We study the basic allocation problem of assigning resources to players so as to maximize fairness. This is one of the few natural problems that enjoys the intriguing status of having a better estimation algorithm than approximation algorithm. Indeed, a certain Configuration-LP can be used to estimate the value of the optimal allocation to within a factor of 4 + ε. In contrast, however, the best known approximation algorithm for the problem has an unspecified large constant guarantee.In this paper we significantly narrow this gap by giving a 13-approximation algorithm for the problem. Our approach develops a local search technique introduced by Haxell [Hax95] for hypergraph matchings, and later used in this context by Asadpour, Feige, and Saberi [AFS12]. For our local search procedure to terminate in polynomial time, we introduce several new ideas such as lazy updates and greedy players. Besides the improved approximation guarantee, the highlight of our approach is that it is purely combinatorial and uses the Configuration-LP only in the analysis.

The Tree Augmentation Problem (TAP) is a fundamental network design problem in which we are given a tree and a set of additional edges, also called links. The task is to find a set of links, of minimum size, whose addition to the tree leads to a 2-edge-connected graph. A long line of results on TAP culminated in the previously best known approximation guarantee of 1.5 achieved by a combinatorial approach due to Kortsarz and Nutov [ACM Transactions on Algorithms 2016], and also by an SDP-based approach by Cheriyan and Gao [Algorithmica 2017]. Moreover, an elegant LP-based (1.5 + ǫ)-approximation has also been found very recently by Fiorini, Groß, Könemann, and Sanitá [SODA 2018]. In this paper, we show that an approximation factor below 1.5 can be achieved, by presenting a 1.458-approximation that is based on several new techniques.By extending prior results of Adjiashvili [SODA 2017], we first present a black-box reduction to a very structured type of instance, which played a crucial role in recent development on the problem, and which we call k-wide. Our main contribution is a new approximation algorithm for O(1)-wide tree instances with approximation guarantee strictly below 1.458, based on one of their fundamental properties: wide trees naturally decompose into smaller subtrees with a constant number of leaves. Previous approaches in similar settings rounded each subtree independently and simply combined the obtained solutions. We show that additionally, when starting with a well-chosen LP, the combined solution can be improved through a new "rewiring" technique, showing that one can replace some pairs of used links by a single link. We can rephrase the rewiring problem as a stochastic version of a matching problem, which may be of independent interest. By showing that large matchings can be obtained in this problem, we obtain that a significant number of rewirings are possible, thus leading to an approximation factor below 1.5.

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 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 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 5 6 to 2 √ 2 − 2 ≈ 0.828 and also prove hardness of approximation results for both cases. *

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 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 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

International audienceThe efficient design of networks has been an important engineering task that involves challenging combinatorial optimization problems. Typically, a network designer has to select among several alternatives which links to establish so that the resulting network satisfies a given set of connectivity requirements and the cost of establishing the network links is as low as possible. The Minimum Spanning Tree problem, which is well-understood, is a nice example.In this paper, we consider the natural scenario in which the connectivity requirements are posed by selfish users who have agreed to share the cost of the network to be established according to a well-defined rule. The design proposed by the network designer should now be consistent not only with the connectivity requirements but also with the selfishness of the users. Essentially, the users are players in a so-called network design game and the network designer has to propose a design that is an equilibrium for this game. As it is usually the case when selfishness comes into play, such equilibria may be suboptimal. In this paper, we consider the following question: can the network designer enforce particular designs as equilibria or guarantee that efficient designs are consistent with users’ selfishness by appropriately subsidizing some of the network links? In an attempt to understand this question, we formulate corresponding optimization problems and present positive and negative results