(P, R) each relationship has a “desire growth rate” ⇝ edge weights g : R → R>0 pain of separation grows by g(e) each day Goal: perpetual schedule S : N0 → 2R that minimizes “heat”: maximal desire ever felt (minimax objective) each person can engage in one relationship per day ⇝ every day t, schedule a matching S(t) A A B B C C D D E E F F G G H H Example polycule with 8 persons: Adam, Brady, Charlie, Daisy, Eli, Frankie, Grace, and Holly Sebastian Wild Polyamorous Scheduling 2024-06-06 2 / 16

(P, R) each relationship has a “desire growth rate” ⇝ edge weights g : R → R>0 pain of separation grows by g(e) each day Goal: perpetual schedule S : N0 → 2R that minimizes “heat”: maximal desire ever felt (minimax objective) each person can engage in one relationship per day ⇝ every day t, schedule a matching S(t) A A B B C C D D E E F F G G H H 40 ♡ 80 ♡ 16 ♡ 20 ♡ 40 ♡ 40 ♡ 40 ♡ 80 ♡ 16 ♡ 80 ♡ g(E−F) = 40 Example polycule with 8 persons: Adam, Brady, Charlie, Daisy, Eli, Frankie, Grace, and Holly Sebastian Wild Polyamorous Scheduling 2024-06-06 2 / 16

(P, R) each relationship has a “desire growth rate” ⇝ edge weights g : R → R>0 pain of separation grows by g(e) each day Goal: perpetual schedule S : N0 → 2R that minimizes “heat”: maximal desire ever felt (minimax objective) each person can engage in one relationship per day ⇝ every day t, schedule a matching S(t) A A B B C C D D E E F F G G H H 40 ♡ 80 ♡ 16 ♡ 20 ♡ 40 ♡ 40 ♡ 40 ♡ 80 ♡ 16 ♡ 80 ♡ g(E−F) = 40 Example polycule with 8 persons: Adam, Brady, Charlie, Daisy, Eli, Frankie, Grace, and Holly Sebastian Wild Polyamorous Scheduling 2024-06-06 2 / 16

A♡C B♡C C♡D A A B B C C D D 40 ♡ 20 ♡ 17 ♡ 25 ♡ Day A♡B A♡C B♡C C♡D 0 ♥ ♥ 1 ♥ 2 ♥ ♥ 3 ♥ 4 ♥ ♥ 5 ♥ Sebastian Wild Polyamorous Scheduling 2024-06-06 3 / 16

A♡C B♡C C♡D A A B B C C D D 40 ♡ 20 ♡ 17 ♡ 25 ♡ Day A♡B A♡C B♡C C♡D 0 ♥ ♥ 1 ♥ 2 ♥ ♥ 3 ♥ 4 ♥ ♥ 5 ♥ Sebastian Wild Polyamorous Scheduling 2024-06-06 3 / 16

A♡C B♡C C♡D A A B B C C D D 40 ♡ 20 ♡ 17 ♡ 25 ♡ Day A♡B A♡C B♡C C♡D 0 ♥ ♥ 1 ♥ 2 ♥ ♥ 3 ♥ 4 ♥ ♥ 5 ♥ 6 ♥ ♥ Sebastian Wild Polyamorous Scheduling 2024-06-06 3 / 16

A♡C B♡C C♡D A A B B C C D D 40 ♡ 20 ♡ 17 ♡ 25 ♡ Day A♡B A♡C B♡C C♡D 0 ♥ ♥ 1 ♥ 2 ♥ ♥ 3 ♥ 4 ♥ ♥ 5 ♥ 6 ♥ ♥ Sebastian Wild Polyamorous Scheduling 2024-06-06 3 / 16

A♡C B♡C C♡D A A B B C C D D 40 ♡ 20 ♡ 17 ♡ 25 ♡ Day A♡B A♡C B♡C C♡D 0 ♥ ♥ 1 ♥ 2 ♥ ♥ 3 ♥ 4 ♥ ♥ 5 ♥ 6 ♥ ♥ 7 ♥ S∗ = {AB, CD}, {AC}, {AB, CD}, {BC} with heat h(S) = 80 Observation: schedules without follows from finite state space loss of generality periodic ∃T ∀t S(t) = S(t + T) Sebastian Wild Polyamorous Scheduling 2024-06-06 3 / 16

