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+є)-approximation has also been found very recently by Fiorini, Groß, K'onemann, 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.

We consider the classic problem of scheduling jobs on unrelated machines so as to minimize the weighted sum of completion times. Recently, for a small constant ∊ > 0, Bansal et al. gave a (3/2 – ∊)-approximation algorithm improving upon the “natural” barrier of 3/2 which follows from independent randomized rounding. In simplified terms, their result is obtained by an enhancement of independent randomized rounding via strong negative correlation properties. In this work, we take a different approach and propose to use the same elegant rounding scheme for the weighted completion time objective as devised by Shmoys and Tardos for optimizing a linear function subject to makespan constraints. Our main result is a 1.21-approximation algorithm for the natural special case where the weight of a job is proportional to its processing time (specifically, all jobs have the same Smith ratio), which expresses the notion that each unit of work has the same weight. In addition, as a direct consequence of the rounding, our algorithm also achieves a bi-criteria 2-approximation for the makespan objective. Our technical contribution is a tight analysis of the expected cost of the solution compared to the one given by the Configuration-LP relaxation - we reduce this task to that of understanding certain worst-case instances which are simple to analyze.

We study the basic allocation problem of assigning resources to players 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 article, we significantly narrow this gap by giving a 13-approximation algorithm for the problem. Our approach develops a local search technique introduced by Haxell [13] for hypergraph matchings and later used in this context by Asadpour, Feige, and Saberi [2]. 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.

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.

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 [11] for hypergraph matchings, and later used in this context by Asadpour, Feige, and Saberi [1]. 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 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.