We study two related problems: finding a set of k vertices and minimum number of edges (kmin) and finding a graph with at least m' edges and minimum number of vertices (mvms). Goldschmidt and Hochbaum GH97 show that the mvms problem is NP-hard and they give a 3-approximation algorithm for the problem. We improve GH97 by giving a ratio of 2. A 2(1+ )-approximation for the problem follows from the work of Carnes and Shmoys CS08 . We improve the approximation ratio to 2. algorithm for the problem. We show that the natural LP for has an integrality gap of 2-o(1). We improve the NP-completeness of GH97 by proving the pronlem are APX-hard unless a well-known instance of the dense k-subgraph admits a constant ratio. The best approximation guarantee known for this instance of dense k-subgraph is O(n^ 2 9 ) BCCFV . We show that for any constant >1, an approximation guarantee of for the problem implies a (1+o(1)) approximation for . Finally, we define we give an exact algorithm for the density version of kmin.
Consider an objective function in which we minimize @math . One can associate a cost @math with each vertex @math and a size @math for each vertex @math , and then the objective is just to minimize @math subject to @math . Carnes and Shmoys @cite_9 give a @math -approximation for the problem. Using this result and the observation that the objective function is at most a factor of @math away from the objective function for the problem, a @math -approximation follows for the problem.
@cite_9
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An algorithm for solving the problem: minimize @math (a convex function) subject to @math , @math , each @math a concave function, is presented. Specifically, the function [ P [ x,t,r_k ] f( x ) + r_k^ - 1 [ g_i ( x ) - t_i ] ^2 ] is minimized over all x, nonnegative t, for a strictly decreasing null sequence @math . This extends the work of T. Pietrzykowski [5]. It is proved that for every @math , there exists a finite point @math which minimizes P, and which solves the convex programming problem as @math . This algorithm is similar to the Sequential Unconstrained Minimization Technique (SUMT) [1] in that it solves the (Wolfe) dual programming problem [6]. It differs from SUMT in that (1) it approaches the optimum from the region of infeasibility (i.e., it is a relaxation technique), (2) it does not require a nonempty interior to the nonlinearly constrained region, (3) no separate feasibilit... We consider a network creation game in which, each player (vertex) has a limited budget to establish links to other players. In our model, each link has a unit cost and each agent tries to minimize its cost which is its local diameter or its total distance to other players in the (undirected) underlying graph of the created network. Two variants of the game are studied: in the MAX version, the cost incurred to a vertex is the maximum distance between that vertex and other vertices, and in the SUM version, the cost incurred to a vertex is the sum of distances between that vertex and other vertices. We prove that in both versions pure Nash equilibria exist, but the problem of finding the best response of a vertex is NP-hard. Next, we study the maximum possible diameter of an equilibrium graph with n vertices in various cases. For infinite numbers of n, we construct an equilibrium tree with diameter Θ(n) in the MAX version. Also, we prove that the diameter of any equilibrium tree is O(log n) in the SUM version and this bound is tight. When all vertices have unit budgets (i.e. can establish link to just one vertex), the diameter in both versions is O(1). We give an example of equilibrium graph in MAX version, such that all vertices have positive budgets and yet the diameter is as large as Ω(√log n). This interesting result shows that the diameter does not decrease necessarily and may increase as the budgets are increased. For the SUM version, we prove that every equilibrium graph has diameter 2O(√log n) when all vertices have positive budgets. Moreover, if the budget of every players is at least k, then every equilibrium graph with diameter more than 3 is k-connected. A method is described for the minimization of a function of n variables, which depends on the comparison of function values at the (n 41) vertices of a general simplex, followed by the replacement of the vertex with the highest value by another point. The simplex adapts itself to the local landscape, and contracts on to the final minimum. The method is shown to be effective and computationally compact. A procedure is given for the estimation of the Hessian matrix in the neighbourhood of the minimum, needed in statistical estimation problems. Consider the problem of choosing a string of actions to optimize an objective function that is string submodular. It was shown in previous papers that the greedy strategy, consisting of a string of actions that only locally maximizes the step-wise gain in the objective function, achieves at least a @math -approximation to the optimal strategy. This paper improves this approximation by introducing additional constraints on curvature, namely, total backward curvature, total forward curvature, and elemental forward curvature. We show that if the objective function has total backward curvature @math , then the greedy strategy achieves at least a @math -approximation of the optimal strategy. If the objective function has total forward curvature @math , then the greedy strategy achieves at least a @math -approximation of the optimal strategy. Moreover, we consider a generalization of the diminishing-return property by defining the elemental forward curvature. We also introduce the notion of string-matroid and consider the problem of maximizing the objective function subject to a string-matroid constraint. We investigate two applications of string submodular functions with curvature constraints: 1) choosing a string of actions to maximize the expected fraction of accomplished tasks; and 2) designing a string of measurement matrices such that the information gain is maximized.