Given two vertex sets S and T in a graph, the ST-diameter is the maximum s-t-distance between vertices s ∈ S and t ∈ T . We study the problem of estimating the ST-diameter of graphs that are subject to a small number of transient edge failures. An f-edge fault-tolerant ST-diameter oracle (f-FDO-ST) is a data structure that preprocesses a graph G, sets S, T, and a positive integer f. When queried with a set F of at most f failing edges, the oracle returns an estimate D of the ST-diameter in G - F . The oracle is said to have stretch σ⩾ 1 if diam (G-F,S,T) ⩽D⩽σ· diam (G-F,S,T) . We design new f-FDO-STs by reducing their construction to that of all-pairs and single-source distance sensitivity oracles (f-DSOs). These are data structures that estimate the pairwise graph distances, or respectively the distances from a distinguished source, under up to f failures. We obtain several new trade-offs between the size of the ST-diameter oracles, their stretch guarantees, query and preprocessing times by combining our black-box reductions with f-DSO results from the literature. We further provide a lower bound on the space requirement of approximate ST-diameter oracles. We prove that there exists a family of graphs for which any f-FDO-ST with sensitivity f ⩾ 2 and stretch better than 5/3 requires Ω (n^3/2) bits of space, regardless of the query time.

Let L and f be positive integers. An (L,f)-replacement path covering (RPC) for a graph G is a family 𝒢 of subgraphs such that, for every set F of at most f edges, there is a subfamily 𝒢_F ⊆𝒢 with the following properties. (1) No subgraph in 𝒢_F contains an edge of F. (2) For each pair of vertices s,t that have a shortest path in G-F with at most L edges, one such path also exists in some subgraph in 𝒢_F. The total number |𝒢| of subgraphs is called the covering value. RPCs are an important tools in the design of fault-tolerant data structures. Weimann and Yuster [TALG 2013] presented an RPC with covering value O(f L^f). Karthik and Parter [TALG 2024] showed that Ω( (L/f)^f ) subgraphs are necessary. Recently, Bilò, Chechik, Choudhary, Cohen, and Schirneck [ICALP 2026] devised a new approach for very small sensitivities f = o(log L) with covering value O(f e^f (L/f)^f+o(1)). They also showed that any RPC in the complementary range f = Ω(log L) must contain Ω( (√(f e^f)/L) · (L/f)^f) subgraphs. This left open the question of what is the true covering value. We give two surprisingly simple constructions that improve both the upper and lower bound. This results in a near-tight covering value of ((L+f)^L+f/L^L f^f) ·𝗉𝗈𝗅𝗒(f) for the much wider range of f = O(L).

Consider the following puzzle: a farmland consists of several fields, each occupied by either a farmer, a fox, a chicken, or a caterpillar. Creatures in neighboring fields can swap positions as long as the fox avoids the farmer, the chicken avoids the fox, and the caterpillar avoids the chicken. The objective is to decide whether there exists a sequence of swaps that rearranges the creatures into a desired final configuration, while avoiding any unwanted encounters. The above puzzle can be cast an instance of the colored token swapping problem with k = 4 colors (i.e., creature types), in which only certain pairs of colors can be swapped. We prove that such problem is 𝖯𝖲𝖯𝖠𝖢𝖤-hard even when the graph representing the farmland is planar and cubic. We also show that the problem is polynomial-time solvable when at most three creature types are involved. We do so by providing a more general algorithm deciding instances with arbitrary values of k, as long as the set of all admissible swaps between creature types induces a spanning star. Our results settle a problem explicitly left open in [Yang and Zhang, IPL 2025], which established 𝖯𝖲𝖯𝖠𝖢𝖤-completeness for eight creature types and left the complexity status unresolved when the number of creature types is between three and seven.

An important tool in the design of fault-tolerant graph data structures are (L,f)-replacement path coverings (RPCs). An RPC is a family 𝒢 of subgraphs of a given graph G such that, for every set F of at most f edges, there is a subfamily 𝒢_F ⊆ 𝒢 with the following properties. (1) No subgraph in 𝒢_F contains an edge of F. (2) For each pair of vertices s,t that have a shortest path in G-F with at most L edges, one such path also exists in some subgraph in 𝒢_F. The covering value of the RPC is the total number |𝒢| of subgraphs. The query time is the time needed to compute the subfamily 𝒢_F given the set F. Weimann and Yuster [TALG'13] devised a randomized RPC with covering value O(fL^f) and query time O(f^2 L^f). This was derandomized by Karthik and Parter [TALG'24], who also reduced the query time to O(f^2 L). Their approach uses some heavy algebraic machinery involving error-correcting codes and an increased covering value of O((cfL log n)^f+1) for some constant c > 1. We instead devise a much simpler derandomization via conditional expectations that lowers the covering value back to O(fL^f+o(1)) and decreases the query time to O(f^5/2L^o(1)), assuming f = o(log L). We also investigate the optimal covering value of any (L,f)-replacement path covering (deterministic or randomized) for different parameter ranges. We provide a new randomized construction as well as improving a known lower bound, also by Karthik and Parter. For example, for f = o(log L), we give an RPC with O( (L/f)^f L^o(1)) subgraphs and show that this is tight up to the L^o(1) term.