D D 40 ♡ 20 ♡ 17 ♡ 25 ♡ Optimization Poly Scheduling Maximal frequencies to remain below heat 80: heat: 80 A♡B A♡C B♡C C♡D 2 4 4 3 frequency = heat growth rate Decision instance: A A B B C C D D 2 ♡ 4 ♡ 4 ♡ 3 ♡ A and C must meet (at least) once in every 4-day period Decision Poly Scheduling Sebastian Wild Polyamorous Scheduling 2024-06-06 4 / 16

D D 40 ♡ 20 ♡ 17 ♡ 25 ♡ Optimization Poly Scheduling Maximal frequencies to remain below heat 100: heat: 100 A♡B A♡C B♡C C♡D 2 5 5 4 frequency = heat growth rate Decision instance: A A B B C C D D 2 ♡ 5 ♡ 5 ♡ 4 ♡ A and C must meet (at least) once in every 5-day period Decision Poly Scheduling Sebastian Wild Polyamorous Scheduling 2024-06-06 4 / 16

D D 40 ♡ 20 ♡ 17 ♡ 25 ♡ Optimization Poly Scheduling Maximal frequencies to remain below heat 79.99: heat: 79.99 A♡B A♡C B♡C C♡D 1 3 4 3 frequency = heat growth rate Decision instance: A A B B C C D D 1 ♡ 3 ♡ 4 ♡ 3 ♡ Decision Poly Scheduling Sebastian Wild Polyamorous Scheduling 2024-06-06 4 / 16

D D 40 ♡ 20 ♡ 17 ♡ 25 ♡ Optimization Poly Scheduling Maximal frequencies to remain below heat 79.99: heat: 79.99 A♡B A♡C B♡C C♡D 1 3 4 3 frequency = heat growth rate Decision instance: A A B B C C D D 1 ♡ 3 ♡ 4 ♡ 3 ♡ Decision Poly Scheduling Sebastian Wild Polyamorous Scheduling 2024-06-06 4 / 16

– or are they facing a computationally hard problem? Let’s consider a simple special case: Non-hierarchical Poly Scheduling (optimization): all edges have weight 1 (decision): all edges have the same frequency f ⇝ feasible schedule for decision problem ⇐⇒ schedule of heat ⩽ f in optimization problem ⇝ Equivalent to Edge minimal #colors χ1 to color all edges s.t. no 2 incident to same vertex have same color Coloring (a.k.a. Chromatic Index) A A B B C C D D E E 1 ♡ 1 ♡ 1 ♡ 1 ♡ 1 ♡ 1 ♡ 1 ♡ 1 ♡ Edge Coloring =⇒ Poly Schedule: Schedule = Round Robin of χ1 color classes heat of schedule χ1 . Edge Coloring ⇐= Poly Schedule: Color = first day edge is scheduled Heat h ⇝ at most h colors Sebastian Wild Polyamorous Scheduling 2024-06-06 5 / 16

– or are they facing a computationally hard problem? Let’s consider a simple special case: Non-hierarchical Poly Scheduling (optimization): all edges have weight 1 (decision): all edges have the same frequency f ⇝ feasible schedule for decision problem ⇐⇒ schedule of heat ⩽ f in optimization problem ⇝ Equivalent to Edge minimal #colors χ1 to color all edges s.t. no 2 incident to same vertex have same color Coloring (a.k.a. Chromatic Index) A A B B C C D D E E 1 ♡ 1 ♡ 1 ♡ 1 ♡ 1 ♡ 1 ♡ 1 ♡ 1 ♡ Edge Coloring =⇒ Poly Schedule: Schedule = Round Robin of χ1 color classes heat of schedule χ1 . Edge Coloring ⇐= Poly Schedule: Color = first day edge is scheduled Heat h ⇝ at most h colors Sebastian Wild Polyamorous Scheduling 2024-06-06 5 / 16

