TSP With Locational Uncertainty: The Adversarial Model

In this paper we study a natural special case of the Traveling Salesman Problem (TSP) with point-locational-uncertainty which we will call the adversarial TSP problem (ATSP). Given a metric space (X, d) and a set of subsets R = {R_1, R_2, ...R_n} : R_i subseteq X, the goal is to devise an ordering of the regions, sigma_R, that the tour will visit such that when a single point is chosen from each region, the induced tour over those points in the ordering prescribed by sigma_R is as short as possible. Unlike the classical locational-uncertainty-TSP problem, which focuses on minimizing the expected length of such a tour when the point within each region is chosen according to some probability distribution, here, we focus on the adversarial model in which once the choice of sigma_R is announced, an adversary selects a point from each region in order to make the resulting tour as long as possible. In other words, we consider an offline problem in which the goal is to determine an ordering of the regions R that is optimal with respect to the ``worst\u27\u27 point possible within each region being chosen by an adversary, who knows the chosen ordering. We give a 3-approximation when R is a set of arbitrary regions/sets of points in a metric space. We show how geometry leads to improved constant factor approximations when regions are parallel line segments of the same lengths, and a polynomial-time approximation scheme (PTAS) for the important special case in which R is a set of disjoint unit disks in the plane

Network Optimization on Partitioned Pairs of Points

Given n pairs of points, S = {{p_1, q_1}, {p_2, q_2}, ..., {p_n, q_n}}, in some metric space, we study the problem of two-coloring the points within each pair, red and blue, to optimize the cost of a pair of node-disjoint networks, one over the red points and one over the blue points. In this paper we consider our network structures to be spanning trees, traveling salesman tours or matchings. We consider several different weight functions computed over the network structures induced, as well as several different objective functions. We show that some of these problems are NP-hard, and provide constant factor approximation algorithms in all cases

SpacTor-T5: Pre-training T5 Models with Span Corruption and Replaced Token Detection

Pre-training large language models is known to be extremely resource intensive and often times inefficient, under-utilizing the information encapsulated in the training text sequences. In this paper, we present SpacTor, a new training procedure consisting of (1) a hybrid objective combining span corruption (SC) and token replacement detection (RTD), and (2) a two-stage curriculum that optimizes the hybrid objective over the initial $\tau$ iterations, then transitions to standard SC loss. We show empirically that the effectiveness of the hybrid objective is tied to the two-stage pre-training schedule, and provide extensive analysis on why this is the case. In our experiments with encoder-decoder architectures (T5) on a variety of NLP tasks, SpacTor-T5 yields the same downstream performance as standard SC pre-training, while enabling a 50% reduction in pre-training iterations and 40% reduction in total FLOPs. Alternatively, given the same amount of computing budget, we find that SpacTor results in significantly improved downstream benchmark performance.Comment: 9+13 pages, 5 figure