Our economy, communication, and even our social life crucially depend on networks. These typically emerge from the interaction of many entities, which is why researchers study agent-based models of network formation. In particular, Bilò et al. [1] recently introduced a model where a network is formed by selfish agents corresponding to nodes in a given host network with edges having labels denoting their availability over time. Each agent strategically selects local, i.e., incident, edges to ensure temporal reachability towards everyone at low cost. We explore two novel conceptual features: agents can create non-incident edges, called the global setting, and agents might only want to ensure reachability of a subset of nodes, called the terminal model. For both, we study the existence, structure, and quality of equilibrium networks. For the terminal model, we prove that many properties depend on the number of terminals and we show how to translate equilibrium constructions from the non-terminal model. For the global setting, we show the surprising result that equilibria in the global and the local model are incomparable and we establish a high lower bound on the Price of Anarchy of the global setting that matches the upper bound of the local model. This shows the counter-intuitive fact that allowing agents more flexibility in edge creation does not improve the quality of equilibrium networks. Finally, all of our results hold for the general case where every edge can have multiple labels.

Visibility problems have been investigated for a long time under different assumptions as they pose challenging combinatorial problems and are connected to robot navigation problems. The mutual-visibility problem in a graph G of n vertices asks to find the largest set of vertices X⊆ V(G) , also called μ -set, such that for any two vertices u,v∈ X , there is a shortest u, v-path P where all internal vertices of P are not in X. This means that u and v are visible w.r.t. X. Variations of this problem are known as total, outer, and dual mutual-visibility problems, depending on the visibility property of vertices inside and/or outside X. The mutual-visibility problem and all its variations are known to be -complete on graphs of diameter 4. In this paper, we design a polynomial-time algorithm that finds a μ -set with size ( √(n/D)) , where D is the average distance between any two vertices of G. Moreover, we show inapproximability results for all visibility problems on graphs of diameter 2 and strengthen the inapproximability ratios for graphs of diameter 3 or larger. More precisely, for graphs of diameter at least 3 and for every constant ε > 0 , we show that mutual-visibility and dual mutual-visibility problems are not approximable within a factor of n^1/3-ε , while outer and total mutual-visibility problems are not approximable within a factor of n^1/2 - ε , unless = . Furthermore, in the extended version of this paper we study the relationship between the mutual-visibility number and the general position number in which no three distinct vertices u, v, w of X belong to any shortest path of G.

We study the minimum Monitoring Edge Geodetic Set (MEG-Set) problem introduced in [Foucaud et al., CALDAM'23]: given a graph G, we say that an edge is monitored by a pair u, v of vertices if all shortest paths between u and v traverse e; the goal is to find a subset M of vertices of G such that each edge of G is monitored by at least one pair of vertices in M, and |M| is minimized. In this paper, we prove that all polynomial-time approximation algorithms for the minimum MEG-Set problem must have an approximation ratio of Omega(log n), unless P = NP. To the best of our knowledge, this is the first non-constant inapproximability result known for this problem. We also strengthen the known NP-hardness of the problem on 2-apex graphs by showing that the same result holds for 1-apex graphs. This leaves open the question of determining whether the problem remains NP-hard on planar (i.e., 0-apex) graphs. On the positive side, we design an algorithm that computes good approximate solutions for hereditary graph classes that admit efficiently computable balanced separators of truly sublinear size. This immediately yields polynomial-time approximation algorithms achieving an approximation ratio of O(n (1/4) root log n) on planar graphs, graphs with bounded genus, and k-apex graphs with k = O(n(1/4)). On graphs with bounded treewidth, we obtain an approximation ratio of O(log(3/2) n). This compares favorably with the best-known approximation algorithm for general graphs, which achieves an approximation ratio of O(root n log n) via a simple reduction to the Set Cover problem. 2012 ACM Subject Classification Theory of computation -> Design and analysis of algorithms

