FPTAS for Additive Success Functions

• FPTAS for additive success functions is a fully polynomial-time approximation scheme that efficiently approximates additive objectives by summing independent contributions in optimization problems.
• It employs structured techniques like dynamic programming, discretization, scaling, and rounding to overcome challenges in team contracts, revenue maximization, and subset selection.
• The schemes guarantee near-optimal solutions with formally bounded error and polynomial complexity, effectively addressing fairness and tail probability estimation in complex scenarios.

An FPTAS (Fully Polynomial-Time Approximation Scheme) for additive success functions provides efficient and provably near-optimal approaches for maximizing, approximating, or contracting over probabilistic or combinatorial objectives that sum independent contributions. These schemes are central in settings such as subset selection under Bernoulli or binomial success models, revenue-maximizing contract design with fairness requirements, and estimating tail probabilities of weighted sums of random variables. Recent literature, notably "Fair Team Contracts" (Castiglioni et al., 22 Dec 2025) and "A Fully Polynomial-Time Approximation Scheme for Approximating a Sum of Random Variables" (Li et al., 2013), articulates the underlying principles, structural properties, dynamic programming strategies, and the scaling/rounding paradigms utilized to ensure polynomial complexity and approximation guarantees.

1. Definitions and Problem Settings

The class of additive success functions encompasses objectives where the overall outcome (such as success probability, revenue, or utility) is a sum of weighted independent variables or agents' contributions. Prototypical domains include:

• Sum of independent random variables: Given $n$ independent, nonnegative, integer-valued random variables $X_1,\dots,X_n$, estimate $F(C) = \Pr[\sum_{i=1}^n X_i \leq C]$ (Li et al., 2013).
• Effort allocation in team production: Agents $i \in N$ choose $e_i \in \{0,1\}$; the success function is $F(e) = \sum_{i=1}^n p_i e_i$, with $p_i$ encoding the marginal gain from agent $i$'s effort (Castiglioni et al., 22 Dec 2025).
• Combinatorial subset selection: For binary variables $e_i$ (select/not select), maximize an objective expressible as a sum or product over item attributes, sometimes under all-or-nothing or knapsack-like constraints (Goldberg et al., 2017).

A central computational challenge is to find optimal or approximately optimal solutions—such as maximum probability, revenue, or fair contract—with additive dependence, within a runtime that scales polynomially in both instance size $n$ and $1/\epsilon$ (the accuracy parameter).

2. Structural Properties of Additive Success Functions

Key structural results facilitate the design of FPTAS algorithms for additive frameworks. In the contract design context with fairness:

• Minimum-share structure: There exists a contract which pays each agent either their break-even compensation $c_i/p_i$ or a "minimum share" $L^*$ ensuring fairness and incentive compatibility.
• $L^*$ for a team $S$ is defined as:

$$L^*_S = \max_{i\in S} \frac{c_i}{p_i} \left( 1 - \frac{p_i}{F(S)} \right)$$

Each contract share is then:

$\alpha_i(S) = \max\{L^*_S, c_i/p_i\}$

(Castiglioni et al., 22 Dec 2025).

For sum-probability estimation, recursion over quantiles and convolution structure allows dynamic programming over discretized probability scales rather than full state spaces (Li et al., 2013). In all-or-nothing subset selection, optimality can be limited to at most one low-reliability item ($p_i < 1/2$), dramatically reducing the effective search space (Goldberg et al., 2017).

3. Algorithmic Methodology and Dynamic Programming

FPTAS algorithms for additive success functions rigorously combine discretization, dynamic programming, and structural characterizations:

• Dynamic programming (DP) table: DP entries $T(i,j)$ or $DP[k,v]$ store, e.g., thresholds or minimal total payments for partial subsets or cumulative probabilities (Li et al., 2013, Castiglioni et al., 22 Dec 2025).
• Recurrence relations: For contract design,

$$DP[k+1, v] = \min\left\{ DP[k, v],\ DP[k, v - \tilde{p}_{i_{k+1}}] + \alpha_{i_{k+1}} \right\}$$

for discretized $p$-values $\tilde{p}_i$.

• Discretization: Probabilities or success increments $p_i$ are quantized to integer multiples of $\delta = \Theta(\epsilon^2/n^2)p_{i^*}$, where $i^*$ indexes the largest $p_i$ in the optimal solution (Castiglioni et al., 22 Dec 2025).

