Many optimization and scheduling problems can be abstracted in terms of a bipartite “assignment graph" G = (L ∪ R, E), where the goal is to select exactly one edge for each right-node. For example, a right-node may correspond to a job, and a left-node to a possible machine assignment. A common strategy to solve such problems is to obtain a fractional relaxation x_e for each edge e, and then have each right-node independently select an edge with probability x_e. However, this may cause the left-nodes to become unevenly loaded, leading to suboptimal solutions for some problems. To address this, a number of algorithms for dependent rounding with strong negative correlation have been developed, e.g. Bansal, Srinivasan Svensson (2021), Im Shadloo (2020), Im Li (2023), Harris (2024), Naor, Srinivasan Wajc (2025). We introduce a new method for this, which we call the Dirichlet mechanism. It is based on having each left-node draw Dirichlet random variables for its edges, and then having each right-node select an edge based on these values. This achieves quantitatively stronger negative correlation than previous algorithms, and is also simpler since it avoids the need for a tie-breaking mechanism. We illustrate the mechanism with improved approximation ratios for two problems. For oblivious online dependent rounding, we achieve a 0.68-approximation which improves upon the previous 0.652-approximation of Naor, Srinivasan Wajc (2025). For the problem of scheduling jobs on unrelated machines to minimize weighted completion time, we achieve a 1.387-approximation which improves upon the 1.398-approximation of Harris (2024). (A recent algorithm of Li (2025) based on iterated rounding also provides a 1.36-approximation if the weights of each job are independent of machine.)
We study dynamic algorithms for maintaining fundamental algebraic properties of matrices, specifically, rank, basis, and full-rank submatrices, with applications to maximum matching on dynamic graphs. Prior dynamic algorithms for rank achieve subquadratic update times but scale with the matrix dimension n, and could not always maintain the corresponding objects such as a basis or maximum full-rank submatrix. We present the first dynamic rank algorithms whose update time scales with the matrix rank r, achieving Õ(r^1.405) time per entry-update and Õ(r^1.528+ z) per column-update, where z is the number of changed entries. This extends to Õ(|M|^1.405) edge-update time to maintain the size |M| of a maximum matching. We also give dynamic algorithms for maintaining a column-basis subject to column-updates and a maximum full-rank submatrix subject to entry-updates.
The rank aggregation problem seeks to combine multiple rank orderings of the same set of candidates into a single consensus ordering. Such problems arise in diverse domains, including web search, employment, college admissions, and voting. In this work we focus on the 1-median objective: given a set of m rankings over [n], the goal is to compute a ranking that minimizes the sum of its distances to all input rankings. We study rank aggregation under several classical distance metrics: Ulam distance, Spearman's footrule, Hamming distance, and Kendall-tau, as well as their weighted variants. Our contributions begin with a novel unified framework that identifies a key structural property: it suffices to focus on a small subset of rankings, where the corresponding local one-median provides a good approximation to the global median. This principle extends across these distance measures, yielding a general algorithmic framework for weighted rank aggregation. Building on this, we present a new approximation algorithm for rank aggregation under the Ulam distance that scales in the Massively Parallel Computation (MPC) model. Our algorithm computes a (2-α)-approximation, for a constant α>0, to the 1-median in a constant number of rounds, using local memory sublinear in n and total memory near-linear in n. We further design new MPC approximation algorithms for Spearman's footrule and for the element-weighted variants of Hamming and Kendall-tau distances. For each metric, we obtain a (2-ζ)-approximation, for a constant ζ>0, to the 1-median in a constant number of rounds, using local memory sublinear in n and total memory linear or near-linear in n. Moreover, for the Ulam distance, we simplify and strengthen the analysis of Chakraborty et al., obtaining an improved 1.968-approximation that further extends to the weighted setting.
