Optimal Budget Allocation: Theoretical Guarantee and Efficient Algorithm Tasuku Soma Naonori Kakimura Kazuhiro Inaba Ken-ichi Kawarabayashi Univ. of Tokyo Univ. of Tokyo Google NII, JST, ERATO Kawarabayashi LargeGraphProject 1 / 16

Budget Allocation Problem (Alon et al. ’11) A mathematical model for company’s ads. source1 source2 source3 Total: 6 cap:3 cap:5 cap:2 2 / 16

Budget Allocation Problem (Alon et al. ’11) A mathematical model for company’s ads. source1 source2 source3 Total: 6 alloc:2 → cap:3 alloc:3 → cap:5 alloc:1 → cap:2 2 / 16

Budget Allocation Problem (Alon et al. ’11) A mathematical model for company’s ads. source1 alloc:2 → cap:3 1st trial: Pr 0.3 2nd trial: Pr 0.7 3rd trial: Pr 0.2 Influence Probs 2 / 16

Budget Allocation Problem (Alon et al. ’11) A mathematical model for company’s ads. source1 alloc:2 → cap:3 1st trial: Pr 0.3 2nd trial: Pr 0.7 3rd trial: Pr 0.2 Influence Probs fail fail success 2 / 16

Budget Allocation Problem (Alon et al. ’11) A mathematical model for company’s ads. source1 alloc:2 → cap:3 1st trial: Pr 0.3 2nd trial: Pr 0.7 3rd trial: Pr 0.2 Influence Probs 2 / 16

Budget Allocation Problem (Alon et al. ’11) A mathematical model for company’s ads. source1 alloc:2 → cap:3 1st trial: Pr 0.3 2nd trial: Pr 0.7 3rd trial: Pr 0.2 Influence Probs fail success success 2 / 16

Budget Allocation Problem (formal) G = (S, T; E): Bipartite graph For each source s ∈ S: • capacity c(s) • Pr of success of the ith trial p(i) s (i = 1, . . . , c(s)) B ∈ Z+: Total budget f(b) := the expected # of influenced nodes by budget allocation b ∈ ZS, which satisfies: • x ≤ y =⇒ f(x) ≤ f(y) (monotonicity) • f(x) + f(y) ≥ f(x ∨ y) + f(x ∧ y) (∀x, y ∈ ZS) (submodularity) 3 / 16

Budget Allocation Problem (formal) G = (S, T; E): Bipartite graph For each source s ∈ S: • capacity c(s) • Pr of success of the ith trial p(i) s (i = 1, . . . , c(s)) B ∈ Z+: Total budget f(b) := the expected # of influenced nodes by budget allocation b ∈ ZS, which satisfies: • x ≤ y =⇒ f(x) ≤ f(y) (monotonicity) • f(x) + f(y) ≥ f(x ∨ y) + f(x ∧ y) (∀x, y ∈ ZS) (submodularity) Maximize f(b) subject to 0 ≤ b(s) ≤ c(s) (s ∈ S) s∈S b(s) ≤ B 3 / 16

Previous Work • NP-hard • If P NP, (1 − 1/e)-approximation is best possible • Previous algorithm (Alon et al. ’11) is polytime, but impractical due to the heavy time complexity • What about more complicated real scenarios? 4 / 16

Our Results 1 Submodular Function Maximization over Integer Lattice: • A gerenal framework including more complicated scenarios • (1 − 1/e)-approximation algorithm 2 Faster Algorithm for Nonincreasing Influence Probabilities: • Speeding up Alon et al.’s algorithm under natural assumption • Almost linear time for graph size • Numerical experiments for real & big data 5 / 16

1 Submodular Function Maximization over Integer Lattice 2 Faster Algorithm for Nonincreasing Influence Probabilities 6 / 16

Submodular Func. Maximization over ZS + f : ZS + → R ... monotone submodular function over integer lattice • x ≤ y =⇒ f(x) ≤ f(y) • f(x) + f(y) ≥ f(x ∨ y) + f(x ∧ y) (∀x, y ∈ ZS) (x ∨ y : elem-wise max, x ∧ y : elem-wise min) 7 / 16

Submodular Func. Maximization over ZS + f : ZS + → R ... monotone submodular function over integer lattice • x ≤ y =⇒ f(x) ≤ f(y) • f(x) + f(y) ≥ f(x ∨ y) + f(x ∧ y) (∀x, y ∈ ZS) (x ∨ y : elem-wise max, x ∧ y : elem-wise min) cf. submodularity for set functions: f(X) + f(Y) ≥ f(X ∪ Y) + f(X ∩ Y) (∀X, Y ⊆ S) 7 / 16

