On the Configuration LP for Maximum Budgeted Allocation

We study the Maximum Budgeted Allocation problem, i.e., the problem of selling a set of $m$ indivisible goods to $n$ players, each with a separate budget, such that we maximize the collected revenue. Since the natural assignment LP is known to have an integrality gap of $\frac{3}{4}$, which matches the best known approximation algorithms, our main focus is to improve our understanding of the stronger configuration LP relaxation. In this direction, we prove that the integrality gap of the configuration LP is strictly better than $\frac{3}{4}$, and provide corresponding polynomial time roundings, in the following restrictions of the problem: (i) 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 (ii) the graph MBA problem, in which an item can be assigned to at most 2 players. Finally, we improve the best known upper bound on the integrality gap for the general case from $\frac{5}{6}$ to $2\sqrt{2}-2\approx 0.828$ and also prove hardness of approximation results for both cases.

💡 Research Summary

The paper investigates the Maximum Budgeted Allocation (MBA) problem, where a set of indivisible items must be assigned to players each equipped with a personal budget. The objective is to maximize the total payment, which for each player is the minimum of his budget and the sum of the values of the items he receives. This problem is NP‑hard, and the classic linear programming relaxation known as the assignment LP has an integrality gap of exactly 3/4. Consequently, the best known approximation algorithms for MBA achieve a 3/4‑approximation, and it is known that improving this ratio for the assignment LP is impossible.

To obtain stronger guarantees, the authors turn to the configuration LP, a more powerful relaxation that explicitly models the fact that each player receives a configuration—a set of items—rather than fractional pieces of many items. The configuration LP introduces a variable (y_{iC}) for every player (i) and every possible subset (C) of items, indicating whether (C) is the set assigned to (i). Constraints enforce that each player receives at most one configuration and each item appears in at most one configuration. Although the LP has exponentially many variables, it can be solved approximately via a polynomial‑time separation oracle for its dual, and any solution can be expressed as a convex combination of a polynomial number of integral configurations using Carathéodory’s theorem.

The paper focuses on two natural restricted settings that capture structural properties of the worst‑case instance for the assignment LP:

1. Restricted Budgeted Allocation (Restricted MBA) – all players share the same budget, and each item has a uniform value (p_j) for the subset of players that can receive it (price is either (p_j) or 0).
2. Graph MBA – each item can be assigned to at most two players; the instance can be represented as a graph where items are edges and players are vertices.

For both settings the authors prove that the configuration LP has an integrality gap strictly larger than 3/4, i.e., there exists a constant (c>0) such that a ((3/4 + c))-approximation algorithm can be obtained directly from the LP solution. This is the first evidence that the configuration LP is genuinely stronger than the assignment LP for MBA.

Graph MBA Rounding

The algorithm for Graph MBA proceeds in two stages. First, items are assigned randomly according to the fractional configuration LP solution, which already guarantees an expected revenue of at least 3/4 of the LP value. Because each item touches at most two players, the random assignment can be interpreted as a random orientation of the edges. In the second stage, any items left unassigned after the first random step are greedily given to players whose budgets have not yet been exceeded. The key technical contribution (Lemma 11) is a careful analysis of how much extra value the second stage contributes, showing that the total expected revenue exceeds the 3/4 barrier by a constant amount.

Restricted MBA Rounding

A naïve random assignment of items based on the configuration LP only recovers a ((1-1/e)) fraction of the LP value, which is insufficient. The authors therefore distinguish between well‑structured LP solutions and the rest. A solution is well‑structured if, for most players, at least half of their budget is already allocated to “expensive” items—items whose price is close to the common budget. For non‑well‑structured solutions, the authors apply a bucket‑based rounding (Algorithm 1) that yields the ((1-1/e)) guarantee.

For well‑structured solutions a novel two‑phase rounding is introduced:

1. Big‑item phase – Expensive items are assigned using a random bipartite matching with negative correlation between the events that two distinct players receive a big item. This is achieved by invoking the negative‑correlation matching theorem of Gandhi, Khuller, Pattabiraman, and Srinivasan (Theorem 2). The negative correlation ensures that the probability that two players both consume a large portion of their budgets is at most the product of the individual probabilities, dramatically reducing the chance of budget overflow.
2. Small‑item phase – After the big items are placed, the remaining budget slack is filled with cheap items using the bucket algorithm, which respects the marginal probabilities of the LP.

The combination of these phases yields an expected revenue of ((3/4 + c)) times the LP optimum for some constant (c>0). The analysis hinges on the fact that the big‑item matching preserves each player’s marginal probability of receiving a big item while simultaneously making the events negatively correlated, a property not shared by ordinary random matchings.

General MBA Integrality Gap

Beyond the two restricted settings, the authors improve the known upper bound on the integrality gap of the configuration LP for the general MBA problem. Previously the best bound was (5/6). By constructing a refined family of instances and analyzing the dual of the configuration LP, they prove that the gap cannot exceed (2\sqrt{2} - 2 \approx 0.828). This result narrows the gap between the known lower bound (3/4) and the upper bound, indicating that the configuration LP is substantially tighter than the assignment LP even in the unrestricted case.

Hardness Results

The paper also establishes matching hardness of approximation for the two restricted problems. For Restricted MBA it is shown that achieving an approximation ratio better than (15/16) is NP‑hard, which aligns with the strongest known hardness for the general MBA problem. This demonstrates that, despite the stronger LP relaxation, the approximation barrier imposed by computational complexity remains high.

Contributions Summary

• Configuration LP Strength: Proved that the configuration LP has an integrality gap strictly better than 3/4 for two natural MBA subclasses.
• New Rounding Techniques: Developed a negative‑correlation bipartite matching method for big items and a bucket‑based rounding for small items, achieving a ((3/4 + c))-approximation.
• Improved General Gap: Tightened the upper bound on the configuration LP integrality gap from (5/6) to (2\sqrt{2} - 2 \approx 0.828).
• Hardness Matching: Showed NP‑hardness of approximating Restricted MBA within (15/16), confirming that the LP‑based improvements cannot be pushed arbitrarily far.

Overall, the work deepens the theoretical understanding of linear‑programming relaxations for budgeted allocation problems, introduces novel probabilistic rounding tools that may be applicable to a broader class of allocation and scheduling problems, and clarifies both the potential and the limits of configuration‑LP‑based algorithms for MBA.