Poly-Time Algorithm for Additive Valuations

• Polynomial-Time Algorithms for Additive Valuations are frameworks that leverage the additive property of agents' valuations to compute fair and efficient resource allocations.
• Key approaches like Controlled Cake Eating, Market Equilibrium, and Constrained Serial Dictatorship use network flow and convex optimization techniques to achieve various fairness guarantees.
• These methods extend to handle variable claims and initial endowments, setting theoretical benchmarks and practical solutions in resource allocation and fair division.

A polynomial-time algorithm for additive valuations is an algorithmic framework that efficiently computes fair and efficient allocations in settings where each agent's value for a set of items (or divisible “cake”) is given by a sum of individual values for those items or subintervals. This paradigm is central to fair division, mechanism design, and resource allocation, with notable applications in both discrete (indivisible items) and continuous (cake cutting) domains. Key methods exploit the additivity and structure of valuations to guarantee strong fairness and/or efficiency properties while remaining computationally tractable.

1. Definitions and Setting

In the additive valuations model, each agent $i$ has a valuation function $v_i(S) = \sum_{x \in S} v_i(x)$ for any set $S$ of items (or subintervals, in cake cutting). This model encompasses both piecewise constant valuations over a continuum (as in cake cutting) and additive valuations over indivisible or divisible items. The objective is to design algorithms that compute allocations $(A_1, \dots, A_n)$ with desirable properties such as robust envy-freeness, proportionality, Pareto optimality, and—where possible—strategyproofness.

A fundamental subdomain is that of piecewise constant (and piecewise uniform) valuations, where the continuous cake is partitioned into subintervals over which each agent’s value density is constant (or either a fixed positive value or zero, respectively). Additivity of valuations facilitates efficient computation since the value of a union is the sum of the parts.

2. Key Algorithmic Frameworks

2.1 Controlled Cake Eating Algorithm (CCEA)

CCEA, inspired by random assignment and network flow algorithms, operates as follows:

1. Partition the cake into intervals at every point where some agent’s value density function changes.
2. Model each interval as a “house” with fractional size.
3. Each agent's preferences over intervals are determined by the heights of their value densities.
4. Use a parametric network flow approach (generalizing the Probabilistic Serial algorithm) to allocate intervals among agents, so that at each infinitesimal step, agents “consume” their favorite available intervals in continuous time, respecting their claims.
5. Translate fractional assignments into contiguous cake pieces for each agent.

This approach efficiently computes allocations that are robustly envy-free, robustly proportional, and non-wasteful for piecewise constant valuations, with a running time of $O(n^5 m^2 \log(n^2/m))$, where $n$ is the number of agents and $m$ is the number of intervals.

2.2 Market Equilibrium Algorithm (MEA)

MEA casts fair division as a Fisher market equilibrium problem, solved by convex optimization:

1. Partition the cake as in CCEA.
2. Formulate a convex program maximizing Nash social welfare, subject to the constraints that all cake is allocated and no agent exceeds their entitled claim:

$$\begin{align} &\text{minimize} \quad -\sum_{j=1}^{n} \log(u_j) \ &\text{subject to} \quad u_j = \sum_{i=1}^{k} v_{ij}x_{ij} \ &\sum_{j=1}^{n} x_{ij} \leq \ell_i \ &x_{ij} \geq 0 \end{align}$$

where $v_{ij}$ is agent $j$’s density in interval $J_i$, $x_{ij}$ the assigned length, and $\ell_i$ the length of $J_i$.

1. Solve the program using polynomial-time convex optimization methods.
2. Allocate cake pieces accordingly.

MEA yields envy-free, proportional, and Pareto optimal allocations for piecewise constant valuations, and runs in polynomial time.