In the coordinated motion planning problem, we are given a graph together with the starting and destination vertices of k robots. At each time step, any subset of robots may move, each traversing an edge of the graph, provided that no two robots collide. The goal is to compute a schedule that routes all robots to their destinations while minimizing some objective function. In this paper, we focus on the well-studied objective of minimizing the total travel length of all robots. This problem is known to be NP-hard, and it has been shown to be fixed-parameter tractable (FPT), when parameterized by the number k of robots, on full grids (SoCG 2023) and on bounded-treewidth graphs (ICALP 2024). We present a fixed-parameter algorithm for coordinated motion planning, parameterized by the number k of robots, on graphs arising from discretizations of simple polygons. Such graphs are of particular interest in real-world applications, where planar motion is often constrained to discretized representations of polygonal environments. Moreover, these graphs generalize rectangular grids; consequently, our result constitutes a significant step toward resolving the parameterized complexity of coordinated motion planning on subgrids and, ultimately, planar graphs – two prominent open problems in the field.
We present parallel algorithms for computing single-source reachability and shortest paths on directed n-vertex m-edge graphs using near-linear Õ(m) work and o(√(n)) depth whenever m≥ n^1+o(1). At the extreme of m=Ω(n^2), our reachability and shortest path algorithms have depth only n^0.136 and n^0.25+o(1), respectively. The state-of-the-art parallel algorithms with near-linear work for both problems require Ω(√(n)) depth in all density regimes.
Given a graph and two fixed vertices s and t, the Replacement Path Problem (RP) is to compute for every edge e, the distance between s and t when e is removed. There are two natural extensions to RP: (1) Single Source Replacement Paths (SSRP): Given a graph G and a source node s, compute for every vertex v and every edge e the s-v distance in G ∖{e}. That is, we do not fix the target anymore. (2) 2-Fault Replacement Paths (2-FRP): Given a graph G and two nodes s and t, compute for every pair of edges e, e' the s-t distance in G ∖{e, e'}. That is, we consider two failures instead of one. Previously, there was no known reduction between SSRP and 2-FRP. It seemed plausible that 2-FRP would be computationally harder because there are no settings where 2-FRP admits a faster algorithm than SSRP. In directed unweighted graphs there is a provable gap in complexity, and in undirected graphs many of the known 2-FRP algorithms in a variety of settings are much slower than those for SSRP in the same setting. The main contribution of this paper is a tight reduction from undirected 2-FRP to undirected SSRP, showing that contrary to prior intuition, 2-FRP is not harder than SSRP. As our reduction is weight-preserving, we obtain the first algorithms for 2-FRP that match the best-known runtimes for SSRP: (1) Õ(M n^ω) for weights in [1, M] [GVW19], improving upon O(Mn^2.87) [CZ24]; (2) n^3/2^Ω(√(log n)) for weights in [1, poly(n)] [GVW19], improving over the previous n^3polylog(n) running time [VWWX22]; (3) Õ(mn^1/2+n^2) combinatorial time for unweighted graphs [CC19], and more generally for rational weights in [1, 2] [CM20], improving upon Õ(n^3-1/18) [CZ24]. We complement these upper bounds with tight lower bounds under fine-grained hypotheses.
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.
A notorious open question in circuit complexity is whether Boolean operations of arbitrary arity can efficiently be expressed using modular counting gates only. Håstad's celebrated switching lemma yields exponential lower bounds for the dual problem - realising modular arithmetic with Boolean gates - but, a similar lower bound for modular circuits computing the Boolean AND function has remained elusive for almost 30 years. We solve this problem for the restricted model of symmetric circuits: We consider MOD_m-circuits of arbitrary depth, and for an arbitrary modulus m ∈ℕ, and obtain subexponential lower bounds for computing the n-ary Boolean AND function, under the assumption that the circuits are syntactically symmetric under all permutations of their n input gates. This lower bound is matched precisely by a construction due to (Idziak, Kawałek, Krzaczkowski, LICS'22), leading to the surprising conclusion that the optimal symmetric circuit size is already achieved with depth 2. Motivated by another construction from (LICS'22), which achieves smaller size at the cost of greater depth, we also prove tight size lower bounds for circuits with a more liberal notion of symmetry characterised by a nested block structure on the input variables.