– or are they facing a computationally hard problem? Let’s consider a simple special case: Non-hierarchical Poly Scheduling (optimization): all edges have weight 1 (decision): all edges have the same frequency f ⇝ feasible schedule for decision problem ⇐⇒ schedule of heat ⩽ f in optimization problem ⇝ Equivalent to Edge minimal #colors χ1 to color all edges s.t. no 2 incident to same vertex have same color Coloring (a.k.a. Chromatic Index) A A B B C C D D E E 1 ♡ 1 ♡ 1 ♡ 1 ♡ 1 ♡ 1 ♡ 1 ♡ 1 ♡ Edge Coloring =⇒ Poly Schedule: Schedule = Round Robin of χ1 color classes heat of schedule χ1 . Edge Coloring ⇐= Poly Schedule: Color = first day edge is scheduled Heat h ⇝ at most h colors Sebastian Wild Polyamorous Scheduling 2024-06-06 5 / 16

– or are they facing a computationally hard problem? Let’s consider a simple special case: Non-hierarchical Poly Scheduling (optimization): all edges have weight 1 (decision): all edges have the same frequency f ⇝ feasible schedule for decision problem ⇐⇒ schedule of heat ⩽ f in optimization problem ⇝ Equivalent to Edge minimal #colors χ1 to color all edges s.t. no 2 incident to same vertex have same color Coloring (a.k.a. Chromatic Index) A A B B C C D D E E 1 ♡ 1 ♡ 1 ♡ 1 ♡ 1 ♡ 1 ♡ 1 ♡ 1 ♡ Edge Coloring =⇒ Poly Schedule: Schedule = Round Robin of χ1 color classes heat of schedule χ1 . Edge Coloring ⇐= Poly Schedule: Color = first day edge is scheduled Heat h ⇝ at most h colors Sebastian Wild Polyamorous Scheduling 2024-06-06 5 / 16

– or are they facing a computationally hard problem? Let’s consider a simple special case: Non-hierarchical Poly Scheduling (optimization): all edges have weight 1 (decision): all edges have the same frequency f ⇝ feasible schedule for decision problem ⇐⇒ schedule of heat ⩽ f in optimization problem ⇝ Equivalent to Edge minimal #colors χ1 to color all edges s.t. no 2 incident to same vertex have same color Coloring (a.k.a. Chromatic Index) A A B B C C D D E E 1 ♡ 1 ♡ 1 ♡ 1 ♡ 1 ♡ 1 ♡ 1 ♡ 1 ♡ Edge Coloring =⇒ Poly Schedule: Schedule = Round Robin of χ1 color classes heat of schedule χ1 . Edge Coloring ⇐= Poly Schedule: Color = first day edge is scheduled Heat h ⇝ at most h colors Sebastian Wild Polyamorous Scheduling 2024-06-06 5 / 16

G = #colors in proper edge coloring ⇝ Obivous: χ1 ⩾ ∆ maximum degree in G Vizing’s Theorem (1965): χ1 ⩽ ∆ + 1 1 ∆ + 1 coloring can always be found in efficiently. running time very recently improved to ˜ O(mn1/3) (arxiv:2405.15449) 2 Computing χ1 exactly (∆ or ∆ + 1!) is NP-hard even for 3-regular graphs! (Holyer 1981) Sebastian Wild Polyamorous Scheduling 2024-06-06 6 / 16

G = #colors in proper edge coloring ⇝ Obivous: χ1 ⩾ ∆ maximum degree in G Vizing’s Theorem (1965): χ1 ⩽ ∆ + 1 1 ∆ + 1 coloring can always be found in efficiently. running time very recently improved to ˜ O(mn1/3) (arxiv:2405.15449) 2 Computing χ1 exactly (∆ or ∆ + 1!) is NP-hard even for 3-regular graphs! (Holyer 1981) Sebastian Wild Polyamorous Scheduling 2024-06-06 6 / 16

G = #colors in proper edge coloring ⇝ Obivous: χ1 ⩾ ∆ maximum degree in G Vizing’s Theorem (1965): χ1 ⩽ ∆ + 1 1 ∆ + 1 coloring can always be found in efficiently. running time very recently improved to ˜ O(mn1/3) (arxiv:2405.15449) 2 Computing χ1 exactly (∆ or ∆ + 1!) is NP-hard even for 3-regular graphs! (Holyer 1981) Sebastian Wild Polyamorous Scheduling 2024-06-06 6 / 16

