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.
Graph Drawing aims to make graphs visually comprehensible while faithfully representing their structure. In layered drawings, each vertex is drawn on one of k given horizontal lines and edges are drawn as y-monotone curves. A key ingredient for constructing such drawings is the One-Sided Bipartite Crossing Minimisation (OBCM) problem: given two layers of a bipartite graph and a fixed horizontal order of the vertices on the first layer, the task is to order the vertices on the second layer to minimise the number of edge crossings. We analyse the performance of simple evolutionary algorithms for OBCM and compare different operators for permutations: exchanging two elements, swapping adjacent elements and jumping an element to a new position. We show that on instances that can be drawn crossing-free, OBCM corresponds to a generalised sorting problem. We provide novel and tight lower bounds of Ω (n^2 log n) for sorting with exchanges and jumps, respectively. This solves a long-standing open problem by Scharnow, Tinnefeld, and Wegener (J. Math. Model. Algorithm 3(4):349–366, 2005). For the simplest and cheapest mutation operator, swap (swapping adjacent elements), we give a tight runtime bound of Θ (n^2) via a parallel BubbleSort algorithm and a delay sequence argument. This proves that the simplest and cheapest mutation operator is also the fastest for sorting and solving planar OBCM instances.
We study the problem of determining whether a given random matching can be implemented as a lottery over weakly stable deterministic matchings – a property known as ex-post stability. This concept arises in randomized allocation mechanisms such as school choice, where stability in each realized outcome is essential for fairness. Despite its importance in practice, the computational complexity of verifying ex-post stability has remained unresolved. We settle this question by showing that testing ex-post stability is NP-complete, even under highly restricted conditions – specifically, when both sides have dichotomous preferences or one of the sides has strict preferences. On the positive side, we present an integer programming formulation that finds a decomposition of a random matching with maximum weight on stable matchings. We also consider stronger versions of ex-post stability (in particular robust ex-post stability and ex-post strong stability) and prove that they can be tested in polynomial time.
Let Γ be an arrangement of Jordan curves in the plane, i.e., simple closed curves in the plane. For any curve γ∈Γ , we denote the bounded region enclosed by γ as γ̃ . We say that Γ is non-piercing if for any two curves α , β∈Γ , both α̃ ∖ β̃ and β̃ ∖ β̃ is connected. A non-piercing arrangement of curves generalizes a set of 2-intersecting curves in which each pair of curves intersect in at most two points. Snoeyink and Hershberger (“Sweeping Arrangements of Curves”, SoCG ’89) proved that if we are given an arrangement Γ of 2-intersecting curves and a sweep curve γ∈Γ , then the arrangement can be swept by γ while always maintaining the 2-intersecting property of the curves in Γ . We generalize the result of Snoeyink and Hershberger to the setting of non-piercing arrangements. Given an arrangement Γ of non-piercing curves, a sweep curve γ∈Γ , and a point P in γ̃ , we show that we can continuously shrink γ to P so that throughout the process, the arrangement remains non-piercing (except at a finite set of points in time where γ crosses other curves), and P lies in γ̃ . We show that our arguments can be modified if P lies outside γ̃ , and we want to sweep γ outwards so that P lies outside γ̃ , and the arrangement remains non-piercing. We give several applications of our results to combinatorial and algorithmic questions including to the multi-hitting set problem involving points and non-piercing regions.
Computing the energy barrier between RNA structures is a classic NP-hard problem in bioinformatics, of which the Independent Set (IS) reconfiguration in bipartite graphs represents a natural generalization. Parameterized algorithms, based on parameters taking limited or bounded values on biological instances, are thus crucial towards practical solutions. In this work, we show that bipartite IS reconfiguration is slice-wise Polynomial (XP) solvable for both the range ρ of IS sizes allowed along the reconfiguration, and the arboricity Φ when the input is restricted to a circle graph. Such a setting is relevant to Bioinformatics as it provides a solution to the RNA energy barrier problem. We propose algorithms based on a divide-and-conquer approach, yielding a O( n^2) -space, O( n^2ρ +2.5) -time algorithm for the range ρ , and a O( n^Φ + 2) -space, O( n^Φ +3) -time algorithm for the arboricity Φ . We demonstrate the practicality of our algorithms on benchmarks respectively consisting of random Erdös-Rényi bipartite graphs, random pairs of RNA structures and experimentally-supported instances of RNA kinetics.