In single-core processors, when multiple processes execute concurrently, they are, in practice, intertwined by a scheduler as a single thread of execution. The language-theoretic operation that corresponds to this is the shuffle of two languages: in general, this is defined as the set of words obtained by interleaving words from the first and second language in an arbitrary fashion. It is well known that regular languages are closed under shuffles, while context-free languages (CFL) are not. Following an established line of research, this paper considers shufflings according to regular "trajectories", that is, subject to scheduling constraints expressed by an automaton. Unsurprisingly, some trajectories, such as "a word from the first language first, then a word from the second", allow for CFLs to be shuffled into CFLs, while some other trajectories do not. This paper provides a robust toolset to show that a given trajectory would always shuffle two nonregular CFLs into a nonCFL. In the case of deterministic CFLs (DCFLs), a salient trichotomy of trajectories depending on how they shuffle DCFLs is provided. These results are based on intricate expressiveness lemmas for CFLs and DCFLs of independent interest, the latter lemma relying on a recent result of Jančar and Šíma (MFCS'2021).
A signed tree model of a graph G is a compact binary structure consisting of a rooted binary tree whose leaves are bijectively mapped to the vertices of G, together with 2-colored edges xy, called transversal pairs, interpreted as bicliques or anti-bicliques whose sides are the leaves of the subtrees rooted at x and at y. We design an algorithm that, given such a representation of an n-vertex graph G with p transversal pairs and a source v ∈ V(G), computes a shortest-path tree rooted at v in G in time O(p log n). A wide variety of graph classes are such that for all n, their n-vertex graphs admit signed tree models with O(n) transversal pairs: for instance, those of bounded symmetric difference, more generally of bounded sd-degeneracy, as well as interval graphs. As applications of our Single-Source Shortest Path algorithm and new techniques, we - improve the runtime of the fixed-parameter algorithm for first-order model checking on graphs given with a witness of low merge-width from cubic [Dreier and Toruńczyk, STOC '25] to quadratic; - give an O(n^2 log n)-time algorithm for All-Pairs Shortest Path (APSP) on graphs given with a witness of low merge-width, generalizing a result known on twin-width [Twin-Width III, SICOMP '24]; - extend and simplify an O(n^2 log n)-time algorithm for multiplying two n × n matrices A, B of bounded twin-width in [Twin-Width V, STACS '23]: now A solely has to be an adjacency matrix of a graph of bounded twin-width and B can be arbitrary; - give an O(n^2 log^2 n)-time algorithm for APSP on graphs of bounded twin-width, bypassing the need for contraction sequences in [Twin-Width III, SICOMP '24; Bannach et al. STACS '24]; - give an O(n^7/3log^2 n)-time algorithm for APSP on graphs of symmetric difference O(n^1/3).
We consider the classic 3SUM problem: given sets of integers A, B, C, determine whether there is a tuple (a, b, c) ∈ A × B × C satisfying a + b + c = 0. The 3SUM Hypothesis, central in fine-grained complexity, states that there does not exist a truly subquadratic time 3SUM algorithm. Given this long-standing barrier, recent work over the past decade has explored 3SUM from a data structural perspective. Specifically, in the 3SUM in preprocessed universes regime, we are tasked with preprocessing sets A, B of size n, to create a space-efficient data structure that can quickly answer queries, each of which is a 3SUM problem of the form A', B', C', where A' ⊆ A and B' ⊆ B. A series of results have achieved Õ(n^2) preprocessing time, Õ(n^2) space, and query time improving progressively from Õ(n^1.9) [CL15] to Õ(n^11/6) [CVX23] to Õ(n^1.5) [KPS25]. Given these series of works improving query time, a natural open question has emerged: can one achieve both truly subquadratic space and truly subquadratic query time for 3SUM in preprocessed universes? We resolve this question affirmatively, presenting a tradeoff curve between query and space complexity. Specifically, we present a simple randomized algorithm achieving Õ(n^1.5 + ε) query time and Õ(n^2 - 2ε/3) space complexity. Furthermore, our algorithm has Õ(n^2) preprocessing time, matching past work. Notably, quadratic preprocessing is likely necessary for our tradeoff as either the preprocessing or the query time must be at least n^2-o(1) under the 3SUM Hypothesis.
