We study online interval scheduling in the irrevocable setting, where each interval must be immediately accepted or rejected upon arrival. The objective is to maximize the total length of accepted intervals while ensuring that no two accepted intervals overlap. We consider this problem in a learning-augmented setting, where the algorithm has access to (machine-learned) predictions. The goal is to design algorithms that leverage these predictions to improve performance while maintaining robust guarantees in the presence of prediction errors. Our main contribution is the SemiTrust-and-Switch framework, which provides a unified approach for combining prediction-based and classical interval scheduling algorithms. This framework applies to both deterministic and randomized algorithms and captures the trade-off between consistency (performance under accurate predictions) and robustness (performance under adversarial inputs). Moreover, we provide lower bounds, proving the tightness of this framework in particular settings. We further design a randomized algorithm that smoothly interpolates between prediction-based and robust algorithms. This algorithm achieves both robustness and smoothness-its performance degrades gracefully with the quality of the prediction.

The anti-Ramsey numbers are a fundamental notion in graph theory, introduced in 1978, by Erdős, Simonovits and Sós. For given graphs G and H the anti-Ramsey number ar ( G , H ) is defined to be the maximum number k such that there exists an assignment of k colors to the edges of G in which every copy of H in G has at least two edges with the same color. Usually, combinatorists study extremal values of anti-Ramsey numbers for various classes of graphs. There are works on the computational complexity of the problem when H is a star. Along this line of research, we study the complexity of computing the anti-Ramsey number ar ( G , P k ) , where P k is a path of length k . First, we observe that when k is close to n (the number of vertices in G ), the problem is hard; hence, the challenging part is the computational complexity of the problem when k is a fixed constant. We provide a characterization of the problem for paths of constant length. Our first main contribution is to prove that computing ar ( G , P k ) for every integer k ≥ 3 is NP-hard. We obtain this by providing several structural properties of such coloring in graphs. We also study the exact complexity of the precolored version and show that there is no subexponential algorithm for the problem unless ETH fails for any fixed constant k .

In the strategic facility location problem, a set of agents report their locations in a metric space and the goal is to use these reports to open a new facility, minimizing an aggregate distance measure from the agents to the facility. However, agents are strategic and may misreport their locations to influence the facility’s placement in their favor. The aim is to design truthful mechanisms, ensuring agents cannot gain by misreporting. This problem was recently revisited through the learning-augmented framework, aiming to move beyond worst-case analysis and design truthful mechanisms that are augmented with (machine-learned) predictions. The focus of this prior work was on mechanisms that are deterministic and augmented with a prediction regarding the optimal facility location. In this paper, we provide a deeper understanding of this problem by exploring the power of randomization as well as the impact of different types of predictions on the performance of truthful learning-augmented mechanisms. We study both the single-dimensional and the Euclidean case and provide upper and lower bounds regarding the achievable approximation of the optimal egalitarian social cost.

We study the online Traveling Salesman Problem (TSP) on the line augmented with machine-learned predictions. In the classical problem, there is a stream of requests released over time along the real line. The goal is to minimize the makespan of the algorithm. We distinguish between the open variant and the closed one, in which we additionally require the algorithm to return to the origin after serving all requests. The state of the art is a 1.64-competitive algorithm and a 2.04-competitive algorithm for the closed and open variants, respectively. In both cases, a tight lower bound is known. In both variants, our primary prediction model involves predicted positions of the requests. We introduce algorithms that (i) obtain a tight 1.5 competitive ratio for the closed variant and a 1.66 competitive ratio for the open variant in the case of perfect predictions, (ii) are robust against unbounded prediction error, and (iii) are smooth, i.e., their performance degrades gracefully as the prediction error increases. Moreover, we further investigate the learning-augmented setting in the open variant by additionally considering a prediction for the last request served by the optimal offline algorithm. Our algorithm for this enhanced setting obtains a 1.33 competitive ratio with perfect predictions while also being smooth and robust, beating the lower bound of 1.44 we show for our original prediction setting for the open variant. Also, we provide a lower bound of 1.25 for this enhanced setting.

