A Simply Exponential Upper Bound on the Maximum Number of Stable Matchings

Stable matching is a classical combinatorial problem that has been the subject of intense theoretical and empirical study since its introduction in 1962 in a seminal paper by Gale and Shapley. In this paper, we provide a new upper bound on $f(n)$, the maximum number of stable matchings that a stable matching instance with $n$ men and $n$ women can have. It has been a long-standing open problem to understand the asymptotic behavior of $f(n)$ as $n\to\infty$, first posed by Donald Knuth in the 1970s. Until now the best lower bound was approximately $2.28^n$, and the best upper bound was $2^{n\log n- O(n)}$. In this paper, we show that for all $n$, $f(n) \leq c^n$ for some universal constant $c$. This matches the lower bound up to the base of the exponent. Our proof is based on a reduction to counting the number of downsets of a family of posets that we call "mixing". The latter might be of independent interest

Approaching Utopia: Strong Truthfulness and Externality-Resistant Mechanisms

We introduce and study strongly truthful mechanisms and their applications. We use strongly truthful mechanisms as a tool for implementation in undominated strategies for several problems,including the design of externality resistant auctions and a variant of multi-dimensional scheduling

Reducing Septic-Reliant Households: How a Comprehensive Legal Scheme Could Improve Water Quality, Environmental, and Human Health

Considering the [...] concerns of septic reliance in Virginia, this paper will present and explicate three issues pertaining to the Maryland Sustainable Growth and Preservation Act (the “Septic Law”). Maryland\u27s Septic Law represents an effort at a statewide legal scheme for approaching the issue of rapid development on septic. It may function as a useful starting point for Virginia to approach the danger that failing septic poses to Virginian citizens and the health of the Chesapeake Bay, a vital contributor to the state economy and cultural heritage. Part II opens with an overview of Maryland’s Septic Law. Part III evaluates three issues with the law: (1) the effects of the limited scope of the Septic law; (2) the impact of geography on the Septic Law with an exploration of current scientific breakthroughs and understanding of pollution methodology; and (3) the lack of enforcement due to the voluntary nature of the Septic Law. In Part IV, this paper discusses and presents statutory schemes from other jurisdictions as examples of how to bulwark the identified weaknesses in Maryland’s Septic Law. Part V identifies some observations and elucidates implications for Virginia should the Commonwealth consider implementing the presented legislative and statutory schemes. Finally, Part VI offers a brief conclusion. This abstract has been taken from the authors\u27 introduction

Stability of Service under Time-of-Use Pricing

We consider "time-of-use" pricing as a technique for matching supply and demand of temporal resources with the goal of maximizing social welfare. Relevant examples include energy, computing resources on a cloud computing platform, and charging stations for electric vehicles, among many others. A client/job in this setting has a window of time during which he needs service, and a particular value for obtaining it. We assume a stochastic model for demand, where each job materializes with some probability via an independent Bernoulli trial. Given a per-time-unit pricing of resources, any realized job will first try to get served by the cheapest available resource in its window and, failing that, will try to find service at the next cheapest available resource, and so on. Thus, the natural stochastic fluctuations in demand have the potential to lead to cascading overload events. Our main result shows that setting prices so as to optimally handle the {\em expected} demand works well: with high probability, when the actual demand is instantiated, the system is stable and the expected value of the jobs served is very close to that of the optimal offline algorithm.Comment: To appear in STOC'1

Minnesrum

En kyrkogård är de bortgångnas bostad men är även de efterlevandes besöksplats. Det här arbetet handlar om att föra in växthusbyggnader i kyrkogårdsmiljö, som en kontemplationsplats för de sörjande. Växthus medför ett gynnsamt klimat för såväl besökare som växter. Det ger den sörjande möjlighet till en värmande och grönskande plats, de månader om året som utemiljön inte kan erbjuda detta. Närvaron av växtlighet i kombination med den visuella kontakt som besökarens ges med begravningsplatsen, är förmodligen positivt för sorgeprocessen. Det viktigaste en anläggare bör tänka på är att ett Minnesrum ska vara neutralt, det vill säga icke-religiöst, i sin utformning. Byggnadsformen bör vara nyskapande och designad så att det inte råder något tvivel om vad det är för något. Byggnadens utformning bör anpassas efter den miljö som den ska placeras i och vara väl avvägd efter antalet besökare. I dagsläget finns endast en liknande anläggning i Sverige men jag hoppas på fler och önskar en friare utformning av våra framtida kyrkogårdar

On-line load balancing

AbstractThe setup for our problem consists of n servers that must complete a set of tasks. Each task can be handled only by a subset of the servers, requires a different level of service, and once assigned cannot be reassigned. We make the natural assumption that the level of service is known at arrival time, but that the duration of service is not. The on-line load balancing problem is to assign each task to an appropriate server in such a way that the maximum load on the servers is minimized. In this paper we derive matching upper and lower bounds for the competitive ratio of the on-line greedy algorithm for this problem, namely, [(3n)23/2](1+o(1)), and derive a lower bound, Ω(n12), for any other deterministic or randomized on-line algorithm