Learning-Augmented Algorithms for Online TSP on the Line

We study the online Traveling Salesman Problem (TSP) on the line augmentedwith machine-learned predictions. In the classical problem, there is a streamof requests released over time along the real line. The goal is to minimize themakespan of the algorithm. We distinguish between the open variant and theclosed one, in which we additionally require the algorithm to return to theorigin after serving all requests. The state of the art is a $1.64$-competitivealgorithm and a $2.04$-competitive algorithm for the closed and open variants,respectively \cite{Bjelde:1.64}. In both cases, a tight lower bound is known\cite{Ausiello:1.75, Bjelde:1.64}. In both variants, our primary prediction model involves predicted positionsof the requests. We introduce algorithms that (i) obtain a tight 1.5competitive ratio for the closed variant and a 1.66 competitive ratio for theopen variant in the case of perfect predictions, (ii) are robust againstunbounded prediction error, and (iii) are smooth, i.e., their performancedegrades gracefully as the prediction error increases. Moreover, we further investigate the learning-augmented setting in the openvariant by additionally considering a prediction for the last request served bythe optimal offline algorithm. Our algorithm for this enhanced setting obtainsa 1.33 competitive ratio with perfect predictions while also being smooth androbust, beating the lower bound of 1.44 we show for our original predictionsetting for the open variant. Also, we provide a lower bound of 1.25 for thisenhanced setting.<br

A Novel Prediction Setup for Online Speed-Scaling

Given the rapid rise in energy demand by data centers and computing systemsin general, it is fundamental to incorporate energy considerations whendesigning (scheduling) algorithms. Machine learning can be a useful approach inpractice by predicting the future load of the system based on, for example,historical data. However, the effectiveness of such an approach highly dependson the quality of the predictions and can be quite far from optimal whenpredictions are sub-par. On the other hand, while providing a worst-caseguarantee, classical online algorithms can be pessimistic for large classes ofinputs arising in practice. This paper, in the spirit of the new area of machine learning augmentedalgorithms, attempts to obtain the best of both worlds for the classical,deadline based, online speed-scaling problem: Based on the introduction of anovel prediction setup, we develop algorithms that (i) obtain provably lowenergy-consumption in the presence of adequate predictions, and (ii) are robustagainst inadequate predictions, and (iii) are smooth, i.e., their performancegradually degrades as the prediction error increases.<br

Nash Social Welfare for 2-value Instances

We study the problem of allocating a set of indivisible goods among agents with 2-value additive valuations. Our goal is to find an allocation with maximum Nash social welfare, i.e., the geometric mean of the valuations of the agents. We give a polynomial-time algorithm to find a Nash social welfare maximizing allocation when the valuation functions are integrally 2-valued, i.e., each agent has a value either $1$ or $p$ for each good, for some positive integer $p$. We then extend our algorithm to find a better approximation factor for general 2-value instances

A Tutorial on Machine Learning for Failure Management in Optical Networks

Failure management plays a role of capital importance in optical networks to avoid service disruptions and to satisfy customers' service level agreements. Machine learning (ML) promises to revolutionize the (mostly manual and human-driven) approaches in which failure management in optical networks has been traditionally managed, by introducing automated methods for failure prediction, detection, localization, and identification. This tutorial provides a gentle introduction to some ML techniques that have been recently applied in the field of the optical-network failure management. It then introduces a taxonomy to classify failure-management tasks and discusses possible applications of ML for these failure management tasks. Finally, for a reader interested in more implementative details, we provide a step-by-step description of how to solve a representative example of a practical failure-management task

Maximizing Nash Social Welfare in 2-Value Instances

We consider the problem of maximizing the Nash social welfare when allocatinga set $\mathcal{G}$ of indivisible goods to a set $\mathcal{N}$ of agents. Westudy instances, in which all agents have 2-value additive valuations: Thevalue of every agent $i \in \mathcal{N}$ for every good $j \in \mathcal{G}$ is$v_{ij} \in \{p,q\}$, for $p,q \in \mathbb{N}$, $p \le q$. Maybe surprisingly,we design an algorithm to compute an optimal allocation in polynomial time if$p$ divides $q$, i.e., when $p=1$ and $q \in \mathbb{N}$ after appropriatescaling. The problem is \classNP-hard whenever $p$ and $q$ are coprime and $p\ge 3$. In terms of approximation, we present positive and negative results forgeneral $p$ and $q$. We show that our algorithm obtains an approximation ratioof at most 1.0345. Moreover, we prove that the problem is \classAPX-hard, witha lower bound of $1.000015$ achieved at $p/q = 4/5$.<br

Maximizing Nash Social Welfare in 2-Value Instances: The Half-Integer Case

We consider the problem of maximizing the Nash social welfare when allocatinga set $G$ of indivisible goods to a set $N$ of agents. We study instances, inwhich all agents have 2-value additive valuations: The value of a good g \inG for an agent $i \in N$ is either $1$ or $s$, where $s$ is an odd multiple of$\frac{1}{2}$ larger than one. We show that the problem is solvable inpolynomial time. Akrami et at. showed that this problem is solvable inpolynomial time if $s$ is integral and is NP-hard whenever $s = \frac{p}{q}$,$p \in \mathbb{N}$ and $q\in \mathbb{N}$ are co-prime and $p > q \ge 3$. Forthe latter situation, an approximation algorithm was also given. It obtains anapproximation ratio of at most $1.0345$. Moreover, the problem is APX-hard,with a lower bound of $1.000015$ achieved at $\frac{p}{q} = \frac{5}{4}$. Thecase $q = 2$ and odd $p$ was left open. In the case of integral $s$, the problem is separable in the sense that theoptimal allocation of the heavy goods (= value $s$ for some agent) isindependent of the number of light goods (= value $1$ for all agents). Thisleads to an algorithm that first computes an optimal allocation of the heavygoods and then adds the light goods greedily. This separation no longer holdsfor $s = \frac{3}{2}$; a simple example is given in the introduction. Thus analgorithm has to consider heavy and light goods together. This complicatesmatters considerably. Our algorithm is based on a collection of improvementrules that transfers any allocation into an optimal allocation and exploits aconnection to matchings with parity constraints.<br

Physarum Multi-Commodity Flow Dynamics

In wet-lab experiments \cite{Nakagaki-Yamada-Toth,Tero-Takagi-etal}, the slime mold Physarum polycephalum has demonstrated its ability to solve shortest path problems and to design efficient networks, see Figure \ref{Wet-Lab Experiments} for illustrations. Physarum polycephalum is a slime mold in the Mycetozoa group. For the shortest path problem, a mathematical model for the evolution of the slime was proposed in \cite{Tero-Kobayashi-Nakagaki} and its biological relevance was argued. The model was shown to solve shortest path problems, first in computer simulations and then by mathematical proof. It was later shown that the slime mold dynamics can solve more general linear programs and that many variants of the dynamics have similar convergence behavior. In this paper, we introduce a dynamics for the network design problem. We formulate network design as the problem of constructing a network that efficiently supports a multi-commodity flow problem. We investigate the dynamics in computer simulations and analytically. The simulations show that the dynamics is able to construct efficient and elegant networks. In the theoretical part we show that the dynamics minimizes an objective combining the cost of the network and the cost of routing the demands through the network. We also give alternative characterization of the optimum solution