We investigate the trade-offs between size and query time that are attainable by fastest-path oracles for temporal trees, i.e., data structures capable of reporting the duration of a fastest point-to-point temporal path, possibly constrained to be within a given time window. For any parameter t ≥ 1 of choice, we provide an oracle of size O(M + M^2/t^2) answering queries in time O(t log (1+L) ) , where L is the maximum number of time-labels of a single edge, and M is the overall number of (not necessarily distinct) time-labels. We also prove a conditional lower bound that relies on the difficulty of the Set Disjointness problem, showing that the trade-offs obtained by our oracles are tight, up to polylogarithmic factors. We also investigate the case of general temporal graphs and we provide some evidences that the problem becomes much more challenging even for the special case of temporal reachability queries with no time constraints.

We design sensitivity oracles for error-prone networks. For a network problem Π, the data structure preprocesses a network G=(V,E) and sensitivity parameter f such that, for any set F of up to f link or node failures, it can report the solution of Π in G-F. We study three network problems Π. - L-Hop Shortest Path: Given s,t in V, is there a shortest s-t-path in G-F with at most L links? - k-Path: Does G-F contain a simple path with k links? - k-Clique: Does G-F contain a clique of k nodes? Our main technical contribution is a new construction of (L,f)-replacement path coverings ((L,f)-RPC) in the parameter realm where f = o(log L). An (L,f)-RPC is a family G' of subnetworks of G which, for every set F of at most f links, has a subfamily G'_F such that (i) no subnetwork in G'_F contains a link of F and (ii) for each s,t in V, if G-F contains a shortest s-t-path with at most L links, then some subnetwork in G'_F retains at least one such path. Our (L,f)-RPC has almost the same size as the one by Weimann and Yuster (2013) but it improves the time to query G'_F from Õ(f^2 L^f) to Õ(f^(5/2) L^o(1)). It also improves over the size and query time of the (L,f)-RPC by Karthik and Parter (2021) by nearly a factor of L. From this construction, we derive oracles for L-Hop Shortest Path, k-Path, and k-Clique. Notably, our solution for k-Path improves the query time of the one by Bilò for f=o(log k).

A distance oracle (DO) for a graph G is a data structure that, when queried with vertices s, t, returns an estimate (d) over cap (s, t) of their distance in G. The oracle has stretch (alpha, beta) if the estimate satisfies d(s, t) <= (d) over cap (s, t) <= alpha.d(s, t) + beta. An f-edge fault-tolerant distance sensitivity oracle (f-DSO) additionally receives a set F of up to f edges and estimates the distance in G-F. Our first contribution is the design of new distance oracles with subquadratic space for undirected graphs. We show that introducing a small additive stretch beta > 0 allows one to make the multiplicative stretch alpha arbitrarily small. This sidesteps a known lower bound of alpha >= 3 (for beta = 0 and subquadratic space) [Thorup & Zwick, JACM 2005]. We present a DO for graphs with edge weights in [0, W] that, for any positive integer l and any c is an element of (0, l/2], has stretch (1 + 1/l, 2W), space (O) over tilde (n(2-c/l)), and query time O(n(c)), generalizing results by Agarwal and Godfrey [SODA 2013] to arbitrarily dense graphs. Our second contribution is a framework that turns an (alpha, beta)-stretch DO for unweighted graphs into an (alpha(1+epsilon), beta)-stretch f-DSO with sensitivity f = o(log(n)/ log log n) retaining subquadratic space. This generalizes a result by Bilo, Chechik, Choudhary, Cohen, Friedrich, Krogmann, and Schirneck [TheoretiCS 2024]. Combining the framework with our new DO gives an f-DSO that, for any gamma is an element of (0, (l+1)/2], has stretch ((1+1/l)(1+epsilon), 2), space n(2-) (gamma/(t+1)(f+1)+o(1))/epsilon(f+2), and query time (O) over tilde (n(gamma)/epsilon(2)). This is the first f-DSO with subquadratic space, near-additive stretch, and sublinear query time. The full version of this work can be found at https://arxiv.org/abs/2408.10014.