Multiple-environment Markov decision processes (MEMDPs) equip an MDP with several probabilistic transition functions (one per possible environment) so that the state is observable but the environment is not. Previous work studies two semantics: (i) the universal semantics, where an adversary picks the environment; and (ii) the prior semantics, where the environment is drawn once before execution from a fixed distribution. We clarify the relation between these semantics. For parity objectives, we show that the qualitative questions, i.e. value one, coincide, and we develop a new algorithm for the general value of MEMDP with prior semantics. In particular, we show that the prior value of an MEMDP with a parity objective can be approximated to any precision with a space efficient algorithm; equivalently, the associated gap problem is decidable in PSPACE when probabilities are given in unary (and in EXPSPACE otherwise). We then prove that the universal value equals the infimum of prior values over all beliefs. This yields a new algorithm for the universal gap problem with the same complexity (PSPACE for unary probabilities, EXPSPACE in general), improving on earlier doubly-exponential-space procedures. Finally, we observe that MEMDPs under the prior semantics form an important tractable subclass of POMDPs: our algorithms exploit the fact that belief entropy never increases, and we establish that any POMDP with this property reduces effectively to a prior-MEMDP, showing that prior-MEMDPs capture a broad and practically relevant subclass of POMDPs.
This paper presents a new deterministic algorithm for single-source shortest paths (SSSP) on real non-negative edge-weighted directed graphs, with running time O(m√(log n)+√(mnlog nloglog n)), which is O(m√(log nloglog n)) for sparse graphs. This improves the recent breakthrough result of O(mlog^2/3 n) time for directed SSSP algorithm [Duan, Mao, Mao, Shu, Yin 2025].
We give a short, self-contained, and easily verifiable proof that determining the outerthickness of a general graph is NP-hard. This resolves a long-standing open problem on the computational complexity of outerthickness. Moreover, our hardness result applies to a more general covering problem P_F, defined as follows. Fix a proper graph class F whose membership is decidable. Given an undirected simple graph G and an integer k, the task is to cover the edge set E(G) by at most k subsets E_1,…,E_k such that each subgraph (V(G),E_i) belongs to F. Note that if F is monotone (in particular, when F is the class of all outerplanar graphs), any such cover can be converted into an edge partition by deleting overlaps; hence, in this case, covering and partitioning are equivalent. Our result shows that for every proper graph class F whose membership is decidable and that satisfies all of the following conditions: (a) F is closed under topological minors, (b) F is closed under 1-sums, and (c) F contains a cycle of length 3, the problem P_F is NP-hard for every fixed integer k≥ 3. In particular: For F equal to the class of all outerplanar graphs, our result settles the long-standing open problem on the complexity of determining outerthickness. For F equal to the class of all planar graphs, our result complements Mansfield's NP-hardness result for the thickness, which applies only to the case k=2. It is also worth noting that each of the three conditions above is necessary. If F is the class of all eulerian graphs, then cond. (a) fails. If F is the class of all pseudoforests, then cond. (b) fails. If F is the class of all forests, then cond. (c) fails. For each of these three classes F, the problem P_F is solvable in polynomial time for every fixed integer k≥ 3, showing that none of the three conditions can be dropped.
We provide proofs certifying that the structure theorem for vertex sets of bounded bidimensionality holds with polynomial bounds. The bidimensionality of vertex sets is a common generalisation of both treewidth and the face-cover-number of vertex sets in planar graphs. As such, it plays a crucial role in extensions of Courcelle's Theorem to H-minor-free graphs. Recently, bidimensionality and similar parameters have emerged as key for extensions of known parameterized algorithms for problems defined on a terminal set R. A prominent example for such a problem is Steiner Tree, which admits efficient algorithms on planar graphs whenever R can be covered with few faces. Key to the algorithmic applications of bidimensionality is a structure theorem that explains how a graph G can be decomposed into pieces where the behaviour of R is highly controlled. One may see this structure theorem as a rooted analogue of Robertson and Seymour's celebrated Grid Theorem. Combining recent advances in obtaining polynomial bounds in the Graph Minors framework with new techniques for handling annotated vertex sets, we show that all parameters in the structure theorem above admit polynomial bounds. As an application, we also provide a sketch showing how our techniques imply polynomial bounds for the structure theorem for graphs excluding an apex minor.