G = #colors in proper edge coloring ⇝ Obivous: χ1 ⩾ ∆ maximum degree in G Vizing’s Theorem (1965): χ1 ⩽ ∆ + 1 1 ∆ + 1 coloring can always be found in efficiently. running time very recently improved to ˜ O(mn1/3) (arxiv:2405.15449) 2 Computing χ1 exactly (∆ or ∆ + 1!) is NP-hard even for 3-regular graphs! (Holyer 1981) Sebastian Wild Polyamorous Scheduling 2024-06-06 6 / 16

G = #colors in proper edge coloring ⇝ Obivous: χ1 ⩾ ∆ maximum degree in G Vizing’s Theorem (1965): χ1 ⩽ ∆ + 1 1 ∆ + 1 coloring can always be found in efficiently. running time very recently improved to ˜ O(mn1/3) (arxiv:2405.15449) 2 Computing χ1 exactly (∆ or ∆ + 1!) is NP-hard even for 3-regular graphs! (Holyer 1981) Implications for (Non-hierarchical) Poly Scheduling 1 Decision Poly Scheduling is NP-hard. 2 Optimization Poly Scheduling does not admit poly-time (1 + δ)-approx for δ < 1/3 unless P = NP. ⇝ such an approximation would decide ∆ vs ∆ + 1 for ∆ = 3. Sebastian Wild Polyamorous Scheduling 2024-06-06 6 / 16

G = #colors in proper edge coloring ⇝ Obivous: χ1 ⩾ ∆ maximum degree in G Vizing’s Theorem (1965): χ1 ⩽ ∆ + 1 1 ∆ + 1 coloring can always be found in efficiently. running time very recently improved to ˜ O(mn1/3) (arxiv:2405.15449) 2 Computing χ1 exactly (∆ or ∆ + 1!) is NP-hard even for 3-regular graphs! (Holyer 1981) Implications for (Non-hierarchical) Poly Scheduling 1 Decision Poly Scheduling is NP-hard. 2 Optimization Poly Scheduling does not admit poly-time (1 + δ)-approx for δ < 1/3 unless P = NP. ⇝ such an approximation would decide ∆ vs ∆ + 1 for ∆ = 3. Sebastian Wild Polyamorous Scheduling 2024-06-06 6 / 16

G = #colors in proper edge coloring ⇝ Obivous: χ1 ⩾ ∆ maximum degree in G Vizing’s Theorem (1965): χ1 ⩽ ∆ + 1 1 ∆ + 1 coloring can always be found in efficiently. running time very recently improved to ˜ O(mn1/3) (arxiv:2405.15449) 2 Computing χ1 exactly (∆ or ∆ + 1!) is NP-hard even for 3-regular graphs! (Holyer 1981) Implications for (Non-hierarchical) Poly Scheduling 1 Decision Poly Scheduling is NP-hard. 2 Optimization Poly Scheduling does not admit poly-time (1 + δ)-approx for δ < 1/3 unless P = NP. ⇝ such an approximation would decide ∆ vs ∆ + 1 for ∆ = 3. Sebastian Wild Polyamorous Scheduling 2024-06-06 6 / 16

G = #colors in proper edge coloring ⇝ Obivous: χ1 ⩾ ∆ maximum degree in G Vizing’s Theorem (1965): χ1 ⩽ ∆ + 1 1 ∆ + 1 coloring can always be found in efficiently. running time very recently improved to ˜ O(mn1/3) (arxiv:2405.15449) 2 Computing χ1 exactly (∆ or ∆ + 1!) is NP-hard even for 3-regular graphs! (Holyer 1981) Implications for (Non-hierarchical) Poly Scheduling 1 Decision Poly Scheduling is NP-hard. 2 Optimization Poly Scheduling does not admit poly-time (1 + δ)-approx for δ < 1/3 unless P = NP. ⇝ such an approximation would decide ∆ vs ∆ + 1 for ∆ = 3. Sebastian Wild Polyamorous Scheduling 2024-06-06 6 / 16

of) MAX-3SAT Given formula φ over m clauses and n variables, can ⩾ k clauses be satisfied simultaneously? Key idea: relation scheduled on odd (even) days = True (False) doesn’t quite work, but with slots module 6 it does. Sebastian Wild Polyamorous Scheduling 2024-06-06 7 / 16

