arXiv subjects

Gianlorenzo D'Angelo

Publications and source records attributed to Gianlorenzo D'Angelo.

At least 19 recordsLinked to original sources

arXiv2026

Towards Dynamic Top-$k$ Simple Shortest Paths

Computing the top-$k$ \textit{simple} shortest paths between two vertices of a graph, i.e., solving the P-KSSP problem, is a fundamental primitive in graph analytics. In modern scenarios, however, queries are rarely one-shot: parameters change as applications adapt their information needs, and recomputing a solution from scratch at every request quickly becomes a bottleneck. We study P-KSSP in a dynamic setting, focusing on changes of the parameter $k$: given a top-$k$ solution for a pair of vertices and a new value $k'$, the goal is to compute a top-$k'$ solution reusing the previously computed paths. Our approach rests on the \textit{pseudo-tree}, a compact representation of a collection of simple $s$--$t$ paths: we show that building it from an arbitrary, unstructured solution amounts to trie construction, and give two constructions running in time linear in the size of the collection. On top of it, we design UPD, a dynamic algorithm that updates a solution upon a change of $k$; we prove its correctness, analyze its complexity, and show that it is a general, solver-agnostic scheme, which starts from the paths alone, is parametric in the routine that generates candidate paths and combinable with known preprocessing techniques, as well as with an online pruning rule that we formalize. While a worst-case update is asymptotically comparable to recomputing from scratch, an extensive experimental evaluation shows that a variant of UPD is the fastest method in the large majority of tested configurations, often by a wide margin (with speedups over recomputation that grow with $k$ and $k'$ and exceed $60\times$), yielding a practical recipe for the configuration to deploy.

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arXiv2026

An Approximation Algorithm for the Connected Maximum Coverage Problem in Directed Graphs

In the Directed rooted Connected Budgeted maximum Coverage problem (\DRCC), we are given a collection of subsets $\mathcal{S}$, defined over a ground set $X$, and a directed graph $G=(V,E)$, where each node is associated with a set of $\mathcal{S}$. Each set in $\mathcal{S}$ has a different cost and each element of $X$ gives a different prize. The goal is to find a subcollection $\mathcal{S}'\subseteq \mathcal{S}$ such that $\mathcal{S}'$ induces an out-tree rooted at a given node, the total cost of the sets in $\mathcal{S}'$ does not exceed a budget $B$, and the total prize of the elements covered by $\mathcal{S}'$ is maximized. In this paper, we provide an algorithm for \DRCC that guarantees an approximation ratio of $O\left(\frac{\sqrt{|V|}\log^2|X|}{ε^2}\right)$, with a budget violation of a factor $1+ε$, where $ε\in (0,1]$. Our algorithm also implies an improved approximation factor for the budgeted node-weighted Steiner problem in directed graphs, a particular case of \DRCC where the prize function is additive, for which we improve from $O\left(\frac{1}{ε^2}|V|^{2/3}\log|V|\right)$ to $O\left(\frac{1}{ε^2}|V|^{1/2}\log^2|V|\right)$.

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arXiv2026

Efficient generation of expected-degree graphs via edge-arrivals

We study the efficient generation of random graphs with a prescribed expected degree sequence, focusing on rank-1 inhomogeneous models in which vertices are assigned weights and edges are drawn independently with probabilities proportional to the product of endpoint weights. We adopt a temporal viewpoint, adding edges to the graph one at a time up to a fixed time horizon, and allowing for self-loops or duplicate edges in the first stage. Then, the simple projection of the resulting multigraph recovers exactly the simple Norros--Reittu random graph, whose expected degrees match the prescribed targets under mild conditions. Building on this representation, we develop an exact generator based on \textit{edge-arrivals} for expected-degree random graphs with running time $O(n+m)$, where $m$ is the number of generated edges, and hence proportional to the output size. This removes the typical vertex sorting used by widely-used fast generator algorithms based on \textit{edge-skipping} for rank-1 expected-degree models, which leads to a total running time of $O(n \log n + m)$. In addition, our algorithm is simpler than those in the literature, easy to implement, and very flexible, thus opening up to extensions to directed and temporal random graphs, generalization to higher-order structures, and improvements through parallelization.

