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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

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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

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Contents

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Motivation - OFDMA

OFDMA=orthogonal frequency division multiple access

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Motivation - OFDMA

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Motivation - OFDMA

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Motivation - OFDMA

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Motivation - OFDMA

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Motivation - OFDMA

16 18 12 14 8 10 4 6 2 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

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Motivation - OFDMA

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Motivation - Infeasible Assignment

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Motivation - Feasible Assignment

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Variant 1: Budget Constraints

Given a complete bipartite graph Kn,n = (Vn, En), a sequence of edge weights w(t) : En → 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

  • t

w(t)(M(t)) .

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Variant 1: Budget Constraints

The 2-phase case: Given a perfect matching M0 in Kn,n, edge weights and an integer k ≥ 0, find a perfect matching of maximum weight that changes at most k edges of M0.

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Variant 1: Budget Constraints

The 2-phase case: Given a perfect matching M0 in Kn,n, edge weights and an integer k ≥ 0, find a perfect matching of maximum weight that changes at most k edges of M0.

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.

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Variant 1: Budget Constraints

The 2-phase case: Given a perfect matching M0 in Kn,n, edge weights and an integer k ≥ 0, find a perfect matching of maximum weight that changes at most k edges of M0.

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

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Variant 1: Budget Constraints

The 2-phase case: Given a perfect matching M0 in Kn,n, edge weights and an integer k ≥ 0, find a perfect matching of maximum weight that changes at most k edges of M0.

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

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Approximation

Theorem 1

There is a 1/2–approximation algorithm for the bipartite matching problem with fixed reconfiguration costs that runs in O(k · n3) time.

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Initial Matching

n = 10, k = 5 M0 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10

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The Algorithm

  • 1. w(M0) ← n

i=1 wii

  • 2. w′

ij ← wij + w(M0) k

− wii+wjj

2

, 1 ≤ i, j ≤ n

  • 3. find a max. weight matching M1 w.r.t. w′ with at most

ℓ := ⌊ k

2⌋ edges

  • 4. compute a matching M0

ALG ⊆ M0 with M0 ALG ∩ M1 = ∅ and

|M0

ALG| = n − k

  • 5. compute a maximum weight perfect matching M1

ALG w.r.t. w

  • n G of nodes not matched by edges in M0

ALG.

  • 6. return M = M0

ALG ∪ M1 ALG

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Example

n = 10, k = 5 M0 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10

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Example

n = 10, k = 5 M1 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10

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Example

n = 10, k = 5 M1 M0

ALG

1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10

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Example

n = 10, k = 5 M0

ALG,M1 ALG

1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 Return: M = M0

ALG ∪ M1 ALG

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Analysis

M = M0

ALG ∪ M1 ∪

M ∪ M2

  • M = {uivi : ui and vi are not end points of edges in M1 ∪ M0

ALG}

M2 arbitrary matching such that M is perfect matching 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10

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Analysis (contd.)

w(M) ≥ w(M)

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Analysis (contd.)

w(M) ≥ w(M) Show: w(M) ≥ ⌊ k

2⌋

k w(Opt) ≈ 1 2w(Opt).

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Analysis (contd.)

w(M) ≥ w(M) Show: w(M) ≥ ⌊ k

2⌋

k w(Opt) ≈ 1 2w(Opt). Use notation: Opt = Opt0 ∪ Opt1 with Opt0 ⊆ M0 and |Opt0| = n − k

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Analysis (contd.)

w(M) ≥ w(M) Show: w(M) ≥ ⌊ k

2⌋

k w(Opt) ≈ 1 2w(Opt). Use notation: Opt = Opt0 ∪ Opt1 with Opt0 ⊆ M0 and |Opt0| = n − k Claim 1: w(M) = w′( M ∪ M1 ∪ M2) Claim 2: w′(M2 ∪ M) ≥ 0. Claim 3: w′(M1) ≥

⌊ k

2 ⌋

k w′(Opt1)

Claim 4: w′(Opt1) = w(Opt).

