Selfish Network Creation with Non-Uniform Edge Cost

Network creation games investigate complex networks from a game-theoretic point of view. Based on the original model by Fabrikant et al. [PODC'03] many variants have been introduced. However, almost all versions have the drawback that edges are treated uniformly, i.e. every edge has the same cost and that this common parameter heavily influences the outcomes and the analysis of these games. We propose and analyze simple and natural parameter-free network creation games with non-uniform edge cost. Our models are inspired by social networks where the cost of forming a link is proportional to the popularity of the targeted node. Besides results on the complexity of computing a best response and on various properties of the sequential versions, we show that the most general version of our model has constant Price of Anarchy. To the best of our knowledge, this is the first proof of a constant Price of Anarchy for any network creation game.Comment: To appear at SAGT'1

The linearization problem of a binary quadratic problem and its applications

We provide several applications of the linearization problem of a binary quadratic problem. We propose a new lower bounding strategy, called the linearization-based scheme, that is based on a simple certificate for a quadratic function to be non-negative on the feasible set. Each linearization-based bound requires a set of linearizable matrices as an input. We prove that the Generalized Gilmore-Lawler bounding scheme for binary quadratic problems provides linearization-based bounds. Moreover, we show that the bound obtained from the first level reformulation linearization technique is also a type of linearization-based bound, which enables us to provide a comparison among mentioned bounds. However, the strongest linearization-based bound is the one that uses the full characterization of the set of linearizable matrices. Finally, we present a polynomial-time algorithm for the linearization problem of the quadratic shortest path problem on directed acyclic graphs. Our algorithm gives a complete characterization of the set of linearizable matrices for the quadratic shortest path problem

The Quadratic Cycle Cover Problem: special cases and efficient bounds

The quadratic cycle cover problem is the problem of finding a set of node-disjoint cycles visiting all the nodes such that the total sum of interaction costs between consecutive arcs is minimized. In this paper we study the linearization problem for the quadratic cycle cover problem and related lower bounds. In particular, we derive various sufficient conditions for the quadratic cost matrix to be linearizable, and use these conditions to compute bounds. We also show how to use a sufficient condition for linearizability within an iterative bounding procedure. In each step, our algorithm computes the best equivalent representation of the quadratic cost matrix and its optimal linearizable matrix with respect to the given sufficient condition for linearizability. Further, we show that the classical Gilmore-Lawler type bound belongs to the family of linearization based bounds, and therefore apply the above mentioned iterative reformulation technique. We also prove that the linearization vectors resulting from this iterative approach satisfy the constant value property. The best among here introduced bounds outperform existing lower bounds when taking both quality and efficiency into account

Routing for analog chip designs at NXP Semiconductors

During the study week 2011 we worked on the question of how to automate certain aspects of the design of analog chips. Here we focused on the task of connecting different blocks with electrical wiring, which is particularly tedious to do by hand. For digital chips there is a wealth of research available for this, as in this situation the amount of blocks makes it hopeless to do the design by hand. Hence, we set our task to finding solutions that are based on the previous research, as well as being tailored to the specific setting given by NXP. This resulted in an heuristic approach, which we presented at the end of the week in the form of a protoype tool. In this report we give a detailed account of the ideas we used, and describe possibilities to extend the approach

Exact and Heuristic Methods for the Weapon Target Assignment Problem

The Weapon Target Assignment (WTA) problem is a fundamental problem arising in defense-related applications of operations research. This problem consists of optimally assigning n weapons to m targets so that the total expected survival value of the targets after all the engagements is minimum. The WTA problem can be formulated as a nonlinear integer programming problem and is known to be NP-complete. There do not exist any exact methods for the WTA problem which can solve even small size problems (for example, with 20 weapons and 20 targets). Though several heuristic methods have been proposed to solve the WTA problem, due to the absence of exact methods, no estimates are available on the quality of solutions produced by such heuristics. In this paper, we suggest linear programming, integer programming, and network flow based lower bounding methods using which we obtain several branch and bound algorithms for the WTA problem. We also propose a network flow based construction heuristic and a very large-scale neighborhood (VLSN) search algorithm. We present computational results of our algorithms which indicate that we can solve moderately large size instances (up to 80 weapons and 80 targets) of the WTA problem optimally and obtain almost optimal solutions of fairly large instances (up to 200 weapons and 200 targets) within a few second

Sources of Superlinearity in Davenport-Schinzel Sequences

A generalized Davenport-Schinzel sequence is one over a finite alphabet that contains no subsequences isomorphic to a fixed forbidden subsequence. One of the fundamental problems in this area is bounding (asymptotically) the maximum length of such sequences. Following Klazar, let Ex(\sigma,n) be the maximum length of a sequence over an alphabet of size n avoiding subsequences isomorphic to \sigma. It has been proved that for every \sigma, Ex(\sigma,n) is either linear or very close to linear; in particular it is O(n 2^{\alpha(n)^{O(1)}}), where \alpha is the inverse-Ackermann function and O(1) depends on \sigma. However, very little is known about the properties of \sigma that induce superlinearity of \Ex(\sigma,n). In this paper we exhibit an infinite family of independent superlinear forbidden subsequences. To be specific, we show that there are 17 prototypical superlinear forbidden subsequences, some of which can be made arbitrarily long through a simple padding operation. Perhaps the most novel part of our constructions is a new succinct code for representing superlinear forbidden subsequences