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arXiv2026

Optimizing Distances for Multi-Broadcast in Temporal Graphs

Temporal graphs represent networks in which connections change over time, with edges available only at specific moments. Motivated by applications in logistics, multi-agent information spreading, and wireless networks, we introduce the D-Temporal Multi-Broadcast (D-TMB) problem, which asks for scheduling the availability of edges so that a predetermined subset of sources reach all other vertices while optimizing the worst-case temporal distance D from any source. We show that D-TMB generalizes ReachFast (arXiv:2112.08797). We then characterize the computational complexity and approximability of D-TMB under six definitions of temporal distance D, namely Earliest-Arrival (EA), Latest-Departure (LD), Fastest-Time (FT), Shortest-Traveling (ST), Minimum-Hop (MH), and Minimum-Waiting (MW). For a single source, we show that D-TMB can be solved in polynomial time for EA and LD, while for the other temporal distances it is NP-hard and hard to approximate within a factor that depends on the adopted distance function. We give approximation algorithms for FT and MW. For multiple sources, if feasibility is not assumed a priori, the problem is inapproximable within any factor unless P = NP, even with just two sources. We complement this negative result by identifying structural conditions that guarantee tractability for EA and LD for any number of sources.

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arXiv2025

Approximating Optimal Labelings for Temporal Connectivity

In a temporal graph the edge set dynamically changes over time according to a set of time-labels associated with each edge that indicates at which time-steps the edge is available. Two vertices are connected if there is a path connecting them in which the edges are traversed in increasing order of their labels. We study the problem of scheduling the availability time of the edges of a temporal graph in such a way that all pairs of vertices are connected within a given maximum allowed time $a$ and the overall number of labels is minimized. The problem, known as \emph{Minimum Aged Labeling} (MAL), has several applications in logistics, distribution scheduling, and information spreading in social networks, where carefully choosing the time-labels can significantly reduce infrastructure costs, fuel consumption, or greenhouse gases. The problem MAL has previously been proved to be NP-complete on undirected graphs and \APX-hard on directed graphs. In this paper, we extend our knowledge on the complexity and approximability of MAL in several directions. We first show that the problem cannot be approximated within a factor better than $O(\log n)$ when $a\geq 2$, unless $\text{P} = \text{NP}$, and a factor better than $2^{\log ^{1-ε} n}$ when $a\geq 3$, unless $\text{NP}\subseteq \text{DTIME}(2^{\text{polylog}(n)})$, where $n$ is the number of vertices in the graph. Then we give a set of approximation algorithms that, under some conditions, almost match these lower bounds. In particular, we show that the approximation depends on a relation between $a$ and the diameter of the input graph. We further establish a connection with a foundational optimization problem on static graphs called \emph{Diameter Constrained Spanning Subgraph} (DCSS) and show that our hardness results also apply to DCSS.

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arXiv2023

On the Cost of Demographic Parity in Influence Maximization

Modeling and shaping how information spreads through a network is a major research topic in network analysis. While initially the focus has been mostly on efficiency, recently fairness criteria have been taken into account in this setting. Most work has focused on the maximin criteria however, and thus still different groups can receive very different shares of information. In this work we propose to consider fairness as a notion to be guaranteed by an algorithm rather than as a criterion to be maximized. To this end, we propose three optimization problems that aim at maximizing the overall spread while enforcing strict levels of demographic parity fairness via constraints (either ex-post or ex-ante). The level of fairness hence becomes a user choice rather than a property to be observed upon output. We study this setting from various perspectives. First, we prove that the cost of introducing demographic parity can be high in terms of both overall spread and computational complexity, i.e., the price of fairness may be unbounded for all three problems and optimal solutions are hard to compute, in some case even approximately or when fairness constraints may be violated. For one of our problems, we still design an algorithm with both constant approximation factor and fairness violation. We also give two heuristics that allow the user to choose the tolerated fairness violation. By means of an extensive experimental study, we show that our algorithms perform well in practice, that is, they achieve the best demographic parity fairness values. For certain instances we additionally even obtain an overall spread comparable to the most efficient algorithms that come without any fairness guarantee, indicating that the empirical price of fairness may actually be small when using our algorithms.