Network Creation Games are a well-known approach for explaining and analyzing the structure, quality and dynamics of real-world networks like the Internet and other infrastructure networks which evolved via the interaction of selfish agents without a central authority. In these games selfish agents which correspond to nodes in a network strategically buy incident edges to improve their centrality. However, past research on these games has only considered the creation of networks with unit-weight edges. In practice, e.g. when constructing a fiber-optic network, the choice of which nodes to connect and also the induced price for a link crucially depends on the distance between the involved nodes and such settings can be modeled via edge-weighted graphs. We incorporate arbitrary edge weights by generalizing the well-known model by Fabrikant et al.[PODC'03] to edge-weighted host graphs and focus on the geometric setting where the weights are induced by the distances in some metric space. In stark contrast to the state-of-the-art for the unit-weight version, where the Price of Anarchy is conjectured to be constant and where resolving this is a major open problem, we prove a tight non-constant bound on the Price of Anarchy for the metric version and a slightly weaker upper bound for the non-metric case. Moreover, we analyze the existence of equilibria, the computational hardness and the game dynamics for several natural metrics. The model we propose can be seen as the game-theoretic analogue of a variant of the classical Network Design Problem. Thus, low-cost equilibria of our game correspond to decentralized and stable approximations of the optimum network design.

In a temporal forest each edge has an associated set of time labels that specify the time instants in which the edges are available. A temporal path from vertex u to vertex v in the forest is a selection of a label for each edge in the unique path from u to v, assuming it exists, such that the labels selected for any two consecutive edges are non-decreasing. We design linear-size data structures that maintain a temporal forest of rooted trees under addition and deletion of both edge labels and singleton vertices, insertion of root-to-node edges, and removal of edges with no labels. Such data structures can answer temporal reachability, earliest arrival, and latest departure queries. All queries and updates are handled in polylogarithmic worst-case time. Our results can be adapted to deal with latencies. More precisely, all the worst-case time bounds are asymptotically unaffected when latencies are uniform. For arbitrary latencies, the update time becomes amortized in the incremental case where only label additions and edge/singleton insertions are allowed as well as in the decremental case in which only label deletions and edge/singleton removals are allowed. To the best of our knowledge, the only previously known data structure supporting temporal reachability queries is due to Brito, Albertini, Casteigts, and Travencolo [Social Network Analysis and Mining, 2021], which can handle general temporal graphs, answers queries in logarithmic time in the worst case, but requires an amortized update time that is quadratic in the number of vertices, up to polylogarithmic factors.

There was a mix-up in Escher's bar and n customers sitting at the same table have each received a beer ordered by somebody else in the party. The drinks can be rearranged by swapping them in pairs, but the eccentric table shape only allows drinks to be exchanged between people sitting on opposite sides of the table. We study the problem of finding the minimum number of swaps needed so that each customer receives its desired beer before it gets warm. Formally, we consider the COLORED TOKEN SWAPPING problem on complete bipartite graphs. This problem is known to be solvable in polynomial time when all ordered drinks are different Yamanaka et al., FUN 2014], but no results are known for the more general case in which multiple people in the party can order the same beer. We prove that COLORED TOKEN SWAPPING on complete bipartite graphs is NP -hard and that it is fixed-parameter tractable when parameterized by the number of distinct types of beer served by the bar.

A spanner is a sparse subgraph of a given graph G which preserves distances, measured w.r.t. some distance metric, up to a multiplicative stretch factor. This paper addresses the problem of constructing graph spanners w.r.t. the group Steiner metric, which generalizes the recently introduced beer distance metric. In such a metric we are given a collection of groups of required vertices, and we measure the distance between two vertices as the length of the shortest path between them that traverses at least one required vertex from each group. We discuss the relation between group Steiner spanners and classic spanners and we show that they exhibit strong ties with sourcewise spanners w.r.t. the shortest path metric. Nevertheless, group Steiner spanners capture several interesting scenarios that are not encompassed by existing spanners. This happens, e.g., for the singleton case, in which each group consists of a single required vertex, thus modeling the setting in which routes need to traverse certain points of interests (in any order). We provide several constructions of group Steiner spanners for both the all-pairs and single-source case, which exhibit various size-stretch trade-offs. Notably, we provide spanners with almost-optimal trade-offs for the singleton case. Moreover, some of our spanners also yield novel trade-offs for classical sourcewise spanners. Finally, we also investigate the query times that can be achieved when our spanners are turned into group Steiner distance oracles with the same size, stretch, and building time.

We study the Solo-Chess problem which has been introduced in [Aravind et al., FUN 2022]. This is a single-player variant of chess in which the player must clear all but one piece from the board via a sequence captures while ensuring that each piece performs at most as many captures as its budget allows. The time complexity of finding a winning sequence of captures has already been pinpointed for several combinations of piece types and initial budgets. We contribute to a better understanding of the computational landscape of Solo-Chess by closing two problems left open in [Aravind et al., FUN 2022]. Namely, we show that Solo-Chess is hard even when all pieces are restricted to only rooks with budget exactly 2, or only knights with budget exactly 11.

