Undermodelling Detection with Sign-Perturbed Sums

Sign-Perturbed Sums (SPS) is a finite sample system identification method that can build exact confidence regions for the unknown parameters of linear systems under mild statistical assumptions. Theoretical studies of the SPS method have assumed so far that the order of the system model is known to the user. In this paper we discuss the implications of this assumption for the applicability of the SPS method, and we propose an extension that, under mild assumptions, i) still delivers guaranteed confidence regions when the model order is correct, and ii) it is guaranteed to detect, in the long run, if the model order is wrong

Weakly-Popular and Super-Popular Matchings with Ties and Their Connection to Stable Matchings

In this paper, we study a slightly different definition of popularity in bipartite graphs $G=(U,W,E)$ with two-sided preferences, when ties are present in the preference lists. This is motivated by the observation that if an agent $u$ is indifferent between his original partner $w$ in matching $M$ and his new partner $w'\ne w$ in matching $N$, then he may probably still prefer to stay with his original partner, as change requires effort, so he votes for $M$ in this case, instead of being indifferent. We show that this alternative definition of popularity, which we call weak-popularity allows us to guarantee the existence of such a matching and also to find a weakly-popular matching in polynomial-time that has size at least $\frac{3}{4}$ the size of the maximum weakly popular matching. We also show that this matching is at least $\frac{4}{5}$ times the size of the maximum (weakly) stable matching, so may provide a more desirable solution than the current best (and tight under certain assumptions) $\frac{2}{3}$-approximation for such a stable matching. We also show that unfortunately, finding a maximum size weakly popular matching is NP-hard, even with one-sided ties and that assuming some complexity theoretic assumptions, the $\frac{3}{4}$-approximation bound is tight. Then, we study a more general model than weak-popularity, where for each edge, we can specify independently for both endpoints the size of improvement the endpoint needs to vote in favor of a new matching $N$. We show that even in this more general model, a so-called $\gamma$-popular matching always exists and that the same positive results still hold. Finally, we define an other, stronger variant of popularity, called super-popularity, where even a weak improvement is enough to vote in favor of a new matching. We show that for this case, even the existence problem is NP-hard

A Simple 1.5-Approximation Algorithm for a Wide Range of Max-SMTI Generalizations

We give a simple approximation algorithm for a common generalization of many previously studied extensions of the stable matching problem with ties. These generalizations include the existence of critical vertices in the graph, amongst whom we must match as much as possible, free edges, that cannot be blocking edges and $\Delta$-stabilities, which mean that for an edge to block, the improvement should be large enough on one or both sides. We also introduce other notions to generalize these even further, which allows our framework to capture many existing and future applications. We show that our edge duplicating technique allows us to treat these different types of generalizations simultaneously, while also making the algorithm, the proofs and the analysis much simpler and shorter then in previous approaches. In particular, we answer an open question by \cite{socialstable} about the existence of a $\frac{3}{2}$-approximation algorithm for the \smti\ problem with free edges. This demonstrates well that this technique can grasp the underlying essence of these problems quite well and have the potential to be able to solve countless future applications as well

Popular and Dominant Matchings with Uncertain, Multilayer and Aggregated Preferences

We study the Popular Matching problem in multiple models, where the preferences of the agents in the instance may change or may be unknown/uncertain. In particular, we study an Uncertainty model, where each agent has a possible set of preferences, a Multilayer model, where there are layers of preference profiles, a Robust model, where any agent may move some other agents up or down some places in his preference list and an Aggregated Preference model, where votes are summed over multiple instances with different preferences. We study both one-sided and two-sided preferences in bipartite graphs. In the one-sided model, we show that all our problems can be solved in polynomial time by utilizing the structure of popular matchings. We also obtain nice structural results. With two-sided preferences, we show that all four above models lead to NP-hard questions for popular matchings. By utilizing the connection between dominant matchings and stable matchings, we show that in the robust and uncertainty model, a certainly dominant matching in all possible prefernce profiles can be found in polynomial-time, whereas in the multilayer and aggregated models, the problem remains NP-hard for dominant matchings too. We also answer an open question about $d$-robust stable matchings

PageRank Optimization by Edge Selection

The importance of a node in a directed graph can be measured by its PageRank. The PageRank of a node is used in a number of application contexts - including ranking websites - and can be interpreted as the average portion of time spent at the node by an infinite random walk. We consider the problem of maximizing the PageRank of a node by selecting some of the edges from a set of edges that are under our control. By applying results from Markov decision theory, we show that an optimal solution to this problem can be found in polynomial time. Our core solution results in a linear programming formulation, but we also provide an alternative greedy algorithm, a variant of policy iteration, which runs in polynomial time, as well. Finally, we show that, under the slight modification for which we are given mutually exclusive pairs of edges, the problem of PageRank optimization becomes NP-hard.Comment: 30 pages, 3 figure