2.3 Constrained Serial Dictatorship (CSD)

CSD extends random serial dictatorship to the case of divisible resources:

1. Enumerate agent orderings (permutations).
2. For each ordering, each agent in turn picks their most preferred unallocated piece of length $1/n$ (or according to their claim), until the cake is allocated.
3. For each subinterval, assign to each agent a piece proportional to the number of orderings in which they obtained that subinterval.
4. The final allocation is a randomized or fractional mixture of such assignments.

While CSD is exponential in $n$ in worst-case time, for constant or two agents, it is polynomial-time and achieves robust proportionality, unanimity, and—when randomization is used—strategyproofness in expectation.

3. Algorithmic Properties and Theoretical Guarantees

A comparative overview is summarized below:

Algorithm	Domain	Robust Envy-Free	Envy-Free	Robust Proportional	Proportional	Group SP	SP	Pareto Opt.	Poly Time
CCEA	Piecewise constant	+	+	+	+	–	–	–	+
CCEA	Piecewise uniform	+	+	+	+	+	+	+	+
MEA	Piecewise constant	–	+	–	+	–	–	+	+
MEA	Piecewise uniform	+	+	+	+	+	+	+	+
CSD	Piecewise constant	–	–	+	+	–	+e*	–	–
CSD	Two agents	+	+	+	+	–	+	–	+

e: strategyproof in expectation.

Impossibility results show that no algorithm can simultaneously achieve all of Pareto optimality, robust proportionality, and strategyproofness, even for piecewise constant additive valuations; thus, the algorithms above are essentially optimal for their domains.

4. Extensions: Variable Claims and Endowments

The frameworks above generalize to richer models:

• Variable Claims: CCEA and MEA can handle assignments where agents have different entitlements by adjusting “eating rates” (CCEA) or budgets (MEA) accordingly, without losing their key fairness and group-strategyproofness properties (for piecewise uniform valuations).
• Initial Endowments: If agents hold initial pieces (“endowments”), these are incorporated into the assignment and optimization frameworks naturally, with properties preserved.

Such extensions enable the application of these algorithms to broader real-world problems with heterogeneous agent entitlements and pre-assigned resources.

5. Computational Considerations and Limitations

The algorithms exploit the additivity and piecewise structure of valuations for efficiency:

• CCEA and MEA: Polynomial time for any number of agents and intervals, with explicit step-by-step procedures or convex optimization back ends.
• CSD: Only polynomial-time for two or constant agents; otherwise, the enumeration of all permutations makes the algorithm exponential.
• Strategyproofness: CCEA and MEA are strategyproof only in the piecewise uniform domain; outside this, misreports can yield higher utility for manipulative agents.
• Pareto Optimality: Achievable for MEA (and CCEA in piecewise uniform case), but not guaranteed for randomized CSD allocations except for two agents.

6. Historical and Scientific Context

The CCEA generalizes random assignment algorithms such as Probabilistic Serial [Bogomolnaia & Moulin] and Controlled Consuming [Athanassoglou & Sethuraman] to the cake cutting (divisible) domain. MEA builds on the Fisher market model for fair division ([Reijnierse & Potters], [Devanur et al.]), applying convex programming techniques to ensure both fairness and efficiency. Mechanism 1 of Chen et al. is equivalent to CCEA and MEA on piecewise uniform valuations. These advances have set a new benchmark for what can be efficiently achieved in additive fair division, sharply delineating the boundaries between computational feasibility and impossibility.

7. Practical Applications and Impact

These polynomial-time algorithms for additive valuations enable robust, scalable fair division in applications such as:

• Scheduling resources or processing time in computational systems.
• Land or time-sharing in legal or estate division.
• Automated division of divisible assets with heterogeneous stakeholders, variable entitlements, and fair division constraints.

Their guarantees regarding fairness, efficiency, and (where possible) strategyproofness directly address the core requirements in resource allocation, law, and collaborative decision-making.

---

Polynomial-time, maximal-property algorithms for additive valuations fundamentally transform the tractability landscape for fair division. In both divisible and indivisible settings (with suitable extensions), these methods provide practically viable tools and clear theoretical frontiers for fair and efficient allocation.