Papers

In the online metric matching problem, we have $n$ servers with known locations in some metric space. Requests arrive one-by-one at certain locations, and upon arrival a request must be matched to a server that was not matched to a previous request. The goal is to minimize the matching cost. For randomized algorithms with an oblivious adversary, the best known competitive ratio is obtained by embedding the metric space into an HST, and then solving the problem in the setting where the metric space is defined by the HST. Bansal et al. (Algorithmica, 2014) introduced a framework for online metric matching where one develops an algorithm in a restricted reassignment model, and then transforms this into a true online algorithm. Using this framework, they obtained an expected competitive ratio of $O(\log n)$ for HSTs; this also gives the best known competitive ratio of $O(\log^2 n)$ for general metrics. In this paper, we revisit this framework with the aim of developing new algorithms. For HSTs where each node has at most $\Delta$ children, we develop an algorithm via this framework with an expected competitive ratio of $O((\log\log \Delta) \cdot \log \Delta)$. In particular, this ratio is independent of $n$, the number of servers/requests. It is near-optimal, as the expected competitive ratio of any algorithm is $\Omega(\log \Delta)$.

We consider the offline problem of parallel relocation of k identical mobile resources. After each request, known in advance, the resources may move simultaneously, and the duration of a step is determined by the longest individual movement. This model is equivalent to the offline version of the time-optimal k-server problem. We prove that the decision version is strongly NP-complete already on metrics of finite subsets of the Euclidean line, or equivalently, on vertex metrics of weighted paths. Thus, the computational hardness persists even under a linear arrangement of the admissible resource locations. As a positive result, we show that the problem can be solved exactly in polynomial time on metrics of undirected unweighted graphs with a universal vertex.

The prize-collecting traveling salesperson problem is a variant of the metric traveling salesperson problem in which vertices may be left unvisited by paying their associated penalties. The objective is to minimize the length of the tour plus the total penalty of the unvisited vertices. Blauth, Klein, and Nägele gave the previously best-known LP-relative $1.599$-approximation. We show that a simpler version of their algorithm, obtained by omitting the splitting-off preprocessing before the tree decomposition, has an LP-relative approximation ratio of $1.555761$. The improvement comes entirely from a new analysis of the parity-correction step: a simple analysis already gives $1.56$, and the stated factor follows from a numerical parameter search with exact verification.

In the one-round discrete Voronoi game a multiset $V$ of $n$ voters on a line is given; player P places $k$ facilities, player Q then places $\ell$, and each voter is won by the nearer facility, ties going to P. P wins if it keeps at least $n/2$ voters. In the vocabulary of competitive location this is the absolute $(\ell|k)$-centroid problem on a path with unit demands, and the responder's problem is the $(\ell|X_k)$-medianoid, whose closed form on a path – the sum of the $\ell$ largest of at most $2k$ explicit marginals – is due to Spoerhase and Wirth. We record this structure, with complete proofs, and draw two consequences that we believe are new. First, we compute the value of the game against a single responding facility, $\Gamma_{k,1}(V)$, together with an optimal strategy for P, in $O(n\log n)$ time for arbitrary positive real demands and every $k$. This improves the $O(kn\log^2 n)$ bound of Lazar and Tamir for the absolute $(1|k)$-centroid on a path. Second, we study the facility advantage $k^*(\ell)$, the least $k$ for which P wins every instance against $\ell$ facilities. We prove $k^*(\ell)\le 2\ell-1$, exhibit instances proving $k^*(\ell)\ge\ell+1$ for $2\le\ell\le6$ (an exact, computer-assisted proof resting on a half-integer discretisation), determine $k^*(1)=1$ and $k^*(2)=3$, and show that on uniform instances $k=\ell$ already suffices, so the extremal instances are weighted and Q wins them by a single voter. We conjecture $k^*(\ell)=\ell+1$ for all $\ell\ge2$.

Let $X$ be uniform on $\{-1,1\}^n$, let $Y$ be obtained by passing its coordinates independently through a binary symmetric channel with crossover probability $p$, and let $g:\{-1,1\}^n\to\{0,1\}$ be a Boolean function. We give a computer-assisted proof of the Courtade–Kumar conjecture $I(g(X);Y)\le1-H_2(p)$, where $H_2$ is binary entropy, with equality attained by dictator functions. The present work builds on the differential-equation method, itself a limiting form of the auxiliary-receiver approach in network information theory using a continuum of degraded receivers. The proof proceeds from a local inequality to a dimension-independent bound on entropy production. Differentiation along the Boolean noise semigroup expresses entropy production as an average of edge costs. The key estimate is therefore an unrestricted Bellman inequality with two mean constraints and two entropy constraints, allowing arbitrary couplings of the edge variables. This paper and its supplement provide the proofs and computational verification records. The document is lengthy because it is designed to be entirely self-contained, deriving all proofs from first principles and reproducing the proofs of cited results. We also give a self-contained expository note explaining the reduction to a low-dimensional inequality and the ideas behind the key lower bounds. The entire proof, including all numerical certificates, has been formally verified in Lean end-to-end, and is available online.

