The Steiner tree problem is one of the most fundamental NP -hard problems: given a weighted undirected graph and a subset of terminal nodes, find a minimum-cost tree spanning the terminals. In a sequence of papers, the approximation ratio for this problem was improved from 2 to 1.55 [Robins and Zelikovsky 2005]. All these algorithms are purely combinatorial. A long-standing open problem is whether there is an LP relaxation of Steiner tree with integrality gap smaller than 2 [Rajagopalan and Vazirani 1999]. In this article we present an LP-based approximation algorithm for Steiner tree with an improved approximation factor. Our algorithm is based on a, seemingly novel, iterative randomized rounding technique. We consider an LP relaxation of the problem, which is based on the notion of directed components. We sample one component with probability proportional to the value of the associated variable in a fractional solution: the sampled component is contracted and the LP is updated consequently. We iterate this process until all terminals are connected. Our algorithm delivers a solution of cost at most ln(4) + ε < 1.39 times the cost of an optimal Steiner tree. The algorithm can be derandomized using the method of limited independence. As a by-product of our analysis, we show that the integrality gap of our LP is at most 1.55, hence answering the mentioned open question.
The Steiner tree problem is one of the most fundamental NP-hard problems: given a weighted undirected graph and a subset of terminal nodes, find a minimum-cost tree spanning the terminals. In a sequence of papers, the approximation ratio for this problem was improved from 2 to the current best 1.55 [Robins,. All these algorithms are purely combinatorial. A long-standing open problem is whether there is an LP-relaxation for Steiner tree with integrality gap smaller than 2 [Vazirani, .In this paper we improve the approximation factor for Steiner tree, developing an LP-based approximation algorithm. Our algorithm is based on a, seemingly novel, iterative randomized rounding technique. We consider a directedcomponent cut relaxation for the k-restricted Steiner tree problem. We sample one of these components with probability proportional to the value of the associated variable in the optimal fractional solution and contract it. We iterate this process for a proper number of times and finally output the sampled components together with a minimum-cost terminal spanning tree in the remaining graph. Our algorithm delivers a solution of cost at most ln(4) times the cost of an optimal k-restricted Steiner tree. This directly implies a ln(4) + ε < 1.39 approximation for Steiner tree.As a byproduct of our analysis, we show that the integrality gap of our LP is at most 1.55, hence answering to the * Extended abstract.
Dependent rounding is a useful technique for optimization problems with hard budget constraints. This framework naturally leads to negative correlation properties. However, what if an application naturally calls for dependent rounding on the one hand, and desires positive correlation on the other? More generally, we develop algorithms that guarantee the known properties of dependent rounding, but also have nearly best-possible behavior -near-independence, which generalizes positive correlation -on "small" subsets of the variables. The recent breakthrough of Li & Svensson for the classical kmedian problem has to handle positive correlation in certain dependent-rounding settings, and does so implicitly. We improve upon Li-Svensson's approximation ratio for k-median from 2.732+ to 2.675+ by developing an algorithm that improves upon various aspects of their work. Our dependent-rounding approach helps us improve the dependence of the runtime on the parameter from Li-Svensson's N O(1/ 2 ) to N O((1/ ) log(1/ )) . Introduction and High-Level DetailsWe consider two notions in combinatorial optimization: a concrete problem (the classical k-median problem) and a formulation of new types of distributions (generalizations of dependent-rounding techniques); the breakthrough of Li & Svensson on the former [24] uses special cases of the latter. We improve the approximation ratio of [24] for the former, and develop efficient samplers for the latter -which, in particular, show that such distributions exist; we then combine the two to improve the run-time of our approximation algorithm for k-median. The ideas developed here also lead to optimal approximations for certain budgeted satisfiability problems, of which the classical budgeted set-cover problem is a special case. We discuss these contributions in further detail in Sections 1.1, 1.2, and 1.3.
We obtain a 1.5-approximation algorithm for the metric uncapacitated facility location problem (UFL), which improves on the previously best known 1.52-approximation algorithm by Mahdian, Ye and Zhang. Note, that the approximability lower bound by Guha and Khuller is 1.463..An algorithm is a (λ f ,λ c )-approximation algorithm if the solution it produces has total cost at most λ f • F * + λ c • C * , where F * and C * are the facility and the connection cost of an optimal solution. Our new algorithm, which is a modification of the (1 + 2/e)-approximation algorithm of Chudak and Shmoys, is a (1.6774,1.3738)-approximation algorithm for the UFL problem and is the first one that touches the approximability limit curve (γ f , 1 + 2e −γ f ) established by Jain, Mahdian and Saberi. As a consequence, we obtain the first optimal approximation algorithm for instances dominated by connection costs. When combined with a (1.11,1.7764)-approximation algorithm proposed by Jain et al., and later analyzed by Mahdian et al., we obtain the overall approximation guarantee of 1.5 for the metric UFL problem. We also describe how to use our algorithm to improve the approximation ratio for the 3-level version of UFL.
We consider the problem of computing additively approximate Nash equilibria in non-cooperative two-player games. We provide a new polynomial time algorithm that achieves an approximation guarantee of 0.36392. Our work improves the previously best known (0.38197 +)-approximation algorithm of Daskalakis, Mehta and Papadimitriou [6]. First, we provide a simpler algorithm, which also achieves 0.38197. This algorithm is then tuned, improving the approximation error to 0.36392. Our method is relatively fast, as it requires solving only one linear program and it is based on using the solution of an auxiliary zero-sum game as a starting point.
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