Let G be a bipartite graph where every vertex has a strict preference order over its neighbors. The preferences of a vertex over its neighbors extend naturally to preferences over matchings. A matching N is more popular than matching M if the vertices that prefer N to M outnumber those that prefer M to N. A matching M is popular if there is no matching more popular than M. Every stable matching is popular, thus popular matchings always exist in G and can be efficiently computed. We consider the following problem: edges in G have utilities and it is only max-utility matchings that are relevant. Our goal is to find a popular max-utility matching, i.e., a max-utility matching M such that there is no max-utility matching more popular than M. We show there always exists a popular max-utility matching; furthermore, such a matching can be efficiently computed. The popular critical matching algorithm is the key subroutine in our algorithm. Given a set of prioritized or critical vertices in G, we are interested in only those matchings that match as many critical vertices as possible. We call such matchings “critical” and seek a popular critical matching, i.e., a critical matching M such that there is no critical matching more popular than M. We show popular critical matchings always exist in G and a min-size/max-size such matching can be efficiently computed. We focus on max-size critical matchings and show a compact extended formulation for the popular max-critical matching polytope, i.e., the polytope of max-size critical matchings that are popular within the set of all max-size critical matchings. Thus we can efficiently solve linear optimization problems over the set of popular max-size critical matchings.
We study the question of whether submodular functions of random variables satisfying various notions of negative dependence satisfy Chernoff-like concentration inequalities. We prove such a concentration inequality for the lower tail when the random variables satisfy negative association or negative regression, partially resolving an open problem raised in ([1]). Previous work showed such concentration results for random variables that come from specific dependent-rounding algorithms ([2, 3]). We discuss some applications of our results to combinatorial optimization and beyond. We also show applications to the concentration of read-k families [4] under certain forms of negative dependence; we further show a simplified proof of the entropy-method approach of [4].
Given a set of pattern strings 𝒫={P_1, P_2,… P_k} and a text string S, the classic dictionary matching problem is to report all occurrences of each pattern in S. We study the dictionary problem in the compressed setting, where the pattern strings and the text string are compressed using run-length encoding, and the goal is to solve the problem without decompression and achieve efficient time and space in the size of the compressed strings. Let m and n be the total length of the patterns 𝒫 and the length of the text string S, respectively, and let m and n be the total number of runs in the run-length encoding of the patterns in 𝒫 and S, respectively. Our main result is an algorithm that achieves O( (m + n)loglog m + occ) expected time, and O(m) space, where occ is the total number of occurrences of patterns in S. This is the first non-trivial solution to the problem. Since any solution must read the input, our time bound is optimal within a loglog m factor. We introduce several new techniques to achieve our bounds, including a new compressed representation of the classic Aho-Corasick automaton and a new efficient string index that supports fast queries in run-length encoded strings.
Practical EMO algorithms like NSGA-II, NSGA-III, and SMS-EMOA combine the dominance relation with diversity criteria to identify promising solutions. Despite many success stories, their theoretical foundation remains underdeveloped, with key questions still unanswered—such as which information obtained throughout the evolution is critical for their success. We explore the limitations of the information provided by the dominance relation between search points encountered so far. We construct an artificial problem with a small Pareto set where almost all pairs of search points are incomparable. For this problem, we prove that any black-box EMO algorithm that only relies on the dominance relation for making decisions and only use variation operators that are invariant to bit values, fails spectacularly, requiring exponential time with high probability. In stark contrast, NSGA-II, NSGA-III, and SMS-EMOA efficiently cover the Pareto front in expected quadratic time by incorporating additional information, such as objective values. Our results highlight the superiority of practical EMO algorithms and the necessity of using information beyond dominance for effective multi-objective optimisation.
In this paper we present algorithms for several string problems in the Congested Clique model. In the Congested Clique model, n nodes (computers) are used to solve some problem. The input to the problem is distributed among the nodes, and the communication between the nodes is conducted in rounds. In each round, every node is allowed to send an O(log n)-bit message to every other node in the network. We consider three fundamental string problems in the Congested Clique model. First, we present an O(1) rounds algorithm for string sorting that supports strings of arbitrary length. Second, we present an O(1) rounds combinatorial pattern matching algorithm. Finally, we present an O(loglog n) rounds algorithm for the computation of the suffix array and the corresponding Longest Common Prefix array of a given string.
We present a simple method based on a variant of Hölder’s inequality to lower-bound the trace norm of Boolean matrices. As the main result, we obtain an exponential separation between the randomized decision tree depth and the spectral norm (i.e. the Fourier L_1 -norm) of a Boolean function. This answers an open question of Cheung, Hatami, Hosseini, and Shirley (CCC 2023). As immediate consequences, we obtain the following results.
