Maximum Budgeted Allocation Problem

• MBA is the problem of assigning indivisible goods to players with limited budgets to maximize capped additive revenue.
• LP relaxations, including the Assignment-LP and Configuration-LP, offer different trade-offs between approximation guarantees and computational efficiency.
• Online and restricted MBA variants in digital advertising illustrate the balance between immediate allocation decisions and future budget utility.

Searching arXiv for the cited MBA and related papers to ground the article with current identifiers and titles. Maximum Budgeted Allocation (MBA) is the problem of allocating a set of indivisible goods to players with separate budgets so as to maximize collected revenue. In the standard formulation, player $i$ has budget $B_i$ and values $v_{ij}$ for items $j$; if $S_i$ is the subset allocated to $i$, the contribution of that player is $\min\!\left\{B_i,\sum_{j\in S_i} v_{ij}\right\}$. The objective is therefore budget-capped additive welfare rather than ordinary additive welfare, which places MBA at the intersection of assignment, knapsack, generalized assignment, combinatorial allocation, and advertising (Kalaitzis et al., 2014, Viswanathan, 26 May 2026).

1. Canonical formulation

MBA is defined on a set of players $I$ and a set of indivisible items $J$. Each player $i\in I$ has a budget $B_i$0, and each item $B_i$1 has player-specific value $B_i$2. An allocation assigns each item to at most one player. If player $B_i$3 receives bundle $B_i$4, then the realized revenue from that player is capped by the budget: $B_i$5 The optimization problem is

$B_i$6

where $B_i$7 is a partition, or partial partition, of the items (Kalaitzis et al., 2014).

A closely related notation writes the same objective as budgeted social welfare. With additive valuations $B_i$8 over indivisible goods $B_i$9, and agent budgets $v_{ij}$0, the objective is

$v_{ij}$1

where $v_{ij}$2 is an allocation (Viswanathan, 26 May 2026). The two formulations are equivalent at the level of the capped additive objective.

The budget cap is the defining structural feature. MBA is therefore not the ordinary assignment problem: allocating additional high-value items to a player can become worthless after the player reaches budget. This budget truncation is exactly what makes LP design, rounding, and online decision-making substantially more delicate than in uncapped additive allocation (Kalaitzis et al., 2014).

2. Assignment-LP and Configuration-LP

A central theme in the MBA literature is the contrast between the Assignment-LP and the Configuration-LP. The Assignment-LP uses variables $v_{ij}$3, interpreted as fractional assignment of item $v_{ij}$4 to player $v_{ij}$5: $v_{ij}$6 This LP is compact and polynomial-size, but it “sees” each item separately and enforces budgets only through linear constraints (Kalaitzis et al., 2014).

The Configuration-LP is stronger. For each player $v_{ij}$7, let $v_{ij}$8 denote feasible configurations $v_{ij}$9 satisfying $j$0, and introduce a variable $j$1 for each configuration. The Configuration-LP is

$j$2

Here the LP reasons directly about bundles rather than isolated items (Kalaitzis et al., 2014).

The Configuration-LP is a tighter relaxation. From a feasible configuration solution, one can define

$j$3

so every configuration solution induces a feasible assignment solution with the same objective value. Consequently,

$j$4

and the Configuration-LP is strictly stronger in the sense relevant to maximization (Kalaitzis, 2015).

This strength comes at a computational price: the Configuration-LP is exponentially large. The standard remedy is column generation, using an ellipsoid-style framework or a restricted master problem together with a pricing subproblem. The pricing problem for a fixed player is essentially a knapsack-type problem, which is why configuration-based methods remain algorithmically viable despite the exponential family of configurations (Kalaitzis, 2015).

3. Approximation guarantees and integrality-gap landscape

Before configuration-based improvements, the best known approximation algorithms for the general MBA problem achieved a $j$5-approximation ratio, and the Assignment-LP was known to have integrality gap exactly $j$6. This established a clear barrier for any black-box rounding of the natural assignment relaxation (Kalaitzis, 2015).

A major step beyond that barrier was the analysis of the Configuration-LP. For the Restricted Budgeted Allocation problem, in which all the players have the same budget and every item has the same value for any player it can be sold to, and for the graph MBA problem, in which an item can be assigned to at most $j$7 players, the integrality gap of the Configuration-LP was shown to be strictly better than $j$8, with corresponding polynomial-time rounding algorithms. For the general case, the best known upper bound on the integrality gap was improved from $j$9 to $S_i$0, together with hardness of approximation results for both restricted cases (Kalaitzis et al., 2014).

A subsequent result gave an algorithm for general MBA that achieves a $S_i$1-approximation ratio for some constant $S_i$2. The algorithm rounds the Configuration-LP, and thereby shows that the Configuration-LP is strictly stronger than the Assignment-LP for MBA in an algorithmic as well as polyhedral sense (Kalaitzis, 2015).

These results establish the current structural picture conveyed by the data. The Assignment-LP captures the basic budgeted assignment geometry but saturates at the $S_i$3 barrier; the Configuration-LP provably improves on that barrier, yet it does not collapse the gap to $S_i$4. A plausible implication is that MBA derives much of its difficulty from correlations among items assigned to the same player, and that bundle-level relaxations are necessary but not obviously sufficient for tight approximations.

4. Online allocation, AdWords, and display advertising

MBA has a prominent online incarnation in display advertising and AdWords-like allocation. In the online budgeted allocation model, items arrive sequentially, bids or values are revealed on arrival, and the decision to allocate must be made immediately and irrevocably. The revenue objective remains budget-capped: $S_i$5 but now the variables $S_i$6 are chosen online (Aggarwal et al., 2010).