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arXiv2023

Improving Fairness in Information Exposure by Adding Links

Fairness in influence maximization has been a very active research topic recently. Most works in this context study the question of how to find seeding strategies (deterministic or probabilistic) such that nodes or communities in the network get their fair share of coverage. Different fairness criteria have been used in this context. All these works assume that the entity that is spreading the information has an inherent interest in spreading the information fairly, otherwise why would they want to use the developed fair algorithms? This assumption may however be flawed in reality -- the spreading entity may be purely \emph{efficiency-oriented}. In this paper we propose to study two optimization problems with the goal to modify the network structure by adding links in such a way that efficiency-oriented information spreading becomes \emph{automatically fair}. We study the proposed optimization problems both from a theoretical and experimental perspective, that is, we give several hardness and hardness of approximation results, provide efficient algorithms for some special cases, and more importantly provide heuristics for solving one of the problems in practice. In our experimental study we then first compare the proposed heuristics against each other and establish the most successful one. In a second experiment, we then show that our approach can be very successful in practice. That is, we show that already after adding a few edges to the networks the greedy algorithm that purely maximizes spread surpasses all fairness-tailored algorithms in terms of ex-post fairness. Maybe surprisingly, we even show that our approach achieves ex-post fairness values that are comparable or even better than the ex-ante fairness values of the currently most efficient algorithms that optimize ex-ante fairness.

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arXiv2022

Approximation algorithms for Node-weighted Steiner Problems: Digraphs with Additive Prizes and Graphs with Submodular Prizes

In the \emph{budgeted rooted node-weighted Steiner tree} problem, we are given a graph $G$ with $n$ nodes, a predefined node $r$, two weights associated to each node modelling costs and prizes. The aim is to find a tree in $G$ rooted at $r$ such that the total cost of its nodes is at most a given budget $B$ and the total prize is maximized. In the \emph{quota rooted node-weighted Steiner tree} problem, we are given a real-valued quota $Q$, instead of the budget, and we aim at minimizing the cost of a tree rooted at $r$ whose overall prize is at least $Q$. For the case of directed graphs with additive prize function, we develop a technique resorting on a standard flow-based linear programming relaxation to compute a tree with good trade-off between prize and cost, which allows us to provide very simple polynomial time approximation algorithms for both the budgeted and the quota problems. For the \emph{budgeted} problem, our algorithm achieves a bicriteria $(1+ε, O(\frac{1}{ε^2}n^{2/3}\ln{n}))$-approximation, for any $ε\in (0, 1]$. For the \emph{quota} problem, our algorithm guarantees a bicriteria approximation factor of $(2, O(n^{2/3}\ln{n}))$. Next, by using the flow-based LP, we provide a surprisingly simple polynomial time $O((1+ε)\sqrt{n} \ln {n})$-approximation algorithm for the node-weighted version of the directed Steiner tree problem, for any $ε>0$. For the case of undirected graphs with monotone submodular prize functions over subsets of nodes, we provide a polynomial time $O(\frac{1}{ε^3}\sqrt{n}\log{n})$-approximation algorithm for the budgeted problem that violates the budget constraint by a factor of at most $1+ε$, for any $ε\in (0, 1]$. Our technique allows us to provide a good approximation also for the quota problem.

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arXiv2022

Budgeted Out-tree Maximization with Submodular Prizes

We consider a variant of the prize collecting Steiner tree problem in which we are given a \emph{directed graph} $D=(V,A)$, a monotone submodular prize function $p:2^V \rightarrow \mathbb{R}^+ \cup \{0\}$, a cost function $c:V \rightarrow \mathbb{Z}^{+}$, a root vertex $r \in V$, and a budget $B$. The aim is to find an out-subtree $T$ of $D$ rooted at $r$ that costs at most $B$ and maximizes the prize function. We call this problem \emph{Directed Rooted Submodular Tree} (\textbf{DRSO}). Very recently, Ghuge and Nagarajan [SODA\ 2020] gave an optimal quasi-polynomial-time $O\left(\frac{\log n'}{\log \log n'}\right)$-approximation algorithm, where $n'$ is the number of vertices in an optimal solution, for the case in which the costs are associated to the edges. In this paper, we give a polynomial-time algorithm for \textbf{DRSO} that guarantees an approximation factor of $O(\sqrt{B}/ε^3)$ at the cost of a budget violation of a factor $1+ε$, for any $ε\in (0,1]$. The same result holds for the edge-cost case, to the best of our knowledge this is the first polynomial-time approximation algorithm for this case. We further show that the unrooted version of \textbf{DRSO} can be approximated to a factor of $O(\sqrt{B})$ without budget violation, which is an improvement over the factor $O(Δ\sqrt{B})$ given in~[Kuo et al.\ IEEE/ACM\ Trans.\ Netw.\ 2015] for the undirected and unrooted case, where $Δ$ is the maximum degree of the graph. Finally, we provide some new/improved approximation bounds for several related problems, including the additive-prize version of \textbf{DRSO}, the maximum budgeted connected set cover problem, and the budgeted sensor cover problem.

