Connectivity in temporal graphs relies on the notion of temporal paths, in which edges follow a chronological order (either strict or non-strict). In this work, we investigate the question of how to make a temporal graph connected. More precisely, we tackle the problem Temporal Connectivity Augmentation (TCA) which consists of finding, among a set of proposed temporal edges, the smallest subset such that its addition makes the graph temporally connected. We study the complexity of this problem and variants, under restricted lifespan of the graph, i.e. the maximum time step in the graph. Our main result on TCA is that for any fixed lifespan at least 2, it is NP-complete in both the strict (paths follows a strict chronological order) and non-strict (paths follow a non-strict chronological order) setting. We additionally provide a set of restrictions in the non-strict setting which makes the problem solvable in polynomial time and design an algorithm achieving this complexity. Interestingly, we prove that the source variant (making a given vertex a source in the augmented graph) is as difficult as TCA. On the opposite, we prove that the version where a list of connectivity demands has to be satisfied is solvable in polynomial time, when the size of the list is fixed. Finally, we highlight a variant of the previous case for which even with two pairs the problem is already NP-hard.

We consider a learning augmented framework for NP-hard permutation problems. The algorithm has access to predictions telling, given a pair $u,v$ of elements, whether $u$ is before $v$ or not in an optimal solution. Building on the work of Braverman and Mossel (SODA 2008), we show that for a class of optimization problems including scheduling, network design and other graph permutation problems, these predictions allow to solve them in polynomial time with high probability, provided that predictions are true with probability at least $1/2+\epsilon$. Moreover, this can be achieved with a parsimonious access to the predictions.

This work deals with resource leveling problems. A set of jobs is given as well as a resource level representing a capacity that may be exceeded at some cost. Jobs have integer processing times, must be scheduled non-preemptively and consume one unit of resource while processed. More precisely, the objective to be maximized is the resource use below the resource level, i.e., the complementary of the total overload cost. Two main families of problems are investigated: either with or without precedence constraints. The case with no precedence constraints is shown to admit an EPTAS; a quasi-linear time approximation algorithm with constant ratio 7 8 is also provided. The case with precedence constraints is shown to be significantly harder to solve as it does not admit a PTAS under some classical complexity assumption. Approximation algorithms with constant ratios are provided for special cases with in-tree precedence graph or with fixed resource level.

We study in this paper single-peakedness on arbitrary graphs. Given a collection of preferences (rankings of alternatives), we aim at determining a connected graph G on which the preferences are single-peaked, in the sense that all the preferences are traversals of G. Note that a collection of preferences is always single-peaked on the complete graph. We propose an Integer Linear Programming formulation (ILP) of the problem of minimizing the number of edges in G or the maximum degree of a vertex in G. We prove that both problems are NP-hard in the general case. However, we show that if the optimal number of edges is m−1 (where m is the number of candidates) then any optimal extreme point solution of the continuous relaxation of the ILP is integer and thus the integrality constraints can be relaxed. This provides an alternative proof of the polynomial time complexity of recognizing single-peaked preferences on a tree. We prove the same result for the case of a path (an axis), providing here also an alternative proof of polynomiality of the recognition problem. Furthermore, we provide a polynomial time procedure to recognize single-peaked preferences on a pseudotree (a connected graph that contains at most one cycle). We also give some experimental results, both on real and synthetic datasets.

A set of voters’ preferences on a set of candidates is 2-Euclidean if candidates and voters can be mapped to the plane so that the preferences of each voter decrease with the Euclidean distance between her position and the positions of candidates. Based on geometric properties, we propose a recognition algorithm, that returns either “yes” (together with a planar positioning of candidates and voters) if the preferences are 2-Euclidean, or “no” if it is able to find a concise certificate that they are not, or “unknown” if a time limit is reached. Our algorithm outperforms a quadratically constrained programming solver achieving the same task, both in running times and the percentage of instances it is able to recognize. In the numerical tests conducted on the PrefLib library of preferences, 91.5% (resp. 4.5%) of the available sets of complete strict orders are proven not to be (resp. to be) 2-Euclidean, and the status of only 4.5% of them could not be decided. Furthermore, for instances involving 5 (resp. 6, 7) candidates, we were able to find planar representations that are compatible with 87.4% (resp. 58.1%, 60.1%) of voters’ preferences.