One particularly tractable regime is the single-bids case, where each agent $S_i$7 uses the same nonzero bid $S_i$8 on every desired item, so $S_i$9. In that setting, online budgeted allocation reduces to online vertex-weighted bipartite matching. The reduction splits each budget $i$0 into copies of weight $i$1, plus a remainder vertex if needed, and translates each arriving item into an online vertex adjacent to the copies of agents that bid on it. This yields an optimal $i$2-competitive randomized algorithm for the single-bid case, even when the bid is comparable to the budget (Aggarwal et al., 2010).

Display-ad allocation leads to a broader online viewpoint. In the model studied in “Online Allocation Rules in Display Advertising,” impressions arrive one by one; on arrival, cost and revenue vectors are revealed, and the impression should be assigned to an advertiser almost immediately. Without any assumption on the distribution or arrival of impressions, the paper proposes a framework to capture the risk to the ad network for each possible allocation, and impressions are allocated so that the risk of the ad network is minimized. In practice, this translates to starting with an initial estimate of dual prices and updating them according to the belief of the ad network toward the future demand and remaining budgets. On a real data set, the Kullback-Leibler divergence risk measure had the best performance in terms of revenue and balanced budget delivery (Shamsi et al., 2014).

This suggests a natural conceptual split inside online MBA. One line of work emphasizes competitive analysis and matching-style reductions; another emphasizes risk-aware pacing, dual pricing, and empirical budget delivery. Both are organized around the same tension: immediate value must be traded against the future value of remaining budget.

5. Structured special cases and neighboring formulations

Several restricted forms of MBA are sufficiently structured to permit sharper analysis. Restricted Budgeted Allocation fixes a common budget across players and requires that every item have the same value for any player it can be sold to. Graph MBA restricts each item to at most two eligible players. Both restrictions materially simplify the interaction between item competition and budget caps, and both were used to demonstrate that the Configuration-LP is strictly better than the $i$3 Assignment-LP barrier (Kalaitzis et al., 2014).

MBA also borders closely on coverage-type budgeted selection problems. The Budgeted Maximum Coverage Problem (BMCP) has a single global budget $i$4, costs on selected sets, and a coverage objective

$i$5

where $i$6 records how many selected sets cover each element. BMCP can be viewed as a special-case MBA with one agent, bundle choices with fixed prices, and valuation function $i$7, which is a monotone submodular coverage function rather than an additive item-by-item valuation (Zhou et al., 2022).

That comparison is useful because it isolates what is specific to MBA. BMCP is a single-budget, single-buyer analogue with submodular coverage valuation; classical MBA has multiple budgets and additive values capped per player. This suggests that MBA sits between generalized assignment and submodular budgeted selection: it preserves additive valuations, but the budget cap induces a nonlinear objective at the player level.

The same neighboring formulations also clarify what MBA is not. It is not simply knapsack, because there are multiple players and exclusivity constraints across items. It is not simply matching, because a player can absorb multiple items until reaching budget. It is not maximum coverage, because the valuation is additive before truncation rather than coverage-based. Much of the literature’s technical apparatus—configuration relaxations, correlated rounding, negative correlation arguments, and online dual pricing—exists precisely to manage that hybrid structure (Kalaitzis et al., 2014, Zhou et al., 2022).

6. Hardness, limitations, and open directions

MBA is APX-hard, and its hardness of approximation has been sharpened over time. A later hardness survey reports that Chakrabarty and Goel improved the hardness to a factor of $i$8, and that, assuming the Unique Games Conjecture, it is NP-hard to approximate the maximum budgeted allocation by a factor better than $i$9. The same survey places this in the standard additive-valuations MBA setting with arbitrary budgets (Viswanathan, 26 May 2026).

The 2026 hardness result uses a dictator test on indivisible-goods instances and derives the MBA gap by choosing budgets

$\min\!\left\{B_i,\sum_{j\in S_i} v_{ij}\right\}$0

so that in the YES case every agent reaches budget, while in the NO case the total utility available to agents without large goods is bounded by $\min\!\left\{B_i,\sum_{j\in S_i} v_{ij}\right\}$1. This yields the $\min\!\left\{B_i,\sum_{j\in S_i} v_{ij}\right\}$2 hardness factor under UGC (Viswanathan, 26 May 2026).

These hardness results are not tight relative to the algorithmic and LP landscape summarized in the data. The same survey notes a best known approximation ratio of $\min\!\left\{B_i,\sum_{j\in S_i} v_{ij}\right\}$3 for some small constant $\min\!\left\{B_i,\sum_{j\in S_i} v_{ij}\right\}$4, and cites $\min\!\left\{B_i,\sum_{j\in S_i} v_{ij}\right\}$5 as the best known integrality gap of the Configuration-LP (Viswanathan, 26 May 2026). A plausible implication is that MBA still has a substantial unresolved interval between known positive results and known impossibility results.

Open questions stated in the MBA literature include the tight integrality gap of the Configuration-LP, whether stronger relaxations such as lift-and-project or SDP hierarchies improve the picture, whether the restricted-case bounds are tight, and how far configuration-based and negative-correlation techniques extend to online and stochastic variants (Kalaitzis et al., 2014). Those questions remain central because MBA continues to serve as a canonical model for budget-constrained allocation: it is simple enough to admit clean LP formulations, yet difficult enough that even its strongest standard relaxations do not settle the approximation frontier.