Approximating Source Location and Star Survivable Network Problems

In Source Location (SL) problems the goal is to select a mini-mum cost source set $S \subseteq V$ such that the connectivity (or flow) $\psi(S,v)$ from $S$ to any node $v$ is at least the demand $d_v$ of $v$. In many SL problems $\psi(S,v)=d_v$ if $v \in S$, namely, the demand of nodes selected to $S$ is completely satisfied. In a node-connectivity variant suggested recently by Fukunaga, every node $v$ gets a "bonus" $p_v \leq d_v$ if it is selected to $S$. Fukunaga showed that for undirected graphs one can achieve ratio $O(k \ln k)$ for his variant, where $k=\max_{v \in V}d_v$ is the maximum demand. We improve this by achieving ratio \min\{p^*\lnk,k\}\cdot O(\ln (k/q^*)) for a more general version with node capacities, where $p^*=\max_{v \in V} p_v$ is the maximum bonus and $q^*=\min_{v \in V} q_v$ is the minimum capacity. In particular, for the most natural case $p^*=1$ considered by Fukunaga, we improve the ratio from $O(k \ln k)$ to $O(\ln^2k)$. We also get ratio $O(k)$ for the edge-connectivity version, for which no ratio that depends on $k$ only was known before. To derive these results, we consider a particular case of the Survivable Network (SN) problem when all edges of positive cost form a star. We give ratio $O(\min\{\ln n,\ln^2 k\})$ for this variant, improving over the best ratio known for the general case $O(k^3 \ln n)$ of Chuzhoy and Khanna

Posimodular Function Optimization

Given a posimodular function $f: 2^V \to \mathbb{R}$ on a finite set $V$, we consider the problem of finding a nonempty subset $X$ of $V$ that minimizes $f(X)$. Posimodular functions often arise in combinatorial optimization such as undirected cut functions. In this paper, we show that any algorithm for the problem requires $\Omega(2^{\frac{n}{7.54}})$ oracle calls to $f$, where $n=|V|$. It contrasts to the fact that the submodular function minimization, which is another generalization of cut functions, is polynomially solvable. When the range of a given posimodular function is restricted to be $D=\{0,1,...,d\}$ for some nonnegative integer $d$, we show that $\Omega(2^{\frac{d}{15.08}})$ oracle calls are necessary, while we propose an $O(n^dT_f+n^{2d+1})$-time algorithm for the problem. Here, $T_f$ denotes the time needed to evaluate the function value $f(X)$ for a given $X \subseteq V$. We also consider the problem of maximizing a given posimodular function. We show that $\Omega(2^{n-1})$ oracle calls are necessary for solving the problem, and that the problem has time complexity $\Theta(n^{d-1}T_f)$ when $D=\{0,1,..., d\}$ is the range of $f$ for some constant $d$.Comment: 18 page

Designing Network Protocols for Good Equilibria

Designing and deploying a network protocol determines the rules by which end users interact with each other and with the network. We consider the problem of designing a protocol to optimize the equilibrium behavior of a network with selfish users. We consider network cost-sharing games, where the set of Nash equilibria depends fundamentally on the choice of an edge cost-sharing protocol. Previous research focused on the Shapley protocol, in which the cost of each edge is shared equally among its users. We systematically study the design of optimal cost-sharing protocols for undirected and directed graphs, single-sink and multicommodity networks, and different measures of the inefficiency of equilibria. Our primary technical tool is a precise characterization of the cost-sharing protocols that induce only network games with pure-strategy Nash equilibria. We use this characterization to prove, among other results, that the Shapley protocol is optimal in directed graphs and that simple priority protocols are essentially optimal in undirected graphs

Applications of network optimization

Includes bibliographical references (p. 41-48).Ravindra K. Ahuja ... [et al.]

Designing Overlapping Networks for Publish-Subscribe Systems

From the publish-subscribe systems of the early days of the Internet to the recent emergence of Web 3.0 and IoT (Internet of Things), new problems arise in the design of networks centered at producers and consumers of constantly evolving information. In a typical problem, each terminal is a source or sink of information and builds a physical network in the form of a tree or an overlay network in the form of a star rooted at itself. Every pair of pub-sub terminals that need to be coordinated (e.g. the source and sink of an important piece of control information) define an edge in a bipartite demand graph; the solution must ensure that the corresponding networks rooted at the endpoints of each demand edge overlap at some node. This simple overlap constraint, and the requirement that each network is a tree or a star, leads to a variety of new questions on the design of overlapping networks. In this paper, for the general demand case of the problem, we show that a natural LP formulation has a non-constant integrality gap; on the positive side, we present a logarithmic approximation for the general demand case. When the demand graph is complete, however, we design approximation algorithms with small constant performance ratios, irrespective of whether the pub networks and sub networks are required to be trees or stars

A Graph-Theoretic Approach to Improved Curriculum Structure and Assessment Placement

Assurance of student learning is an important issue in modern education. Accrediting bodies and government entities alike have begun requiring proof that students actually learn the material taught in the classroom. To comply with this mandate, education programs are actively engaged in incorporating assessment procedures into their curricula. Unfortunately, there are no generally agreed upon methods on how to do this. This paper addresses the issue and answers several key questions concerning where to locate topic coverage, assessment data collection, and corrective action within the curriculum. The approach described and developed in this paper cast the curriculum problem into an abstract graph representation. Graph theoretic metrics are calculated and visualization software is utilized to create a picture that helps answer the research questions. The result is a systematic, easy to understand approach that produces defensible output which should greatly aid faculty and administrators tasked with creating an assessment scheme

On green routing and scheduling problem

The vehicle routing and scheduling problem has been studied with much interest within the last four decades. In this paper, some of the existing literature dealing with routing and scheduling problems with environmental issues is reviewed, and a description is provided of the problems that have been investigated and how they are treated using combinatorial optimization tools