∆ + 1 ∆ -approx for unweighted instances any valid schedule must reach heat h ⩾ ∆ on a degree-∆ vertex ⇝ in general: ∆ + 1 ∆ · gmax gmin -approx and (simultaneously) (∆ + 1) could be Ω(n) ... -approx unbounded any schedule reaches heat h ⩾ ∆ · gmin on a degree-∆ vertex any schedule reaches heat h ⩾ gmax Colors Round Robin schedule achieves h ⩽ (∆ + 1)gmax Sebastian Wild Polyamorous Scheduling 2024-06-06 8 / 16

∆ + 1 ∆ -approx for unweighted instances any valid schedule must reach heat h ⩾ ∆ on a degree-∆ vertex ⇝ in general: ∆ + 1 ∆ · gmax gmin -approx and (simultaneously) (∆ + 1) could be Ω(n) ... -approx unbounded any schedule reaches heat h ⩾ ∆ · gmin on a degree-∆ vertex any schedule reaches heat h ⩾ gmax Colors Round Robin schedule achieves h ⩽ (∆ + 1)gmax Sebastian Wild Polyamorous Scheduling 2024-06-06 8 / 16

∆ + 1 ∆ -approx for unweighted instances any valid schedule must reach heat h ⩾ ∆ on a degree-∆ vertex ⇝ in general: ∆ + 1 ∆ · gmax gmin -approx and (simultaneously) (∆ + 1) could be Ω(n) ... -approx unbounded any schedule reaches heat h ⩾ ∆ · gmin on a degree-∆ vertex any schedule reaches heat h ⩾ gmax Colors Round Robin schedule achieves h ⩽ (∆ + 1)gmax Sebastian Wild Polyamorous Scheduling 2024-06-06 8 / 16

∆ + 1 ∆ -approx for unweighted instances any valid schedule must reach heat h ⩾ ∆ on a degree-∆ vertex ⇝ in general: ∆ + 1 ∆ · gmax gmin -approx and (simultaneously) (∆ + 1) could be Ω(n) ... -approx unbounded any schedule reaches heat h ⩾ ∆ · gmin on a degree-∆ vertex any schedule reaches heat h ⩾ gmax Colors Round Robin schedule achieves h ⩽ (∆ + 1)gmax Sebastian Wild Polyamorous Scheduling 2024-06-06 8 / 16

∆ + 1 ∆ -approx for unweighted instances any valid schedule must reach heat h ⩾ ∆ on a degree-∆ vertex ⇝ in general: ∆ + 1 ∆ · gmax gmin -approx and (simultaneously) (∆ + 1) could be Ω(n) ... -approx unbounded any schedule reaches heat h ⩾ ∆ · gmin on a degree-∆ vertex any schedule reaches heat h ⩾ gmax Colors Round Robin schedule achieves h ⩽ (∆ + 1)gmax Sebastian Wild Polyamorous Scheduling 2024-06-06 8 / 16

Layers i = 0, . . . L − 1 contain edges with gmax 2i+1 ⩽ g(e) ⩽ gmax 2i Layer L the rest, i.e., g(e) ⩽ gmax 2L ⇝ within one layer gmax /gmin ⩽ 2 ⇝ Color Round Robin within layer gives good approximation. Also schedule layers Round Robin ⇝ Heat of overall schedule ⩽ h with ∆i = max degree in layer i h = maxi∈[0..L] (L + 1) · (∆i + 1) gmax 2i ⇝ each layer (in isolation) induces lower bound, so h∗ ⩾ h with h = max maxi∈[0..L] ∆i · gmax 2i + 1 , gmax growth rate A B C D E F G H A B C D E F G H A B C D E F G H A B C D E F G H gmax gmax /2 gmax /22 gmax /23 gmax /24 Sebastian Wild Polyamorous Scheduling 2024-06-06 9 / 16

Layers i = 0, . . . L − 1 contain edges with gmax 2i+1 ⩽ g(e) ⩽ gmax 2i Layer L the rest, i.e., g(e) ⩽ gmax 2L ⇝ within one layer gmax /gmin ⩽ 2 ⇝ Color Round Robin within layer gives good approximation. Also schedule layers Round Robin ⇝ Heat of overall schedule ⩽ h with ∆i = max degree in layer i h = maxi∈[0..L] (L + 1) · (∆i + 1) gmax 2i ⇝ each layer (in isolation) induces lower bound, so h∗ ⩾ h with h = max maxi∈[0..L] ∆i · gmax 2i + 1 , gmax growth rate A B C D E F G H A B C D E F G H A B C D E F G H A B C D E F G H gmax gmax /2 gmax /22 gmax /23 gmax /24 Sebastian Wild Polyamorous Scheduling 2024-06-06 9 / 16

