On the Configuration LP for Maximum Budgeted Allocation

Kalaitzis, Christos; Mądry, Aleksander; Newman, Alantha; Poláček, Lukáš; Svensson, Ola

doi:10.1007/978-3-319-07557-0_28

Cited by 4 publications

(15 citation statements)

References 26 publications

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“…• As mentioned earlier, c = 3/4 for MBA, which is also the integrality gap of the assignment LP formulation of the problem. Thus, our results essentially (i.e., barring the 3/4 + δ approximation via the configuration LP in [17]) generalize the state of the art of MBA to instances with general monotone concave functions. Note that for all functions, the total curvature c defined for the SMBC problem will tend towards 1 as the total sum of bid values becomes sufficiently large, and therefore its the approximation ratio guaranteed will tend towards 1 − 1/e for all functions and bid values shown above.…”

Section: Contributionsmentioning

confidence: 51%

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Approximations for Allocating Indivisible Items with Concave-Additive Valuations

Kell¹,

Sun²

2021

Preprint

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We study a general allocation setting where agent valuations are specified by an arbitrary set of monotone concave functions. In this model, a collection of items must be uniquely assigned to a set of agents, where all agent-item pairs have a specified bid value. The objective is to maximize the sum of agent valuations, each of which is a nondecreasing concave function of the agent's total spend. This setting was studied by Devanur and Jain (STOC 2012) in the online setting for fractional assignments. In this paper, we generalize the state of the art in the offline setting for indivisible allocations. In particular, for a collection of monotone concave functions and maximum bid value b, we define the notion of a local curvature bound c ∈ (0, 1], which intuitively measures the largest multiplicative gap between any valuation function and a local linear lower bound with x-width b. For our main contribution, we define an efficient primal-dual algorithm that achieves an approximation of (1 − )c , and provide a matching c integrality gap, showing our algorithm is optimal among those that utilize the natural assignment CP.Additionally, we show our techniques have an interesting extension to the Smooth Nash Social Welfare problem (Fain et al. EC 2018, Fluschnik et al. AAAI 2019, which is a variant of NSW that reduces the extent to which the objective penalizes under-allocations by adding a smoothing constant to agent utilities. This objective can be viewed as a sum of logs optimization, which therefore falls in the above setting. However, we also show how our techniques can be extended to obtain an approximation for the canonical product objective by obtaining an additive guarantee for the log version of the problem.

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Section: Contributionsmentioning

confidence: 51%

“…These 3/4-approximation algorithms consider the standard LP relaxation known as the assignment LP. Most recently, Kalaitzis et al [17] analyzed a stronger relaxation LP known as the configuration LP. For two special cases which they call graph MBA and restricted MBA, they give (3/4 + δ)-approximation algorithms for some constant δ > 0.…”

Section: Related Workmentioning

confidence: 99%

Approximations for Allocating Indivisible Items with Concave-Additive Valuations

Kell¹,

Sun²

2021

Preprint

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“…In a setting with high-capacity vehicles, however, this ceases to be true, and it is unclear if a stochastic model of our system would exhibit the rapid mixing property with which low-capacity ride-sharing models are endowed, and which allows for these attractive guarantees. Randomized rounding for resource allocation problems: Our methodological approach is inspired by the use of configuration programs for improved approximations for a number of combinatorial optimization problems [21,26,41]. At a high level, the approximation algorithms proposed in this line of work reformulate the resource allocation problem as an exponential-size integer program that optimizes over all feasible sets of resources; the LP relaxation of this program can be (approximately) solved in polynomial time, and used to produce approximately optimal solutions to the original problem via rounding.…”

Section: Related Workmentioning

confidence: 99%

Real-Time Approximate Routing for Smart Transit Systems

Banerjee¹,

Hssaine²,

Périvier³

et al. 2021

Preprint

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We study real-time routing policies in smart transit systems, where the platform has a combination of cars and high-capacity vehicles (e.g., buses or shuttles) and seeks to serve a set of incoming trip requests. The platform can use its fleet of cars as a feeder to connect passengers to its high-capacity fleet, which operates on fixed routes. Our goal is to find the optimal set of (bus) routes and corresponding frequencies to maximize the social welfare of the system in a given time window. This generalizes the Line Planning Problem, a widely studied topic in the transportation literature, for which existing solutions are either heuristic (with no performance guarantees), or require extensive computation time (and hence are impractical for real-time use). To this end, we develop a 1 − 1 𝑒 − 𝜀 approximation algorithm for the Real-Time Line Planning Problem, using ideas from randomized rounding and the Generalized Assignment Problem. Our guarantee holds under two assumptions: (𝑖) no inter-bus transfers and (𝑖𝑖) access to a pre-specified set of feasible bus lines. We moreover show that these two assumptions are crucial by proving that, if either assumption is relaxed, the Real-Time Line Planning Problem does not admit any constant-factor approximation. Finally, we demonstrate the practicality of our algorithm via numerical experiments on real-world and synthetic datasets, in which we show that, given a fixed time budget, our algorithm outperforms Integer Linear Programming-based exact methods.

