Stabilizing Weighted Graphs

An edge-weighted graph G=(V,E) is called stable if the value of a maximum-weight matching equals the value of a maximum-weight fractional matching. Stable graphs play an important role in some interesting game theory problems, such as network bargaining games and cooperative matching games, because they characterize instances which admit stable outcomes. Motivated by this, in the last few years many researchers have investigated the algorithmic problem of turning a given graph into a stable one, via edge- and vertex-removal operations. However, all the algorithmic results developed in the literature so far only hold for unweighted instances, i.e., assuming unit weights on the edges of G. We give the first polynomial-time algorithm to find a minimum cardinality subset of vertices whose removal from G yields a stable graph, for any weighted graph G. The algorithm is combinatorial and exploits new structural properties of basic fractional matchings, which are of independent interest. In particular, one of the main ingredients of our result is the development of a polynomial-time algorithm to compute a basic maximum-weight fractional matching with minimum number of odd cycles in its support. This generalizes a fundamental and classical result on unweighted matchings given by Balas more than 30 years ago, which we expect to prove useful beyond this particular application. In contrast, we show that the problem of finding a minimum cardinality subset of edges whose removal from a weighted graph G yields a stable graph, does not admit any constant-factor approximation algorithm, unless P=NP. In this setting, we develop an O(Delta)-approximation algorithm for the problem, where Delta is the maximum degree of a node in G

Simple games versus weighted voting games: Bounding the critical threshold value

A simple game $(N,v)$ is given by a set $N$ of $n$ players and a partition of~$2^N$ into a set~$\mathcal{L}$ of losing coalitions~$L$ with value $v(L)=0$ that is closed under taking subsets and a set $\mathcal{W}$ of winning coalitions $W$ with $v(W)=1$. Simple games with $\alpha= \min_{p\geq 0}\max_{W\in {\cal W}, L\in {\cal L}} \frac{p(L)}{p(W)}<1$ are exactly the weighted voting games. We show that $\alpha\leq \frac{1}{4}n$ for every simple game $(N,v)$, confirming the conjecture of Freixas and Kurz (IJGT, 2014). For complete simple games, Freixas and Kurz conjectured that $\alpha=O(\sqrt{n})$. We prove this conjecture up to a $\ln n$ factor. We also prove that for graphic simple games, that is, simple games in which every minimal winning coalition has size~2, computing $\alpha$ is \NP-hard, but polynomial-time solvable if the underlying graph is bipartite. Moreover, we show that for every graphic simple game, deciding if $\alpha0$.Comment: 10 pages; the paper is a follow-up and merge of arXiv:1805.02192 and arXiv:1806.0317

Vertex Stabilizers for Network Bargaining Games

Network bargaining games form a prominent class of examples of game theory problems defined on graphs, where vertices represent players, and edges represent their possible interactions. An instance of a \emph{network bargaining game} is given by a graph $G = (V, E)$ with edge weights $w \in \mathbb{R}^E_{+}$ and vertex capacities $c \in \mathbb{Z}^V_+$. A \emph{solution} to an instance $(G, w, c)$ of a network bargaining game is data $(M, z)$, where $M \subset E$ is a $c$-matching, and $z \in \mathbb{R}^{2E}_{\geq 0}$ is a vector which assigns each edge $uv \in E$ a pair of values $z_{uv}$ and $z_{vu}$, such that $z_{uv} + z_{vu} = w_{uv}$ if $uv \in M$, and $z_{uv} = z_{vu} = 0$ otherwise. An instance $(G, w, c)$ is said to be \emph{stable} if it admits a so-called `stable' solution, which represents a solution where a player has no incentive to deviate. Not all instances of a network bargaining game have a stable solution, and this naturally motivates the problem of how to modify the underlying graph such that the instance becomes stable. In recent years, researchers have investigated various modifications, typically by adding or removing edges or vertices. The natural algorithmic question which stems from this is whether these modifications can be performed efficiently. The answer varies, depending on the modification in question, and on whether the edges/vertices have been restricted to be unit weight/capacity. In this work, we consider the vertex-removal setting for a general instance $(G, w, c)$ of a network bargaining game. A set of vertices whose deletion from $G$ results in a stable instance of the induced network bargaining game is called a \emph{vertex stabilizer}. We demonstrate in this work that the algorithmic problem of finding a minimum cardinality vertex stabilizer is $NP$-complete, and give an efficient $2$-approximation algorithm. Further, we show that no efficient $(2 - \epsilon)$-approximation for this problem exists for any $\epsilon > 0$, assuming the Unique Games Conjecture holds. These results hold even in the case when all edges are of unit-weight. In contrast, if we are given an instance $(G, w, c)$ together with a maximum weight $c$-matching $M$, we show that the problem of finding a minimum cardinality vertex stabilizer that avoids $M$ can be solved efficiently. We provide a polynomial time algorithm for solving this problem