Layers i = 0, . . . L − 1 contain edges with gmax 2i+1 ⩽ g(e) ⩽ gmax 2i Layer L the rest, i.e., g(e) ⩽ gmax 2L ⇝ within one layer gmax /gmin ⩽ 2 ⇝ Color Round Robin within layer gives good approximation. Also schedule layers Round Robin ⇝ Heat of overall schedule ⩽ h with ∆i = max degree in layer i h = maxi∈[0..L] (L + 1) · (∆i + 1) gmax 2i ⇝ each layer (in isolation) induces lower bound, so h∗ ⩾ h with h = max maxi∈[0..L] ∆i · gmax 2i + 1 , gmax growth rate A B C D E F G H A B C D E F G H A B C D E F G H A B C D E F G H gmax gmax /2 gmax /22 gmax /23 gmax /24 Sebastian Wild Polyamorous Scheduling 2024-06-06 9 / 16

Layers i = 0, . . . L − 1 contain edges with gmax 2i+1 ⩽ g(e) ⩽ gmax 2i Layer L the rest, i.e., g(e) ⩽ gmax 2L ⇝ within one layer gmax /gmin ⩽ 2 ⇝ Color Round Robin within layer gives good approximation. Also schedule layers Round Robin ⇝ Heat of overall schedule ⩽ h with ∆i = max degree in layer i h = maxi∈[0..L] (L + 1) · (∆i + 1) gmax 2i ⇝ each layer (in isolation) induces lower bound, so h∗ ⩾ h with h = max maxi∈[0..L] ∆i · gmax 2i + 1 , gmax growth rate A B C D E F G H A B C D E F G H A B C D E F G H A B C D E F G H gmax gmax /2 gmax /22 gmax /23 gmax /24 Sebastian Wild Polyamorous Scheduling 2024-06-06 9 / 16

1989): Given k tasks (each with frequency ai ∈ N) Is there a perpetual schedule for a (single) agent who can perform one task per day such that every section of length ai contains at least one instance of task i? Example: A = [3, 4, 5, 8] k=4 Task 1 2 3 4 Frequency 3 4 5 8 S = 1, 2, 4, 1, 3, 2, 1, 3 3 4 8 3 5 4 3 5 Sebastian Wild Polyamorous Scheduling 2024-06-06 11 / 16

1989): Given k tasks (each with frequency ai ∈ N) Is there a perpetual schedule for a (single) agent who can perform one task per day such that every section of length ai contains at least one instance of task i? Example: A = [3, 4, 5, 8] k=4 Task 1 2 3 4 Frequency 3 4 5 8 S = 1, 2, 4, 1, 3, 2, 1, 3 3 4 8 3 5 4 3 5 Sebastian Wild Polyamorous Scheduling 2024-06-06 11 / 16

1989): Given k tasks (each with frequency ai ∈ N) Is there a perpetual schedule for a (single) agent who can perform one task per day such that every section of length ai contains at least one instance of task i? Example: A = [3, 4, 5, 8] k=4 Task 1 2 3 4 Frequency 3 4 5 8 S = 1, 2, 4, 1, 3, 2, 1, 3 3 4 8 3 5 4 3 5 Decision version of Bamboo Garden Trimming (Bilò et al., FUN 2020/21) Sebastian Wild Polyamorous Scheduling 2024-06-06 11 / 16

a fraction yM of each day on matching M ⇝ without loss of generality: all days the same! ⇝ Best fractional solution given by LP: M = set of maximal matchings min ¯ h s. t. M∈M yM ⩽ 1 1 M∈M:e∈M yM · ge ⩽ ¯ h ∀e ∈ R yM ∈ [0, 1] ∀M ∈ M Sebastian Wild Polyamorous Scheduling 2024-06-06 13 / 16