A mobile agent, starting from a node $s$ of a simple undirected connected graph $G=(V,E)$, has to explore all nodes and edges of $G$ using the minimum number of edge traversals. To do so, the agent uses a deterministic algorithm that allows it to gain information on $G$ as it traverses its edges. During its exploration, the agent must always respect the constraint of knowing a path of length at most $D$ to go back to node $s$. The upper bound $D$ is fixed as being equal to $(1+\alpha)r$, where $r$ is the eccentricity of node $s$ (i.e., the maximum distance from $s$ to any other node) and $\alpha$ is any positive real constant. This task has been introduced by Duncan et al. [ACM Trans. Algorithms 2006] and is known as \emph{distance-constrained exploration}. The \emph{penalty} of an exploration algorithm running in $G$ is the number of edge traversals made by the agent in excess of $|E|$. Panaite and Pelc [J. Algorithms 1999] gave an algorithm for solving exploration without any constraint on the moves that is guaranteed to work in every graph $G$ with a (small) penalty in $\mathcal{O}(|V|)$. Hence, a natural question is whether we could obtain a distance-constrained exploration algorithm with the same guarantee as well. In this paper, we provide a negative answer to this question. We also observe that an algorithm working in every graph $G$ with a linear penalty in $|V|$ cannot be obtained for the task of \emph{fuel-constrained exploration}, another variant studied in the literature. This solves an open problem posed by Duncan et al. [ACM Trans. Algorithms 2006] and shows a fundamental separation with the task of exploration without constraint on the moves.
Zigzag filtrations of simplicial complexes generalize the usual filtrations by allowing simplex deletions in addition to simplex insertions. The barcodes computed from zigzag filtrations encode the evolution of homological features. Although one can locate a particular feature at any index in the filtration using existing algorithms, the resulting representatives may not be compatible with the zigzag: a representative cycle at one index may not map into a representative cycle at its neighbor. For this, one needs to compute compatible representative cycles along each bar in the barcode. It is known that the barcode for a zigzag filtration with m insertions and deletions can be computed in O(m^ω) time, where ω< 2.373 is the matrix multiplication exponent. However, it is not known how to compute the compatible representatives so efficiently. For a non-zigzag filtration, the classical matrix-based algorithm provides representatives in O(m^3) time, which can be improved to O(m^ω). However, no known algorithm for zigzag filtrations computes the representatives with the O(m^3) time bound. We present an O(m^2n) time algorithm for this problem, where n≤ m is the size of the largest complex in the filtration.
Fairly allocating indivisible goods is a frequently occurring task in everyday life. Given an initial allocation of the goods, we consider the problem of reforming it via a sequence of exchanges to attain fairness in the form of envy-freeness up to one good (EF1). We present a vast array of results on the complexity of determining whether it is possible to reach an EF1 allocation from the initial allocation and, if so, the minimum number of exchanges required. In particular, we uncover several distinctions based on the number of agents involved and their utility functions. Furthermore, we derive essentially tight bounds on the worst-case number of exchanges needed to achieve EF1 when the initial allocation is balanced.
We present an improved approximation algorithm for three related clustering problems. In the Minimum Sum of Radii clustering problem (MSR), we are to select k balls in a metric space to cover all points while minimizing the sum of their radii. In the Minimum Sum of Diameters clustering problem (MSD), we are to pick k clusters to cover all the points such that sum of diameters of all the clusters is minimized. Finally, in the Minimum Sum of Squared Radii problem (MSSR), the goal is to choose k balls, similar to MSR but the goal is to minimize the sum of squares of radii of the balls. We present a 3.389-approximation for MSR and a 6.546-approximation for MSD, improving over respective 3.504 and 7.008 developed by Charikar and Panigrahy (2001). In particular, our guarantee for MSD is better than twice our guarantee for MSR. Note that since our result, Buchem et al. proved a 3+ϵ -approximation for MSR. In the case of MSSR, the best known approximation guarantee is 4· (540)^2 based on the work of Bhowmick, Inamdar, and Varadarajan in their general analysis of the t-Metric Multicover Problem. Furthermore, our analysis yields an 11.078-approximation algorithm for the Minimum Sum of Squared Radii problem
We consider a truthful facility location problem with agents that have private positions on the line of real numbers and known optional preferences over two obnoxious facilities that must be placed at locations chosen from a given set of candidate ones. Each agent wants to maximize the sum of distances from the facilities that affect her, and our goal is to design mechanisms that decide where to place the facilities so as to maximize the total happiness of the agents as well as provide the right incentives to them to truthfully report their positions. We consider separately the setting in which all agents are affected by both facilities (i.e., they have non-optional preferences) and the general optional setting. We show tight bounds on the approximation ratio of deterministic strategyproof mechanisms for both settings, and almost tight bounds for randomized mechanisms.