Submodular Func. Maximization over ZS + f : ZS + → R ... monotone submodular function over integer lattice • x ≤ y =⇒ f(x) ≤ f(y) • f(x) + f(y) ≥ f(x ∨ y) + f(x ∧ y) (∀x, y ∈ ZS) (x ∨ y : elem-wise max, x ∧ y : elem-wise min) cf. submodularity for set functions: f(X) + f(Y) ≥ f(X ∪ Y) + f(X ∩ Y) (∀X, Y ⊆ S) c ∈ ZS + , w ∈ ZS + , B ∈ Z≥0 Maximize f(b) subject to 0 ≤ b ≤ c s∈S w(s)b(s) ≤ B (knapsack constraint) b ∈ ZS 7 / 16

Submodular Func. Maximization over ZS + General Framework Including: • Optimal budget allocation with various unit costs • Optimal budget allocation with competitor • Maximum coverage, Sensor placement, Text summarization, ... 8 / 16

Submodular Func. Maximization over ZS + General Framework Including: • Optimal budget allocation with various unit costs • Optimal budget allocation with competitor • Maximum coverage, Sensor placement, Text summarization, ... Pseudo Polytime (1 − 1/e)-Approximation Algorithm: Theorem A (1 − 1/e)-approximate solution can be found in O(B5|S|4θ) time (θ: the running time of oracle for f) for a monotone submodular function maximization over the integer lattice subject to knapsack constraint. • Extends the algorithm for optimal budget allocation (Alon et al. ’11) 8 / 16

1 Submodular Function Maximization over Integer Lattice 2 Faster Algorithm for Nonincreasing Influence Probabilities 9 / 16

Nonincreasing Influence Probabilities Under a natural assumption, Alon et al.’s algorithm can be accelerated. Assumption: Influence probabilities of each ad source are nonincreasing: p(1) s ≥ p(2) s ≥ · · · ≥ p(c(s)) s (∀s ∈ S) (i.e., Effectiveness of each ad is nonincreasing with time) 10 / 16

Nonincreasing Influence Probabilities Under a natural assumption, Alon et al.’s algorithm can be accelerated. Assumption: Influence probabilities of each ad source are nonincreasing: p(1) s ≥ p(2) s ≥ · · · ≥ p(c(s)) s (∀s ∈ S) (i.e., Effectiveness of each ad is nonincreasing with time) Alon et al.’s algorithm: O(B6|S|5|T|) time Our algorithm: O(B(|S| + |T| + |E|)) time (almost linear for graph size) 10 / 16

Algorithm Under the assumption, f satisfies the following diminishing marginal return property: f(b + 2es ) − f(b + es ) ≤ f(b + es ) − f(b) (b ∈ ZS, s ∈ S) Note: This property is NOT implied by submodularity! 11 / 16

Algorithm Under the assumption, f satisfies the following diminishing marginal return property: f(b + 2es ) − f(b + es ) ≤ f(b + es ) − f(b) (b ∈ ZS, s ∈ S) Note: This property is NOT implied by submodularity! Algorithm 1: b := 0 2: while s∈S b(s) < B do 3: Among s with b(s) < c(s), choose one of f(b +es )−f(b) maximum. 4: Set b := b + es. 5: end while 6: return b 11 / 16

Algorithm One more trick: For optimal budget allocation, f(b + es ) − f(b) can be computed in O(1) time. Theorem A (1 − 1/e)-approximate solution can be found in O(B(|V| + |E|)) time if the influence probabilities of each ad source are nonincreasing. • If B = O(1), runs in linear time for graph size. 12 / 16

Experiments Our algorithm for nonincreasing probabilities vs Heuristics. Graphs: • Real Data (Yahoo! Webscope Dataset) ≈ 10,000 nodes & 50,000 edges • Random Graph ≈ 2M nodes & 8M edges Heuristics: • Degree-prob ... allocate to nodes with higher degree and prob. • Degree ... allocate to nodes with higher degree • Random ... allocate at random (baseline) Machine: Xeon E5-2690 2.9GHz CPU, 64GB RAM 13 / 16

Real Data (10k nodes & 50k edges) 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 0 100 200 300 400 500 600 700 800 900 1000 Influenced nodes: f (b) Budget: B Greedy (Ours) Degree-prob Degree Random • Maximum 15% outperforming 14 / 16

Random Graph (2M nodes & 8M edges) 0 200000 400000 600000 800000 1000000 1200000 1400000 0 100 200 300 400 500 600 700 800 900 1000 Influenced nodes: f (b) Budget: B Greedy (Ours) Degree-prob Degree Random • Our algorithm finds an approx solution in a few seconds 15 / 16

Our Results 1 Submodular Function Maximization over Integer Lattice: • A gerenal framework including more complicated scenarios • (1 − 1/e)-approximation algorithm 2 Faster Algorithm for Nonincreasing Influence Probabilities: • Speeding up Alon et al.’s algorithm under natural assumption • Almost linear time for graph size • Numerical experiments for real & big data 16 / 16