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“…Since, given an allocation of items to players, computing optimal prices that respect the agents' budgets is easy, it becomes clear that the most difficult aspect of the problem is finding such an allocation. The MBA problem is a generic allocation problem, which has In this direction, the Configuration-LP was employed by Kalaitzis et al [9] to provide a 3/4 + capproximation algorithm (for some small constant c > 0) for a restricted version of the MBA problem (which they call Restricted MBA), in which all the players have uniform budgets, and the items have uniform prices. The use of the Configuration-LP for the MBA problem was first proposed by Chakrabarty and Goel [5].…”

Section: Introductionmentioning

confidence: 99%

“…The use of the Configuration-LP for the MBA problem was first proposed by Chakrabarty and Goel [5]. The worst known upper bound on the integrality gap of the Configuration-LP is 2( √ 2 − 1) [9].…”

Section: Introductionmentioning

confidence: 99%

An Improved Approximation Guarantee for the Maximum Budgeted Allocation Problem

Kalaitzis

2015

Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms

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We study the Maximum Budgeted Allocation problem, which is the problem of assigning indivisible items to players with budget constraints. In its most general form, an instance of the MBA problem might include many different prices for the same item among different players, and different budget constraints for every player. So far, the best approximation algorithms we know for the MBA problem achieve a 3/4-approximation ratio, and employ a natural LP relaxation, called the Assignment-LP. In this paper, we give an algorithm for MBA, and prove that it achieves a 3/4 + c-approximation ratio, for some constant c > 0. This algorithm works by rounding solutions to an LP called the Configuration-LP, therefore also showing that the Configuration-LP is strictly stronger than the Assignment-LP (for which we know that the integrality gap is 3/4) for the MBA problem. 1 received significant attention from the research community. The reason is that it arises naturally in the context of multiple practical scenarios, such as when we want to maximize the revenue in an auction with budget-constrained agents. However, it is an NP-hard problem, and therefore hard to solve optimally. Therefore, one natural approach is to take up approximation algorithms in the course of tackling it.So far, the best algorithms we know achieve a 3/4-approximation ratio and are due to Srinivasan [14] and Chakrabarty and Goel [5]. These algorithms work by rounding solutions to the Assignment-LP relaxation, a relaxation for which we know the integrality gap is no better than 3/4. Since our goal is to improve upon the 3/4 ratio, we will have to use a stronger relaxation. Configuration LP-s have often been used in the past to provide good approximation algorithms (e.g., for the Bin Packing problem [10,12]); towards such an end, Chakrabarty and Goel [5] proposed the use of the Configuration-LP for the MBA problem, and conjectured that its integrality gap should be better than 3/4. At this point, it was already observed [5] that the restriction of items having uniform prices (i.e., an item can be assigned to a subset of the players, and every player is willing to pay the same price for that item) is one of the most natural restrictions of MBA that do not make the problem too easy. This is in line, for example, with what was already observed for the Generalized Assignment Problem (GAP), i.e., the problem of finding an allocation of items to bins of size 1 which maximizes the total value, where every item has a separate size and value for each bin. Indeed, in their work on designing an algorithm with a better than 1 − 1/e-approximation ratio for GAP, Feige and Vondrák [6] show how to deal with uniform instances, and then focus on the rest; interestingly, their techniques rely on the use of the Configuration-LP. Furthermore, these facts highlight some structural differences between different allocation problems: while problems such as GAP and MBA might not get substantially harder when the item values are non-uniform, this is not necessarily the case for other al...

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On the Configuration LP for Maximum Budgeted Allocation

Cited by 4 publications

References 26 publications

Approximations for Allocating Indivisible Items with Concave-Additive Valuations

Approximations for Allocating Indivisible Items with Concave-Additive Valuations

Real-Time Approximate Routing for Smart Transit Systems

An Improved Approximation Guarantee for the Maximum Budgeted Allocation Problem

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