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arXiv2022

Sparse Temporal Spanners with Low Stretch

A temporal graph is an undirected graph $G=(V,E)$ along with a function that assigns a time-label to each edge in $E$. A path in $G$ with non-decreasing time-labels is called temporal path and the distance from $u$ to $v$ is the minimum length (i.e., the number of edges) of a temporal path from $u$ to $v$. A temporal $α$-spanner of $G$ is a (temporal) subgraph $H$ that preserves the distances between any pair of vertices in $V$, up to a multiplicative stretch factor of $α$. The size of $H$ is the number of its edges. In this work we study the size-stretch trade-offs of temporal spanners. We show that temporal cliques always admit a temporal $(2k-1)-$spanner with $\tilde{O}(kn^{1+\frac{1}{k}})$ edges, where $k>1$ is an integer parameter of choice. Choosing $k=\lfloor\log n\rfloor$, we obtain a temporal $O(\log n)$-spanner with $\tilde{O}(n)$ edges that has almost the same size (up to logarithmic factors) as the temporal spanner in [Casteigts et al., JCSS 2021] which only preserves temporal connectivity. We then consider general temporal graphs. Since $Ω(n^2)$ edges might be needed by any connectivity-preserving temporal subgraph [Axiotis et al., ICALP'16], we focus on approximating distances from a single source. We show that $\tilde{O}(n/\log(1+\varepsilon))$ edges suffice to obtain a stretch of $(1+\varepsilon)$, for any small $\varepsilon>0$. This result is essentially tight since there are temporal graphs for which any temporal subgraph preserving exact distances from a single-source must use $Ω(n^2)$ edges. We extend our analysis to prove an upper bound of $\tilde{O}(n^2/β)$ on the size of any temporal $β$-additive spanner, which is tight up to polylogarithmic factors. Finally, we investigate how the lifetime of $G$, i.e., the number of its distinct time-labels, affects the trade-off between the size and the stretch of a temporal spanner.

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arXiv2022

Computation and Bribery of Voting Power in Delegative Simple Games

Following Zhang and Grossi~(AAAI 2021), we study in more depth a variant of weighted voting games in which agents' weights are induced by a transitive support structure. This class of simple games is notably well suited to study the relative importance of agents in the liquid democracy framework. We first propose a pseudo-polynomial time algorithm to compute the Banzhaf and Shapley-Shubik indices for this class of game. Then, we study a bribery problem, in which one tries to maximize/minimize the voting power/weight of a given agent by changing the support structure under a budget constraint. We show that these problems are computationally hard and provide several parameterized complexity results.

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arXiv2021

Fairness in Influence Maximization through Randomization

The influence maximization paradigm has been used by researchers in various fields in order to study how information spreads in social networks. While previously the attention was mostly on efficiency, more recently fairness issues have been taken into account in this scope. In this paper, we propose to use randomization as a mean for achieving fairness. Similar to previous works like Fish et al. (WWW '19) and Tsang et al. (IJCAI '19), we study the maximin criterion for (group) fairness. In contrast to their work however, we model the problem in such a way that, when choosing the seed sets, probabilistic strategies are possible rather than only deterministic ones. We introduce two different variants of this probabilistic problem, one that entails probabilistic strategies over nodes (node-based problem) and a second one that entails probabilistic strategies over sets of nodes (set-based problem). While the original deterministic problem involving the maximin criterion has been shown to be inapproximable, interestingly, we show that both probabilistic variants permit approximation algorithms that achieve a constant multiplicative factor of 1-1/e plus an additive arbitrarily small error that is due to the simulation of the information spread. For an experimental study, we provide implementations of multiplicative-weight routines for both problems and compare the achieved fairness values to existing methods. Maybe non-surprisingly, we show that the ex-ante values of the computed probabilistic strategies are significantly larger than the (ex-post) fairness values of previous methods. This indicates that studying fairness via randomization is a worthwhile path to follow. Interestingly and maybe more surprisingly, we observe that even the ex-post fairness values computed by our routines, dominate over the fairness achieved by previous methods on most of the instances tested.

