Constrained resource assignments: Fast algorithms and applications - PowerPoint PPT Presentation

Constrained resource assignments: Fast algorithms and applications in wireless networks Andr´ e Berger, James Gross, Tobias Harks, and Simon Tenbusch Aussois, Workshop on Scheduling Tobias Harks: Constrained resource assignments

Contents Tobias Harks: Constrained resource assignments

Motivation - OFDMA OFDMA=orthogonal frequency division multiple access Tobias Harks: Constrained resource assignments

Motivation - OFDMA Tobias Harks: Constrained resource assignments

Motivation - OFDMA Tobias Harks: Constrained resource assignments

Motivation - OFDMA Tobias Harks: Constrained resource assignments

Motivation - OFDMA Tobias Harks: Constrained resource assignments

Motivation - OFDMA 18 16 14 12 10 8 6 4 2 0 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 Tobias Harks: Constrained resource assignments

Motivation - OFDMA Tobias Harks: Constrained resource assignments

Motivation - Infeasible Assignment Tobias Harks: Constrained resource assignments

Motivation - Feasible Assignment Tobias Harks: Constrained resource assignments

Variant 1: Budget Constraints Given a complete bipartite graph K n , n = ( V n , E n ), a sequence of edge weights w ( t ) : E n → R + ( t = 1 , . . . , T ), and a positive integer k , find a sequence of perfect matchings M ( t ) such that for all 2 ≤ t ≤ T | M ( t − 1) ∩ M ( t ) | ≥ n − k that maximizes the total net weight � w ( t ) ( M ( t ) ) t . Tobias Harks: Constrained resource assignments

Variant 1: Budget Constraints The 2-phase case: Given a perfect matching M 0 in K n , n , edge weights and an integer k ≥ 0, find a perfect matching of maximum weight that changes at most k edges of M 0 . Tobias Harks: Constrained resource assignments

Variant 1: Budget Constraints The 2-phase case: Given a perfect matching M 0 in K n , n , edge weights and an integer k ≥ 0, find a perfect matching of maximum weight that changes at most k edges of M 0 . Theorem (Berger, Bonifaci, Grandoni, Sch¨ afer 2011) There is a PTAS for the budgeted matching problem, i.e. given a graph G with edge weights, edge costs, and and a budget B, find a matching of maximum weight whose cost does not exceed B. Tobias Harks: Constrained resource assignments

Variant 1: Budget Constraints The 2-phase case: Given a perfect matching M 0 in K n , n , edge weights and an integer k ≥ 0, find a perfect matching of maximum weight that changes at most k edges of M 0 . Theorem (Berger, Bonifaci, Grandoni, Sch¨ afer 2011) There is a PTAS for the budgeted matching problem, i.e. given a graph G with edge weights, edge costs, and and a budget B, find a matching of maximum weight whose cost does not exceed B. Problems: ◮ the running time is too high Tobias Harks: Constrained resource assignments

Variant 1: Budget Constraints The 2-phase case: Given a perfect matching M 0 in K n , n , edge weights and an integer k ≥ 0, find a perfect matching of maximum weight that changes at most k edges of M 0 . Theorem (Berger, Bonifaci, Grandoni, Sch¨ afer 2011) There is a PTAS for the budgeted matching problem, i.e. given a graph G with edge weights, edge costs, and and a budget B, find a matching of maximum weight whose cost does not exceed B. Problems: ◮ the running time is too high ◮ no PTAS is known to find perfect matchings Tobias Harks: Constrained resource assignments

Approximation Theorem 1 There is a 1 / 2 –approximation algorithm for the bipartite matching problem with fixed reconfiguration costs that runs in O ( k · n 3 ) time. Tobias Harks: Constrained resource assignments

Initial Matching n = 10 , k = 5 M 0 10 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 10 Tobias Harks: Constrained resource assignments

The Algorithm 1. w ( M 0 ) ← � n i =1 w ii ij ← w ij + w ( M 0 ) − w ii + w jj 2. w ′ , 1 ≤ i , j ≤ n 2 k 3. find a max. weight matching M 1 w.r.t. w ′ with at most ℓ := ⌊ k 2 ⌋ edges ALG ⊆ M 0 with M 0 4. compute a matching M 0 ALG ∩ M 1 = ∅ and | M 0 ALG | = n − k 5. compute a maximum weight perfect matching M 1 ALG w.r.t. w on G of nodes not matched by edges in M 0 ALG . 6. return M = M 0 ALG ∪ M 1 ALG Tobias Harks: Constrained resource assignments

Example n = 10 , k = 5 M 0 10 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 10 Tobias Harks: Constrained resource assignments

Example n = 10 , k = 5 M 1 10 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 10 Tobias Harks: Constrained resource assignments

Example n = 10 , k = 5 M 1 M 0 ALG 10 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 10 Tobias Harks: Constrained resource assignments

Example n = 10 , k = 5 M 0 ALG , M 1 ALG 10 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 10 Return: M = M 0 ALG ∪ M 1 ALG Tobias Harks: Constrained resource assignments

Analysis ALG ∪ M 1 ∪ � M = M 0 M ∪ M 2 M = { u i v i : u i and v i are not end points of edges in M 1 ∪ M 0 � ALG } M 2 arbitrary matching such that M is perfect matching 10 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 10 Tobias Harks: Constrained resource assignments

