A Characterization of Undirected Graphs Admitting Optimal Cost Shares

In a seminal paper, Chen, Roughgarden and Valiant studied cost sharing protocols for network design with the objective to implement a low-cost Steiner forest as a Nash equilibrium of an induced cost-sharing game. One of the most intriguing open problems to date is to understand the power of budget-balanced and separable cost sharing protocols in order to induce low-cost Steiner forests. In this work, we focus on undirected networks and analyze topological properties of the underlying graph so that an optimal Steiner forest can be implemented as a Nash equilibrium (by some separable cost sharing protocol) independent of the edge costs. We term a graph efficient if the above stated property holds. As our main result, we give a complete characterization of efficient undirected graphs for two-player network design games: an undirected graph is efficient if and only if it does not contain (at least) one out of few forbidden subgraphs. Our characterization implies that several graph classes are efficient: generalized series-parallel graphs, fan and wheel graphs and graphs with small cycles.Comment: 60 pages, 69 figures, OR 2017 Berlin, WINE 2017 Bangalor

Distributed Design for Decentralized Control using Chordal Decomposition and ADMM

We propose a distributed design method for decentralized control by exploiting the underlying sparsity properties of the problem. Our method is based on chordal decomposition of sparse block matrices and the alternating direction method of multipliers (ADMM). We first apply a classical parameterization technique to restrict the optimal decentralized control into a convex problem that inherits the sparsity pattern of the original problem. The parameterization relies on a notion of strongly decentralized stabilization, and sufficient conditions are discussed to guarantee this notion. Then, chordal decomposition allows us to decompose the convex restriction into a problem with partially coupled constraints, and the framework of ADMM enables us to solve the decomposed problem in a distributed fashion. Consequently, the subsystems only need to share their model data with their direct neighbours, not needing a central computation. Numerical experiments demonstrate the effectiveness of the proposed method.Comment: 11 pages, 8 figures. Accepted for publication in the IEEE Transactions on Control of Network System

Behavioral Communities and the Atomic Structure of Networks

We develop a theory of `behavioral communities' and the `atomic structure' of networks. We define atoms to be groups of agents whose behaviors always match each other in a set of coordination games played on the network. This provides a microfoundation for a method of detecting communities in social and economic networks. We provide theoretical results characterizing such behavior-based communities and atomic structures and discussing their properties in large random networks. We also provide an algorithm for identifying behavioral communities. We discuss applications including: a method of estimating underlying preferences by observing behavioral conventions in data, and optimally seeding diffusion processes when there are peer interactions and homophily. We illustrate the techniques with applications to high school friendship networks and rural village networks

Error-Sensitive Proof-Labeling Schemes

Proof-labeling schemes are known mechanisms providing nodes of networks with certificates that can be verified locally by distributed algorithms. Given a boolean predicate on network states, such schemes enable to check whether the predicate is satisfied by the actual state of the network, by having nodes interacting with their neighbors only. Proof-labeling schemes are typically designed for enforcing fault-tolerance, by making sure that if the current state of the network is illegal with respect to some given predicate, then at least one node will detect it. Such a node can raise an alarm, or launch a recovery procedure enabling the system to return to a legal state. In this paper, we introduce error-sensitive proof-labeling schemes. These are proof-labeling schemes which guarantee that the number of nodes detecting illegal states is linearly proportional to the edit-distance between the current state and the set of legal states. By using error-sensitive proof-labeling schemes, states which are far from satisfying the predicate will be detected by many nodes, enabling fast return to legality. We provide a structural characterization of the set of boolean predicates on network states for which there exist error-sensitive proof-labeling schemes. This characterization allows us to show that classical predicates such as, e.g., acyclicity, and leader admit error-sensitive proof-labeling schemes, while others like regular subgraphs don\u27t. We also focus on compact error-sensitive proof-labeling schemes. In particular, we show that the known proof-labeling schemes for spanning tree and minimum spanning tree, using certificates on O(log n) bits, and on O(log^2 n) bits, respectively, are error-sensitive, as long as the trees are locally represented by adjacency lists, and not just by parent pointers

On width measures and topological problems on semi-complete digraphs

Under embargo until: 2021-02-01The topological theory for semi-complete digraphs, pioneered by Chudnovsky, Fradkin, Kim, Scott, and Seymour [10], [11], [12], [28], [43], [39], concentrates on the interplay between the most important width measures — cutwidth and pathwidth — and containment relations like topological/minor containment or immersion. We propose a new approach to this theory that is based on outdegree orderings and new families of obstacles for cutwidth and pathwidth. Using the new approach we are able to reprove the most important known results in a unified and simplified manner, as well as provide multiple improvements. Notably, we obtain a number of efficient approximation and fixed-parameter tractable algorithms for computing width measures of semi-complete digraphs, as well as fast fixed-parameter tractable algorithms for testing containment relations in the semi-complete setting. As a direct corollary of our work, we also derive explicit and essentially tight bounds on duality relations between width parameters and containment orderings in semi-complete digraphs.acceptedVersio

Robustness: a New Form of Heredity Motivated by Dynamic Networks

We investigate a special case of hereditary property in graphs, referred to as {\em robustness}. A property (or structure) is called robust in a graph $G$ if it is inherited by all the connected spanning subgraphs of $G$. We motivate this definition using two different settings of dynamic networks. The first corresponds to networks of low dynamicity, where some links may be permanently removed so long as the network remains connected. The second corresponds to highly-dynamic networks, where communication links appear and disappear arbitrarily often, subject only to the requirement that the entities are temporally connected in a recurrent fashion ({\it i.e.} they can always reach each other through temporal paths). Each context induces a different interpretation of the notion of robustness. We start by motivating the definition and discussing the two interpretations, after what we consider the notion independently from its interpretation, taking as our focus the robustness of {\em maximal independent sets} (MIS). A graph may or may not admit a robust MIS. We characterize the set of graphs \forallMIS in which {\em all} MISs are robust. Then, we turn our attention to the graphs that {\em admit} a robust MIS (\existsMIS). This class has a more complex structure; we give a partial characterization in terms of elementary graph properties, then a complete characterization by means of a (polynomial time) decision algorithm that accepts if and only if a robust MIS exists. This algorithm can be adapted to construct such a solution if one exists

Algorithms for Graph Connectivity and Cut Problems - Connectivity Augmentation, All-Pairs Minimum Cut, and Cut-Based Clustering

We address a collection of related connectivity and cut problems in simple graphs that reach from the augmentation of planar graphs to be k-regular and c-connected to new data structures representing minimum separating cuts and algorithms that smoothly maintain Gomory-Hu trees in evolving graphs, and finally to an analysis of the cut-based clustering approach of Flake et al. and its adaption to dynamic scenarios