We study the Online Traveling Salesperson Problem (OLTSP) with predictions. In OLTSP, a sequence of initially unknown requests arrive over time at points (locations) of a metric space. The goal is, starting from a particular point of the metric space (the origin), to serve all these requests while minimizing the total time spent. The server moves with unit speed or is "waiting" (zero speed) at some location. We consider two variants: in the open variant, the goal is achieved when the last request is served. In the closed one, the server additionally has to return to the origin. We adopt a prediction model, introduced for OLTSP on the line [Gouleakis et al., 2023], in which the predictions correspond to the locations of the requests and extend it to more general metric spaces. We first propose an oracle-based algorithmic framework, inspired by previous work [Bampis et al., 2023]. This framework allows us to design online algorithms for general metric spaces that provide competitive ratio guarantees which, given perfect predictions, beat the best possible classical guarantee (consistency). Moreover, they degrade gracefully along with the increase in error (smoothness), but always within a constant factor of the best known competitive ratio in the classical case (robustness). Having reduced the problem to designing suitable efficient oracles, we describe how to achieve this for general metric spaces as well as specific metric spaces (rings, trees and flowers), the resulting algorithms being tractable in the latter case. The consistency guarantees of our algorithms are tight in almost all cases, and their smoothness guarantees only suffer a linear dependency on the error, which we show is necessary. Finally, we provide robustness guarantees improving previous results.

We consider the problem of guaranteeing maximin-share ($\MMS$) when allocating a set of indivisible items to a set of agents with fractionally subadditive ($\XOS$) valuations. For $\XOS$ valuations, it has been previously shown that for some instances no allocation can guarantee a fraction better than $1/2$ of maximin-share to all the agents. Also, a deterministic allocation exists that guarantees $0.219225$ of the maximin-share of each agent. Our results involve both deterministic and randomized allocations. On the deterministic side, we improve the best approximation guarantee for fractionally subadditive valuations to $3/13 = 0.230769$. We develop new ideas on allocating large items in our allocation algorithm which might be of independent interest. Furthermore, we investigate randomized algorithms and the Best-of-both-worlds fairness guarantees. We propose a randomized allocation that is $1/4$-$\MMS$ ex-ante and $1/8$-$\MMS$ ex-post for $\XOS$ valuations. Moreover, we prove an upper bound of $3/4$ on the ex-ante guarantee for this class of valuations.

We consider the problem of maximizing the Nash social welfare when allocating a set G of indivisible goods to a set N of agents. We study instances, in which all agents have 2-value additive valuations: The value of every agent for every good is either p or q, where p and q are integers and p2. In terms of approximation, we present positive and negative results for general p and q. We show that our algorithm obtains an approximation ratio of at most 1.0345. Moreover, we prove that the problem is APX-hard, with a lower bound of 1.000015 achieved at p/q = 4/5.

The anti-Ramsey numbers are a fundamental notion in graph theory, introduced in 1978, by Erdös, Simonovits and Sós. For given graphs G and H the anti-Ramsey number ar(G,H) is defined to be the maximum number k such that there exists an assignment of k colors to the edges of G in which every copy of H in G has at least two edges with the same color. Usually, combinatorists study extremal values of anti-Ramsey numbers for various classes of graphs. There are works on the computational complexity of the problem when H is a star. Along this line of research, we study the complexity of computing the anti-Ramsey number ar(G,P_k), where P_k is a path of length k. First, we observe that when k is close to n, the problem is hard; hence, the challenging part is the computational complexity of the problem when k is a fixed constant. We provide a characterization of the problem for paths of constant length. Our first main contribution is to prove that computing ar(G,P_k) for every integer k > 2 is NP-hard. We obtain this by providing several structural properties of such coloring in graphs. We investigate further and show that approximating ar(G,P₃) to a factor of n^{-1/2 - ε} is hard already in 3-partite graphs, unless P = NP. We also study the exact complexity of the precolored version and show that there is no subexponential algorithm for the problem unless ETH fails for any fixed constant k. Given the hardness of approximation and parametrization of the problem, it is natural to study the problem on restricted graph families. Along this line, we first introduce the notion of color connected coloring, and, employing this structural property, we obtain a linear time algorithm to compute ar(G,P_k), for every integer k, when the host graph, G, is a tree.