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.

Many systems have to be maintained while the underlying constraints, costs and/or profits change over time. Although the state of a system may evolve during time, a non-negligible transition cost is incurred for transitioning from one state to another. In order to model such situations, we look at a recently introduced multistage model where the input is a sequence of instances (one for each time step), and the goal is to find a sequence of solutions (one for each time step) that are both (i) near optimal for each time step and (ii) as stable as possible. We propose a PTAS for the Multistage Knapsack problem. This is the first approximation scheme for a combinatorial optimization problem in the considered multistage setting, and its existence contrasts with the inapproximability results for other combinatorial optimization problems that are even polynomial-time solvable in the static case.

Numerous combinatorial optimization problems (knapsack, maximum-weight matching, etc.) can be expressed as subset maximization problems: One is given a ground set $$N=\{1,\dots ,n\}$$ , a collection $$\mathcal {F}\subseteq 2^N$$ of subsets thereof such that $$\emptyset \in \mathcal {F}$$ , and an objective (profit) function $$p:\mathcal {F}\rightarrow \mathbb {R}_+$$ . The task is to choose a set $$S\in \mathcal {F}$$ that maximizes p(S). We consider the multistage version (Eisenstat et al., Gupta et al., both ICALP 2014) of such problems: The profit function $$p_t$$ (and possibly the set of feasible solutions $$\mathcal {F}_t$$ ) may change over time. Since in many applications changing the solution is costly, the task becomes to find a sequence of solutions that optimizes the trade-off between good per-time solutions and stable solutions taking into account an additional similarity bonus. As similarity measure for two consecutive solutions, we consider either the size of the intersection of the two solutions or the difference of n and the Hamming distance between the two characteristic vectors. We study multistage subset maximization problems in the online setting, that is, $$p_t$$ (along with possibly $$\mathcal {F}_t$$ ) only arrive one by one and, upon such an arrival, the online algorithm has to output the corresponding solution without knowledge of the future. We develop general techniques for online multistage subset maximization and thereby characterize those models (given by the type of data evolution and the type of similarity measure) that admit a constant-competitive online algorithm. When no constant competitive ratio is possible, we employ lookahead to circumvent this issue. When a constant competitive ratio is possible, we provide almost matching lower and upper bounds on the best achievable one.

Liquid democracy is a collective decision making paradigm which lies between direct and representative democracy. One main feature of liquid democracy is that voters can delegate their votes in a transitive manner so that: A delegates to B and B delegates to C leads to A delegates to C. Unfortunately, because voters' preferences over delegates may be conflicting, this process may not converge. There may not even exist a stable state (also called equilibrium). In this paper, we investigate the stability of the delegation process in liquid democracy when voters have restricted types of preference on the agent representing them (e.g., single-peaked preferences). We show that various natural structures of preference guarantee the existence of an equilibrium and we obtain both tractability and hardness results for the problem of computing several equilibria with some desirable properties.

We investigate the issue of manipulability for social ranking rules, where the goal is to rank individuals given the ranking of coalitions formed by them and each individual prefers to reach the highest positions in the social ranking. This problem lies at the intersection of computational social choice and the algorithmic theory of power indices. Different social ranking rules have been recently proposed and studied from an axiomatic point of view. In this paper, we focus on rules representing three classical approaches in social choice theory: the marginal contribution approach, the lexicographic approach and the (ceteris paribus) majority one. We first consider some particular members of these families analysing their resistance to a malicious behaviour of individuals. Then, we analyze the computational complexity of manipulation, and complete our theoretical results with simulations in order to analyse the manipulation frequencies and to assess the effects of manipulations.

A common way of dealing with the paradoxes of preference aggregation consists in restricting the domain of admissible preferences. The most well-known such restriction is single-peakedness. In this paper we focus on the problem of determining whether a given profile is single-peaked with respect to some axis, and on the computation of such an axis. This problem has already been considered in [2]; we give here a more efficient algorithm and address some related issues, such as the number of orders that may be compatible with a given profile, or the communication complexity of preference aggregation under the single-peakedness assumption.