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arXiv2021

Finding single-source shortest $p$-disjoint paths: fast computation and sparse preservers

Let $G$ be a directed graph with $n$ vertices, $m$ edges, and non-negative edge costs. Given $G$, a fixed source vertex $s$, and a positive integer $p$, we consider the problem of computing, for each vertex $t\neq s$, $p$ edge-disjoint paths of minimum total cost from $s$ to $t$ in $G$. Suurballe and Tarjan~[Networks, 1984] solved the above problem for $p=2$ by designing a $O(m+n\log n)$ time algorithm which also computes a sparse \emph{single-source $2$-multipath preserver}, i.e., a subgraph containing $2$ edge-disjoint paths of minimum total cost from $s$ to every other vertex of $G$. The case $p \geq 3$ was left as an open problem. We study the general problem ($p\geq 2$) and prove that any graph admits a sparse single-source $p$-multipath preserver with $p(n-1)$ edges. This size is optimal since the in-degree of each non-root vertex $v$ must be at least $p$. Moreover, we design an algorithm that requires $O(pn^2 (p + \log n))$ time to compute both $p$ edge-disjoint paths of minimum total cost from the source to all other vertices and an optimal-size single-source $p$-multipath preserver. The running time of our algorithm outperforms that of a natural approach that solves $n-1$ single-pair instances using the well-known \emph{successive shortest paths} algorithm by a factor of $Θ(\frac{m}{np})$ and is asymptotically near optimal if $p=O(1)$ and $m=Θ(n^2)$. Our results extend naturally to the case of $p$ vertex-disjoint paths.

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arXiv2021

Improved Approximation Factor for Adaptive Influence Maximization via Simple Greedy Strategies

In the adaptive influence maximization problem, we are given a social network and a budget $k$, and we iteratively select $k$ nodes, called seeds, in order to maximize the expected number of nodes that are reached by an influence cascade that they generate according to a stochastic model for influence diffusion. Differently from the non-adaptive influence maximization problem, where all the seeds must be selected beforehand, here nodes are selected sequentially one by one, and the decision on the $i$th seed is based on the observed cascade produced by the first $i-1$ seeds. We focus on the myopic feedback model, in which we can only observe which neighbors of previously selected seeds have been influenced and on the independent cascade model, where each edge is associated with an independent probability of diffusing influence. Previous works showed that the adaptivity gap is at most $4$, which implies that the non-adaptive greedy algorithm guarantees an approximation factor of $\frac{1}{4}\left(1-\frac{1}{e}\right)$ for the adaptive problem. In this paper, we improve the bounds on both the adaptivity gap and on the approximation factor. We directly analyze the approximation factor of the non-adaptive greedy algorithm, without passing through the adaptivity gap, and show that it is at least $\frac{1}{2}\left(1-\frac{1}{e}\right)$. Therefore, the adaptivity gap is at most $\frac{2e}{e-1}\approx 3.164$. To prove these bounds, we introduce a new approach to relate the greedy non-adaptive algorithm to the adaptive optimum. The new approach does not rely on multi-linear extensions or random walks on optimal decision trees, which are commonly used techniques in the field. We believe that it is of independent interest and may be used to analyze other adaptive optimization problems.

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arXiv2021

When Can Liquid Democracy Unveil the Truth?

In this paper, we investigate the so-called ODP-problem that has been formulated by Caragiannis and Micha [10]. Here, we are in a setting with two election alternatives out of which one is assumed to be correct. In ODP, the goal is to organise the delegations in the social network in order to maximize the probability that the correct alternative, referred to as ground truth, is elected. While the problem is known to be computationally hard, we strengthen existing hardness results by providing a novel strong approximation hardness result: For any positive constant $C$, we prove that, unless $P=NP$, there is no polynomial-time algorithm for ODP that achieves an approximation guarantee of $α\ge (\ln n)^{-C}$, where $n$ is the number of voters. The reduction designed for this result uses poorly connected social networks in which some voters suffer from misinformation. Interestingly, under some hypothesis on either the accuracies of voters or the connectivity of the network, we obtain a polynomial-time $1/2$-approximation algorithm. This observation proves formally that the connectivity of the social network is a key feature for the efficiency of the liquid democracy paradigm. Lastly, we run extensive simulations and observe that simple algorithms (working either in a centralized or decentralized way) outperform direct democracy on a large class of instances. Overall, our contributions yield new insights on the question in which situations liquid democracy can be beneficial.