a fraction yM of each day on matching M ⇝ without loss of generality: all days the same! ⇝ Best fractional solution given by LP: M = set of maximal matchings min ¯ h s. t. M∈M yM ⩽ 1 1 M∈M:e∈M yM · ge ⩽ ¯ h ∀e ∈ R yM ∈ [0, 1] ∀M ∈ M Sebastian Wild Polyamorous Scheduling 2024-06-06 13 / 16

a fraction yM of each day on matching M ⇝ without loss of generality: all days the same! ⇝ Best fractional solution given by LP: M = set of maximal matchings min ¯ h s. t. M∈M yM ⩽ 1 1 M∈M:e∈M yM · ge ⩽ ¯ h ∀e ∈ R yM ∈ [0, 1] ∀M ∈ M Sebastian Wild Polyamorous Scheduling 2024-06-06 13 / 16

a fraction yM of each day on matching M ⇝ without loss of generality: all days the same! ⇝ Best fractional solution given by LP: M = set of maximal matchings min ¯ h s. t. M∈M yM ⩽ 1 1 M∈M:e∈M yM · ge ⩽ ¯ h ∀e ∈ R yM ∈ [0, 1] ∀M ∈ M Sebastian Wild Polyamorous Scheduling 2024-06-06 13 / 16

a fraction yM of each day on matching M ⇝ without loss of generality: all days the same! ⇝ Best fractional solution given by LP: after change of variable ¯ h = 1 ℓ and slight massaging M = set of maximal matchings min ¯ h s. t. M∈M yM ⩽ 1 1 M∈M:e∈M yM · ge ⩽ ¯ h ∀e ∈ R yM ∈ [0, 1] ∀M ∈ M 1 ℓ 1 ℓ Sebastian Wild Polyamorous Scheduling 2024-06-06 13 / 16

a fraction yM of each day on matching M ⇝ without loss of generality: all days the same! ⇝ Best fractional solution given by LP: after change of variable ¯ h = 1 ℓ and slight massaging M = set of maximal matchings min ¯ h s. t. M∈M yM ⩽ 1 1 M∈M:e∈M yM · ge ⩽ ¯ h ∀e ∈ R yM ∈ [0, 1] ∀M ∈ M ℓ M∈M:e∈M yM 1 ge ⩾ 1 ℓ Sebastian Wild Polyamorous Scheduling 2024-06-06 13 / 16

a fraction yM of each day on matching M ⇝ without loss of generality: all days the same! ⇝ Best fractional solution given by LP: after change of variable ¯ h = 1 ℓ and slight massaging M = set of maximal matchings min ¯ h s. t. M∈M yM ⩽ 1 1 M∈M:e∈M yM · ge ⩽ ¯ h ∀e ∈ R yM ∈ [0, 1] ∀M ∈ M ℓ M∈M:e∈M yM 1 ge ⩾ ℓ max Sebastian Wild Polyamorous Scheduling 2024-06-06 13 / 16

s. t. M∈M yM ⩽ 1 1 ge M∈M:e∈M yM ⩾ ℓ ∀e ∈ R yM ⩾ 0 ∀M ∈ M LP has exponentially many variables ... but few constraints ⇝ Consider the dual LP: min x s. t. e∈R ze ⩾ 1 e∈M ze ge ⩽ x ∀M ∈ M ze ⩾ 0 ∀e ∈ R ⇝ any feasible solution (x,⃗ z) lower bounds optimal heat h∗! Theorem (Fractional lower bound) For any ze ∈ [0, 1], for e ∈ R, with e∈R ze = 1, we have h∗ ⩾ ¯ h(z) = 1 maxM∈M e∈M ze ge . Sebastian Wild Polyamorous Scheduling 2024-06-06 14 / 16

practical problem but it’s hard NP-hard hard to approximate better than 4/3 best known polytime approximation only O(log ∆) = O(log n) Some questions: 1 Best possible approximation ratio? 2 Better results for restricted versions (e.g., special types of graphs) 3 Density threshold that guarantees schedule? Sebastian Wild Polyamorous Scheduling 2024-06-06 15 / 16

Ware, Pause08, and Madebyoliver from www.flaticon.com. Vector graphics from Pressfoto, brgfx, macrovector and Jannoon028 on freepik.com Other photos from www.pixabay.com. Sebastian Wild Polyamorous Scheduling 2024-06-06 17 / 16