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Analysis (contd.)

w(M) ≥ w(M) Show: w(M) ≥ ⌊ k

2⌋

k w(Opt) ≈ 1 2w(Opt). Use notation: Opt = Opt0 ∪ Opt1 with Opt0 ⊆ M0 and |Opt0| = n − k Claim 1: w(M) = w′( M ∪ M1 ∪ M2) Claim 2: w′(M2 ∪ M) ≥ 0. Claim 3: w′(M1) ≥

⌊ k

2 ⌋

k w′(Opt1)

Claim 4: w′(Opt1) = w(Opt). w(M) Claim 1 = w′(M1 ∪ M2 ∪ M) = w′(M1) + w′(M2 ∪ M)

Claim 2

≥ w′(M1)

Claim 3

≥ ⌊ k

2⌋

k w′(Opt1) Claim 4 = ⌊ k

2⌋

k w(Opt).

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Proof of Claim 1

w′(M1 ∪ M2 ∪ M) =

  • uivj∈M1∪M2∪

M

  • wij + w(M0)

k − wii + wjj 2

  • = w(M1 ∪ M2 ∪

M) + k · w(M0) k −

  • (i,i)/

∈M0

ALG

wii = w(M1 ∪ M2 ∪ M) + w(M0

ALG) = w(M).

1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10

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Online Variant

Theorem 2

There is an online algorithm for the bipartite matching problem with fixed reconfiguration costs that has competitive ratio k/2n. The competitive ratio of any deterministic online algorithm is at most (k − 1)/n.

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The online algorithm

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The online algorithm

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The online algorithm

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The online algorithm

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The online algorithm

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Variant 2: Elastic Reconfiguration Costs

Given a complete bipartite graph Kn,n = (Vn, En), a sequence of edge weights w(t) : En → R+ (t = 1, . . . , T), find a sequence of perfect matchings M(t) that maximizes the total net weight

  • t

w(t)(M(t))

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Variant 2: Elastic Reconfiguration Costs

Given a complete bipartite graph Kn,n = (Vn, En), a sequence of edge weights w(t) : En → R+ (t = 1, . . . , T), find a sequence of perfect matchings M(t) that maximizes the total net weight

  • t

w(t)(M(t)) · (c + (1 − c)|M(t−1) ∩ M(t)|/n) .

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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(n4) time.

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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(n4) 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.

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Computational Study: Setting

◮ N = 96 frequencies and 96 clients ◮ during one downlink phase 7 symbols can be transmitted ◮ one symbol can signal up to 32 edge changes ◮ at most 3 of the 7 symbols are used for signalling ◮ we considered 2000 downlink phases for 7 different scenarios

in which the movement of the clients varies (and therefore the correlation of edge weights)

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Computational Study: Budget Constraints

1 2 10 20 30 50 100 Movement Speed [m/s] 190 200 210 220 230 240 250 260 270 Throughput [kbit/s]

Greedy-MIP 1/2-Approximation Lagrange Upper Bound

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Computational Study: Elastic Reconfiguration Costs

1 2 10 20 30 50 100 Movement Speed [m/s] 160 180 200 220 240 260 280 300 320 Throughput [kbit/s]

Greedy-MIP 1/2-Approximation

  • Max. Weight Matching

Upper Bound

Tobias Harks: Constrained resource assignments

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Results

Net Throughput Runtime (ms per phase) Greedy-MIP 86% 573 1/2–Approximation 85% 4.8 max weight matching 75% 0.4

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Conclusions

◮ In real-time systems such as for wireless data transmission

very fast algorithms are needed.

◮ We provide such algorithms with low running times and high

performance in various settings.

◮ The models and algorithms can easily be extended to other

settings.

◮ The theoretical complexity of the underlying 2-phase problems

is still unknown.

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Thank you!

Tobias Harks: Constrained resource assignments