A survey of parameterized algorithms and the complexity of edge modification

The survey is a comprehensive overview of the developing area of parameterized algorithms for graph modification problems. It describes state of the art in kernelization, subexponential algorithms, and parameterized complexity of graph modification. The main focus is on edge modification problems, where the task is to change some adjacencies in a graph to satisfy some required properties. To facilitate further research, we list many open problems in the area.publishedVersio

Approximation algorithms for regret minimization in vehicle routing problems

In this thesis, we present new approximation algorithms as well as hardness of approximation results for NP-hard vehicle routing problems related to public transportation. We consider two different problem classes that also occur frequently in areas such as logistics, robotics, or distribution systems. For the first problem class, the goal is to visit as many locations in a network as possible subject to timing or cost constraints. For the second problem class, a given set of locations is to be visited using a minimum-cost set of routes under some constraints. Due to the relevance of both problem classes for public transportation, a secondary objective must be taken into account beyond a low operation cost: namely, it is crucial to design routes that optimize customer satisfaction in order to encourage customers to use the service. Our measure of choice is the regret of a customer, that is the time comparison of the chosen route with the shortest path to a destination. From the first problem class, we investigate variants and extensions of the Orienteering problem that asks to find a short walk maximizing the profit obtained from visiting distinct locations. We give approximation algorithms for variants in which the walk has to respect constraints on the regret of the visited vertices. Additionally, we describe a framework to extend approximation algorithms for Orienteering problems to consider also a second budget constraint, namely node demands, that have to be satisfied in order to collect the profit. We obtain polynomial time approximation schemes for the Capacitated Orienteering problem on trees and Euclidean metrics. Furthermore, we study variants of the School Bus problem (SBP). In SBP, a given set of locations is to be connected to a destination node with both low operation cost and a low maximum regret. We note that the Orienteering problem can be seen as the pricing problem for SBP and it often appears as subroutine in algorithms for SBP. For tree-shaped networks, we describe algorithms with a small constant approximation factor and complement them by showing hardness of approximation results. We give an overview of the known results in arbitrary networks and we prove that a general variant cannot be approximated unless P = NP. Finally, we describe an integer programming approach to solve School Bus problems in practice and present an improved bus schedule for a private school in the lake Geneva region