We provide explicit and polynomial bounds for the Homogeneous Wall Lemma which occurred for the first time implicitly in the 13th entry of Robertson and Seymour's Graph Minors Series [JCTB 1990] and has since become a cornerstone in the algorithmic theory of graph minors. A wall where each brick is assigned a set of colours is said to be homogeneous if each brick is assigned the same set of colours. The Homogeneous Wall Lemma says that there exists a function h that, given non-negative integers q and k and an h(q,k)-wall W where each brick is assigned a, possibly empty, subset of { 1, … , q } contains a k-wall W' as a subgraph such that, if one assigns to each brick B of W' the union of the sets assigned to the bricks of W in its interior, then W' is homogeneous. It is well-known that h(q,k) ∈ k^𝒪(q). The Homogeneous Wall Lemma plays a key role in most applications of the Irrelevant Vertex Technique where an exponential dependency of h on q usually causes non-uniform dependencies on meta-parameters at best and additional exponential blow-ups at worst. By proving that h(q,k) ∈𝒪(q^4 · k^6), we provide a positive answer to a problem raised by Sau, Stamoulis, and Thilikos [ICALP 2020].
We give unconditional parameterized complexity lower bounds on pure dynamic programming algorithms - as modeled by tropical circuits - for connectivity problems such as the Traveling Salesperson Problem. Our lower bounds are higher than the currently fastest algorithms that rely on algebra and give evidence that these algebraic aspects are unavoidable for competitive worst case running times. Specifically, we study input graphs with a small width parameter such as treewidth and pathwidth and show that for any $k$ there exists a graph $G$ of pathwidth at most $k$ and $k^{O(1)}$ vertices such that any tropical circuit calculating the optimal value of a Traveling Salesperson round tour uses at least $2^{Ω(k \log \log k)}$ gates. We establish this result by linking tropical circuit complexity to the nondeterministic communication complexity of specific compatibility matrices. These matrices encode whether two partial solutions combine into a full solution, and Raz and Spieker [Combinatorica 1995] previously proved a lower bound for this complexity measure.
3SUM-Indexing is a preprocessing variant of the 3SUM problem that has recently received a lot of attention. The best known time-space tradeoff for the problem is T S^3 = n^6 (up to logarithmic factors), where n is the number of input integers, S is the length of the preprocessed data structure, and T is the running time of the query algorithm. This tradeoff was achieved in [KP19, GGHPV20] using the Fiat-Naor generic algorithm for Function Inversion. Consequently, [GGHPV20] asked whether this algorithm can be improved by leveraging the structure of 3SUM-Indexing. In this paper, we exploit the structure of 3SUM-Indexing to give a time-space tradeoff of T S = n^2.5, which is better than the best known one in the range n^3/2≪ S ≪ n^7/4. We further extend this improvement to the kSUM-Indexing problem-a generalization of 3SUM-Indexing-and to the related kXOR-Indexing problem, where addition is replaced with XOR. Additionally, we improve the best known time-space tradeoffs for the Gapped String Indexing and Jumbled Indexing problems, which are well-known data structure problems related to 3SUM-Indexing. Our improvement comes from an alternative way to apply the Fiat-Naor algorithm to 3SUM-Indexing. Specifically, we exploit the structure of the function to be inverted by decomposing it into "sub-functions" with certain properties. This allows us to apply an improvement to the Fiat-Naor algorithm (which is not directly applicable to 3SUM-Indexing), obtained in [GGPS23] in a much larger range of parameters. We believe that our techniques may be useful in additional application-dependent optimizations of the Fiat-Naor algorithm.
