 42
Fractional Arborescence Packing D = (V, A): a digraph with a root node s T : collection of spanning arborescences rooted at s A fractional s-arborescence packing in D subject to given arc-capacity z 2 R A + is a set of k pairs (T j, λ j ) 2 T R + satisfying that 1jk,e2T j λ j z (e), 8e 2 A. The value of this packing is k j=1 λ j. Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 38 / 42
Maximum Fractional Arborescence Packing Maximum Fractional Arborescence Packing: nding a fractional s-arborescence packing in D subject to a given arc-capacity z 2 R A + whose value is is maximized. Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 39 / 42
Maximum Fractional Arborescence Packing Maximum Fractional Arborescence Packing: nding a fractional s-arborescence packing in D subject to a given arc-capacity z 2 R A + whose value is is maximized. Gabow-Manu algorithm a greedy algorithm using at most jaj spanning s-arborescences. Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 39 / 42
A Min-Max Relation m ow (u, z): the value of a maximum s u ow in D subject to z, 8u 2 V n fsg Theorem The value of a maximum fractional s-arborescence packing in D subject to z is equal to min m ow (u, z). u2v nfsg Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 40 / 42
MLRS for Aggregation Step 1: Compute an optimal arc-capacity z 2 R A + by solving the LP: max s.t. L e2δ out (v ) c (e) z (e) b (v), 8v 2 V val (f u ) = L, 8u 2 V n fsg f u 2 F u, 8u 2 V n fsg f u z, 8u 2 V n fsg Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 41 / 42
MLRS for Aggregation Step 1: Compute an optimal arc-capacity z 2 R A + by solving the LP: max s.t. L e2δ out (v ) c (e) z (e) b (v), 8v 2 V val (f u ) = L, 8u 2 V n fsg f u 2 F u, 8u 2 V n fsg f u z, 8u 2 V n fsg Step 2: Compute a maximum fractional packing of spanning inward s-arborescences subject to z using the Gabow-Manu algorithm. Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 41 / 42
Correctness opt: life of a max-life routing schedule L: the value of the LP in Step 1 Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 42 / 42
Correctness opt: life of a max-life routing schedule L: the value of the LP in Step 1 Then, 1 opt L Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 42 / 42
Correctness opt: life of a max-life routing schedule L: the value of the LP in Step 1 Then, 1 opt L 2 the life of the output solution in Step 2 L by the min-max relation, Peng-Jun Wan (wan@cs.iit.edu) Maximum-Life Routing Schedule 42 / 42