Computing the Nucleolus of Matching and b-Matching Games

In the classical weighted matching problem the optimizer is given a graph with edge weights and their goal is to find a matching which maximizes the sum of the weights of edges in the matching. It is typically assumed in this process that the optimizer has unilateral control over the decision to take each edge. Where cooperative game theory intersects combinatorial optimization this assumption is subverted. In a cooperative matching game each vertex of the graph is controlled by a distinct player, and an edge can only be taken into a matching with the cooperation of the players at each of its vertices. One can think of the weight of an edge as representing the value the players of that edge generate by collaborating in partnership. In this setting the question is more than simply can we find an optimal matching, as in the classic matching problem, but also how should the players share the total value of the matching amongst themselves. The players should share the value they generate in a way that fairly respects the contributions of each player, and which encourages as well as possible the stable participation of every player in the network. Cooperative game theory formulates such fair distributions of wealth as solution concepts. One classical and beautiful solution concept is the nucleolus. Intuitively the nucleolus distributes value so that the worst off groups of players are as satisfied as possible, and subject to that the second worst off groups, and so on. Here we think of satisfaction as the difference between how much value the players were distributed versus how much they could have generated on their own had they seceded from the grand coalition. This thesis studies the nucleolus of matching games, and their generalization to b-matching games where each player can take on multiple partnerships simultaneously, from a computational perspective. We study when the nucleolus of a b-matching game can be computed efficiently and when it is intractable to do so. Chapter 2 describes an algorithm for computing the nucleolus of any weighted cooperative matching game in polynomial time. Chapter 3 studies the computational complexity of b-matching games. We show that computing the nucleolus of such games is NP-hard even when every vertex has b-value 3, the graph is unweighted, bipartite, and of maximum degree 7. Finally, in Chapter 4 we show that when the problem of determining the worst off coalition under a given allocation in a cooperative game can be formulated as a dynamic program then the nucleolus of the game can be computed in time which is only a polynomial factor larger than the time it takes to solve said dynamic program. We apply this result to show that nucleolus of b-matching games can be computed in polynomial time on graphs of bounded treewidth

Shared Mobility - Operations and Economics

In the last decade, ubiquity of the internet and proliferation of smart personal devices have given rise to businesses that are built on the foundation of the sharing economy. The mobility market has implemented the sharing economy model in many forms, including but not limited to, carsharing, ride-sourcing, carpooling, taxi-sharing, ridesharing, bikesharing, and scooter sharing. Among these shared-use mobility services, ridesharing services, such as peer-to-peer (P2P) ridesharing and ride-pooling systems, are based on sharing both the vehicle and the ride between users, offering several individual and societal benefits. Despite these benefits, there are a number of operational and economic challenges that hinder the adoption of various forms of ridesharing services in practice. This dissertation attempts to address these challenges by investigating these systems from two different, but related, perspectives. The successful operation of ridesharing services in practice requires solving large-scale ride-matching problems in short periods of time. However, the high computational complexity and inherent supply and demand uncertainty present in these problems immensely undermines their real-time application. In the first part of this dissertation, we develop techniques that provide high-quality, although not necessarily optimal, system-level solutions that can be applied in real time. More precisely, we propose a distributed optimization technique based on graph partitioning to facilitate the implementation of dynamic P2P ridesharing systems in densely populated metropolitan areas. Additionally, we combine the proposed partitioning algorithm with a new local search algorithm to design a proactive framework that exploits historical demand data to optimize dynamic dispatching of a fleet of vehicles that serve on-demand ride requests. The main purpose of these methods is to maximize the social welfare of the corresponding ridesharing services. Despite the necessity of developing real-time algorithmic tools for operation of ridesharing services, solely maximizing the system-level social welfare cannot result in increasing the penetration of shared mobility services. This fact motivated the second stream of research in this dissertation, which revolves around proposing models that take economic aspects of ridesharing systems into account. To this end, the second part of this dissertation studies the impact of subsidy allocation on achieving and maintaining a critical mass of users in P2P ridesharing systems under different assumptions. First, we consider a community-based ridesharing system with ride-back guarantee, and propose a traveler incentive program that allocates subsidies to a carefully selected set of commuters to change their travel behavior, and thereby, increase the likelihood of finding more compatible and profitable matches. We further introduce an approximate algorithm to solve large-scale instances of this problem efficiently. In a subsequent study for a cooperative ridesharing market with role flexibility, we show that there may be no stable outcome (a collusion-free pricing and allocation scheme). Hence, we introduced a mathematical formulation that yields a stable outcome by allocating the minimum amount of external subsidy. Finally, we propose a truthful subsidy scheme to determine matching, scheduling, and subsidy allocation in a P2P ridesharing market with incomplete information and a budget constraint on payment deficit. The proposed mechanism is shown to guarantee important economic properties such as dominant-strategy incentive compatibility, individual rationality, budget-balance, and computational efficiency. Although the majority of the work in this dissertation focuses on ridesharing services, the presented methodologies can be easily generalized to tackle related issues in other types of shared-use mobility services.PHDCivil EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/169843/1/atafresh_1.pd