Analysis (contd.) w ( M ) ≥ w ( M ) Tobias Harks: Constrained resource assignments

Analysis (contd.) w ( M ) ≥ w ( M ) Show: w ( M ) ≥ ⌊ k 2 ⌋ k w ( Opt ) ≈ 1 2 w ( Opt ) . Tobias Harks: Constrained resource assignments

Analysis (contd.) w ( M ) ≥ w ( M ) Show: w ( M ) ≥ ⌊ k 2 ⌋ k w ( Opt ) ≈ 1 2 w ( Opt ) . Use notation: Opt = Opt 0 ∪ Opt 1 with Opt 0 ⊆ M 0 and | Opt 0 | = n − k Tobias Harks: Constrained resource assignments

Analysis (contd.) w ( M ) ≥ w ( M ) Show: w ( M ) ≥ ⌊ k 2 ⌋ k w ( Opt ) ≈ 1 2 w ( Opt ) . Use notation: Opt = Opt 0 ∪ Opt 1 with Opt 0 ⊆ M 0 and | Opt 0 | = n − k w ( M ) = w ′ ( � Claim 1: M ∪ M 1 ∪ M 2 ) w ′ ( M 2 ∪ � M ) ≥ 0 . Claim 2: ⌊ k 2 ⌋ w ′ ( M 1 ) ≥ k w ′ ( Opt 1 ) Claim 3: w ′ ( Opt 1 ) = w ( Opt ). Claim 4: Tobias Harks: Constrained resource assignments

Analysis (contd.) w ( M ) ≥ w ( M ) Show: w ( M ) ≥ ⌊ k 2 ⌋ k w ( Opt ) ≈ 1 2 w ( Opt ) . Use notation: Opt = Opt 0 ∪ Opt 1 with Opt 0 ⊆ M 0 and | Opt 0 | = n − k w ( M ) = w ′ ( � Claim 1: M ∪ M 1 ∪ M 2 ) w ′ ( M 2 ∪ � M ) ≥ 0 . Claim 2: ⌊ k 2 ⌋ w ′ ( M 1 ) ≥ k w ′ ( Opt 1 ) Claim 3: w ′ ( Opt 1 ) = w ( Opt ). Claim 4: w ( M ) Claim 1 w ′ ( M 1 ∪ M 2 ∪ � M ) = w ′ ( M 1 ) + w ′ ( M 2 ∪ � = M ) ⌊ k ⌊ k 2 ⌋ 2 ⌋ Claim 2 Claim 3 k w ′ ( Opt 1 ) Claim 4 w ′ ( M 1 ) ≥ ≥ = k w ( Opt ) . Tobias Harks: Constrained resource assignments

Proof of Claim 1 � � � w ij + w ( M 0 ) − w ii + w jj w ′ ( M 1 ∪ M 2 ∪ � M ) = 2 k u i v j ∈ M 1 ∪ M 2 ∪ � M � M ) + k · w ( M 0 ) = w ( M 1 ∪ M 2 ∪ � − w ii k ∈ M 0 ( i , i ) / ALG = w ( M 1 ∪ M 2 ∪ � M ) + w ( M 0 ALG ) = w ( M ) . 10 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 10 Tobias Harks: Constrained resource assignments

Online Variant Theorem 2 There is an online algorithm for the bipartite matching problem with fixed reconfiguration costs that has competitive ratio k / 2 n. The competitive ratio of any deterministic online algorithm is at most ( k − 1) / n. Tobias Harks: Constrained resource assignments

The online algorithm Tobias Harks: Constrained resource assignments

The online algorithm Tobias Harks: Constrained resource assignments

The online algorithm Tobias Harks: Constrained resource assignments

The online algorithm Tobias Harks: Constrained resource assignments

The online algorithm Tobias Harks: Constrained resource assignments

Variant 2: Elastic Reconfiguration Costs Given a complete bipartite graph K n , n = ( V n , E n ), a sequence of edge weights w ( t ) : E n → R + ( t = 1 , . . . , T ), find a sequence of perfect matchings M ( t ) that maximizes the total net weight � w ( t ) ( M ( t ) ) t Tobias Harks: Constrained resource assignments

Variant 2: Elastic Reconfiguration Costs Given a complete bipartite graph K n , n = ( V n , E n ), a sequence of edge weights w ( t ) : E n → R + ( t = 1 , . . . , T ), find a sequence of perfect matchings M ( t ) that maximizes the total net weight � w ( t ) ( M ( t ) ) · ( c + (1 − c ) | M ( t − 1) ∩ M ( t ) | / n ) t . Tobias Harks: Constrained resource assignments

Variant 2: Elastic Reconfiguration Costs Theorem 3 There is a 1 / 2 –approximation algorithm for the bipartite matching problem with elastic reconfiguration costs that runs in O ( n 4 ) time. Tobias Harks: Constrained resource assignments

Variant 2: Elastic Reconfiguration Costs Theorem 3 There is a 1 / 2 –approximation algorithm for the bipartite matching problem with elastic reconfiguration costs that runs in O ( n 4 ) time. Theorem 4 For n ≥ 3 there is an online algorithm for the bipartite matching problem with elastic reconfiguration costs that has competitive ratio 1 / 9 . The competitive ratio of any deterministic online algorithm is at most 1 / 9 . Tobias Harks: Constrained resource assignments