Reoptimization is a setting in which we are given a good approximate solution of an optimization problem instance and a local modification that slightly changes the instance. The main goal is that of finding a good approximate solution of the modified instance. We investigate one of the most studied scenarios in reoptimization known as Steiner tree reoptimization. Steiner tree reoptimization is a collection of strongly $$\textsf {NP}$$ NP -hard optimization problems that are defined on top of the classical Steiner tree problem and for which several constant-factor approximation algorithms have been designed in the last decades. In this paper we improve upon all these results by developing a novel technique that allows us to design polynomial-time approximation schemes. Remarkably, prior to this paper, no approximation algorithm better than recomputing a solution from scratch was known for the elusive scenario in which the cost of a single edge decreases. Our results are best possible since none of the problems addressed in this paper admits a fully polynomial-time approximation scheme, unless $$\textsf {P}=\textsf {NP}$$ P = NP

Given an undirected connected graph G = (V(G), E(G)) on n vertices, the minimum Monitoring Edge-Geodetic Set (MEG-set) problem asks to find a subset M ⊆ V(G) of minimum cardinality such that, for every edge e ∈ E(G), there exist x,y ∈ M for which all shortest paths between x and y in G traverse e. We show that, for any constant c < 1/2, no polynomial-time (c log n)-approximation algorithm for the minimum MEG-set problem exists, unless 𝖯 = 𝖭𝖯.

An $f$-edge fault-tolerant distance sensitive oracle ($f$-DSO) with stretch $\sigma \geq 1$ is a data structure that preprocesses an input graph $G$. When queried with the triple $(s,t,F)$, where $s, t \in V$ and $F \subseteq E$ contains at most $f$ edges of $G$, the oracle returns an estimate $\widehat{d}_{G-F}(s,t)$ of the distance $d_{G-F}(s,t)$ between $s$ and $t$ in the graph $G-F$ such that $d_{G-F}(s,t) \leq \widehat{d}_{G-F}(s,t) \leq \sigma d_{G-F}(s,t)$. For any positive integer $k \ge 2$ and any $0 < \alpha < 1$, we present an $f$-DSO with sensitivity $f = o(\log n/\log\log n)$, stretch $2k-1$, space $O(n^{1+\frac{1}{k}+\alpha+o(1)})$, and an $\widetilde{O}(n^{1+\frac{1}{k} - \frac{\alpha}{k(f+1)}})$ query time. Prior to our work, there were only three known $f$-DSOs with subquadratic space. The first one by Chechik et al. [Algorithmica 2012] has a stretch of $(8k-2)(f+1)$, depending on $f$. Another approach is storing an $f$-edge fault-tolerant $(2k-1)$-spanner of $G$. The bottleneck is the large query time due to the size of any such spanner, which is $\Omega(n^{1+1/k})$ under the Erd\H{o}s girth conjecture. Bil\`o et al. [STOC 2023] gave a solution with stretch $3+\varepsilon$, query time $O(n^{\alpha})$ but space $O(n^{2-\frac{\alpha}{f+1}})$, approaching the quadratic barrier for large sensitivity. In the realm of subquadratic space, our $f$-DSOs are the first ones that guarantee, at the same time, large sensitivity, low stretch, and non-trivial query time. To obtain our results, we use the approximate distance oracles of Thorup and Zwick [JACM 2005], and the derandomization of the $f$-DSO of Weimann and Yuster [TALG 2013], that was recently given by Karthik and Parter [SODA 2021].

Despite extensive research on distance oracles, there are still large gaps between the best constructions for spanners and distance oracles. Notably, there exist sparse spanners with a multiplicative stretch of $1+\varepsilon$ plus some additive stretch. A fundamental open problem is whether such a bound is achievable for distance oracles as well. Specifically, can we construct a distance oracle with multiplicative stretch better than 2, along with some additive stretch, while maintaining subquadratic space complexity? This question remains a crucial area of investigation, and finding a positive answer would be a significant step forward for distance oracles. Indeed, such oracles have been constructed for sparse graphs. However, in the more general case of dense graphs, it is currently unknown whether such oracles exist. In this paper, we contribute to the field by presenting the first distance oracles that achieve a multiplicative stretch of $1+\varepsilon$ along with a small additive stretch while maintaining subquadratic space complexity. Our results represent an advancement particularly for constructing efficient distance oracles for dense graphs. In addition, we present a whole family of oracles that, for any positive integer $k$, achieve a multiplicative stretch of $2k-1+\varepsilon$ using $o(n^{1+1/k})$ space.