In probability estimation, DP is executed over "quantile levels" tracking for each $i$ and quantized probability $Q^{-j}$ the minimum threshold $T(i,j)$ achieving that level.

4. Scaling, Rounding, and Enumeration for Efficiency

FPTAS schemes leverage scaling and rounding for computational tractability:

• Each $p_i$ is replaced by $\tilde{p}_i = \lfloor p_i/\delta \rfloor \delta$ for some small $\delta$, and DP states run over multiples of $\delta$.
• Triple enumeration is used in contract FPTAS: (1) guess $i^*$ (largest $p_i$), (2) guess the contract-share-defining agent $\bar{\imath}$, (3) enumerate the total success probability $x$ over the discretized grid (Castiglioni et al., 22 Dec 2025).
• Such discretization guarantees that the optimal team is not missed by more than an additive $O(\epsilon)$ slack, provable via careful bounding (Castiglioni et al., 22 Dec 2025).

In lower-tail probability FPTAS, probability levels are tracked on a geometric scale $Q = 1+\epsilon/n$; at most $O((n/\epsilon)\log(1/\Delta))$ quantile levels are considered (Li et al., 2013).

5. Approximation Guarantees and Complexity Analysis

Approximation and computational guarantees are explicit:

Scheme	Approximation Ratio	Running Time Complexity
Contract FPTAS (Castiglioni et al., 22 Dec 2025)	Revenue $\geq (1-\epsilon)\cdot$OPT	$O(n^7/\epsilon^4)$ (polynomial in $n, 1/\epsilon$)
Sum-probability FPTAS (Li et al., 2013)	$(1\pm\epsilon)$ relative error	$\text{poly}(n,1/\epsilon,\log C,\log 1/\Delta)$
All-or-nothing subset (Goldberg et al., 2017)	$(1-\epsilon)$ relative error	$O(n^4/\epsilon)$

For all algorithms, careful handling of rounding losses, offset by adjusting enumeration or grid slack, ensures that the total effect is bounded within the prescribed accuracy.

6. Applicability and Comparative Examples

FPTAS techniques for additive success functions apply to a diversity of domains:

• Team contracts and revenue optimization: Designs for efficient, fair revenue-maximizing contracts under additive success are established using the structural minimum-share principle and dynamic programming on discretized support (Castiglioni et al., 22 Dec 2025).
• Lower tail probability of sums: The approach underpins practical estimation of Poisson-binomial, knapsack probability, and generalized subset sum distributions with strong accuracy guarantees (Li et al., 2013).
• All-or-nothing maximal expected value: In this non-linear scenario, FPTAS proceeds by profit-scaling for high-reliability items, enumeration over at most one low-reliability inclusion, and standard DP for the scaled instance, yielding effective selection (Goldberg et al., 2017).

A unifying conclusion is that the structure of the additive function, together with suitable rounding and decomposability, enables polynomial-time $(1-\epsilon)$-approximation algorithms across both estimation and optimization regimes.

7. Key Formulas and Summary

For reference, the principal formulas underlying FPTAS for additive success functions include:

• Additive success: $F(e) = \sum_{i=1}^n p_i e_i$.
• Minimum-share contract: $\alpha_i(S) = \max\{ L^*_S, c_i/p_i \}$, with $L^*_S = \max_{i\in S} (c_i/p_i)(1 - p_i/F(S))$.
• Discretization: $\delta = \Theta(\epsilon^2/n^2)p_{i^*}$, $\tilde{p}_i = \lfloor p_i/\delta \rfloor \delta$.
• DP recurrence: $$DP[k+1,v] = \min\{DP[k,v], DP[k,v-\tilde{p}_{i_{k+1}}] + \alpha_{i_{k+1}}\}$$.
• Quantile DP (sum-probability): $T(i,j)$ as a table of minimal thresholds for probability level $Q^{-j}$.

These algorithms remain fundamental tools in theoretical computer science and algorithmic economics for efficiently handling additive probabilistic models, guaranteeing near-optimal solutions with formally bounded computational effort (Castiglioni et al., 22 Dec 2025, Li et al., 2013, Goldberg et al., 2017).