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arXiv2020

Group-Harmonic and Group-Closeness Maximization -- Approximation and Engineering

Centrality measures characterize important nodes in networks. Efficiently computing such nodes has received a lot of attention. When considering the generalization of computing central groups of nodes, challenging optimization problems occur. In this work, we study two such problems, group-harmonic maximization and group-closeness maximization both from a theoretical and from an algorithm engineering perspective. On the theoretical side, we obtain the following results. For group-harmonic maximization, unless $P=NP$, there is no polynomial-time algorithm that achieves an approximation factor better than $1-1/e$ (directed) and $1-1/(4e)$ (undirected), even for unweighted graphs. On the positive side, we show that a greedy algorithm achieves an approximation factor of $λ(1-2/e)$ (directed) and $λ(1-1/e)/2$ (undirected), where $λ$ is the ratio of minimal and maximal edge weights. For group-closeness maximization, the undirected case is $NP$-hard to be approximated to within a factor better than $1-1/(e+1)$ and a constant approximation factor is achieved by a local-search algorithm. For the directed case, however, we show that, for any $ε<1/2$, the problem is $NP$-hard to be approximated within a factor of $4|V|^{-ε}$. From the algorithm engineering perspective, we provide efficient implementations of the above greedy and local search algorithms. In our experimental study we show that, on small instances where an optimum solution can be computed in reasonable time, the quality of both the greedy and the local search algorithms come very close to the optimum. On larger instances, our local search algorithms yield results with superior quality compared to existing greedy and local search solutions, at the cost of additional running time. We thus advocate local search for scenarios where solution quality is of highest concern.

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arXiv2020

Election Control through Social Influence with Unknown Preferences

The election control problem through social influence asks to find a set of nodes in a social network of voters to be the starters of a political campaign aiming at supporting a given target candidate. Voters reached by the campaign change their opinions on the candidates. The goal is to shape the diffusion of the campaign in such a way that the chances of victory of the target candidate are maximized. Previous work shows that the problem can be approximated within a constant factor in several models of information diffusion and voting systems, assuming that the controller, i.e., the external agent that starts the campaign, has full knowledge of the preferences of voters. However this information is not always available since some voters might not reveal it. Herein we relax this assumption by considering that each voter is associated with a probability distribution over the candidates. We propose two models in which, when an electoral campaign reaches a voter, this latter modifies its probability distribution according to the amount of influence it received from its neighbors in the network. We then study the election control problem through social influence on the new models: In the first model, under the Gap-ETH, election control cannot be approximated within a factor better than $1/n^{o(1)}$, where $n$ is the number of voters; in the second model, which is a slight relaxation of the first one, the problem admits a constant factor approximation algorithm.

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arXiv2020

Better Bounds on the Adaptivity Gap of Influence Maximization under Full-adoption Feedback

In the influence maximization (IM) problem, we are given a social network and a budget $k$, and we look for a set of $k$ nodes in the network, called seeds, that maximize the expected number of nodes that are reached by an influence cascade generated by the seeds, according to some stochastic model for influence diffusion. In this paper, we study the adaptive IM, where the nodes are selected sequentially one by one, and the decision on the $i$th seed can be based on the observed cascade produced by the first $i-1$ seeds. We focus on the full-adoption feedback in which we can observe the entire cascade of each previously selected seed and on the independent cascade model where each edge is associated with an independent probability of diffusing influence. Our main result is the first sub-linear upper bound that holds for any graph. Specifically, we show that the adaptivity gap is upper-bounded by $\lceil n^{1/3}\rceil $, where $n$ is the number of nodes in the graph. Moreover, we improve over the known upper bound for in-arborescences from $\frac{2e}{e-1}\approx 3.16$ to $\frac{2e^2}{e^2-1}\approx 2.31$. Finally, we study $α$-bounded graphs, a class of undirected graphs in which the sum of node degrees higher than two is at most $α$, and show that the adaptivity gap is upper-bounded by $\sqrtα+O(1)$. Moreover, we show that in 0-bounded graphs, i.e. undirected graphs in which each connected component is a path or a cycle, the adaptivity gap is at most $\frac{3e^3}{e^3-1}\approx 3.16$. To prove our bounds, we introduce new techniques to relate adaptive policies with non-adaptive ones that might be of their own interest.

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