Given an undirected, unweighted graph $G = (V,E)$ with $n$ vertices and $m$ edges, the triangle counting problem seeks the number of three-cycles in it. Triangle and subgraph counting are classical problems in graph algorithms, central to applications such as community detection, computing the clustering coefficient, motif discovery in protein networks, and social network analysis. In many of these applications, the graph datasets are so voluminous that we model them as streams of updates to an underlying graph. There are a number of foundational results for streaming triangle counting, both theoretical and practical. There is, however, one major drawback to all previous sublinear-space algorithms: to achieve both a constant factor approximation and the sublinear space guarantees, one needs to know a priori a constant factor approximation of the triangle count $T$, an inherently circular requirement. We initiate the study of parameter-free streaming triangle counting, without any a priori knowledge of $T$ or any quantities depending on $T$, provided $m$, the length of the stream. We describe a family of $O(p)$ pass parameter-free triangle counting algorithms that guarantee a mixed multiplicative and additive approximation of $T$ and use $\widetilde{O}(\frac{m+T}{\sqrt{T}})$ expected space. Moreover, this family leads to an $O(\log\log(n))$ pass algorithm that gives a $(1+\eps)$ multiplicative approximation of $T$ with the same space complexity. These algorithms rely on the notion of a verified parametrized algorithm: an algorithm parametrized by $\tau$ that either provides an approximation of $T$ when $\tau \le T$, or declares that $T < \tau$. Furthermore, we prove a lower bound: any parameter-free algorithm that provides a multiplicative approximation for all values of $T$ must use $\Theta(m)$ space, even on streams where the triangle count is moderately large.

We show that isomorphism of $F$-free tournaments can be solved in FPT time $f(k) \cdot n^{O(1)}$, where $k$ denotes the size of $F$, and $n$ denotes the size of the input tournaments. Our result extends on a previous FPT isomorphism test for tournaments of bounded twin-width [Grohe, Neuen 2024], as well as XP isomorphism tests parameterized by the VC dimension or the chromatic number [Raßmann, Schweitzer 2026]. It also implies that every non-trivial hereditary class of tournaments admits a polynomial-time isomorphism test. Our algorithm builds on a novel combination of spectral, geometric, combinatorial and group-theoretic tools.

In the planted clique problem, one observes either an Erdős–Rényi graph on $n$ vertices or such a graph with a clique added to $k = k(n)$ vertices, and seeks to detect or recover the clique. It is widely believed that $k = \Theta(\sqrt{n})$ is the smallest clique size for which polynomial-time algorithms exist for these tasks. We develop new algorithms in this regime using color-coding to estimate signed subgraph counts, further accelerated with fast matrix multiplication. We first show that, for each $t \geq 1$, for $c(t)$ a constant associated to the order of growth of the number of connected graphs of treewidth at most $t$, cliques of size $k = \lambda\sqrt{n}$ planted in a random location with $\lambda > 1 / \sqrt{c(t)}$ can be detected and recovered in time $n^{t + 1 + o(1)}$. For instance, since $c(1) = e$, this recovers by counting signed trees the performance of the $\widetilde{O}(n^2)$-time message-passing algorithm of Deshpande–Montanari (2015) that succeeds when $\lambda > 1 / \sqrt{e} \approx 0.6066$. For $t \geq 3$, the exact value of $c(t)$ is not known, but lower bounds on it give a hierarchy of slower polynomial-time algorithms that succeed for smaller $\lambda$. We further show that the above algorithm for $t = 2$ can be implemented in time $n^{\omega + o(1)}$ for $\omega$ the constant of square matrix multiplication and succeeds when $\lambda > 0.3320$; under the folklore conjecture that $\omega = 2$, this runs in the nearly-linear time of the algorithm of Deshpande–Montanari while finding smaller cliques. Second, we show that the above algorithm for $t = 1$ can be combined with the boosting scheme of Alon–Krivelevich–Sudakov (1998) using rectangular matrix multiplication, giving improved runtimes for smaller $\lambda$. Taken together, our results achieve the best known tradeoff between runtime and signal strength $\lambda$.

Braess paradox is a well-known phenomenon that originates when latency at Wardrop equilibrium in traffic networks decreases because of removing edges. The possibility of having the paradox was called vulnerability by Roughgarden in 2006 and was characterized later on by graph-theoretical notions, both for undirected and for directed graphs. In this paper we provide an algorithm for the incremental case of checking vulnerability for dynamically evolving graphs. The crucial idea to keep the amortized cost linear for every edge addition is that we do not need to run the vulnerability algorithm on the whole graph, but only on a well-identified subgraph, determined by the edge that we are adding. Overall, to add m edges, we pay a cost of O(m2); this aligns with the O(m2) cost of the state-of-the-art static algorithm for vulnerability.

We establish a strongly sublinear counterpart of a recent result of Chudnovsky, E S, and Lokshtanov (arXiv 2025) on treewidth and tree-independence number. Namely, we prove that a hereditary graph class has strongly sublinear tree-independence number if and only if, for every fixed clique bound, its graphs of bounded clique number have strongly sublinear treewidth. In fact, this is part of a broader equivalence theorem. For hereditary classes, these conditions are also equivalent to having clique-dependent polynomial expansion, to admitting balanced separators whose size is bounded by $K\omega(G)^s |V(G)|^{1-\beta}$ for fixed $K,s,\beta>0$, and to admitting balanced clique-based separators of strongly sublinear size (equivalently, weight). Thus, we show that all these properties, which arose independently in the study of subexponential-time exact algorithms and polynomial-time approximation schemes, in fact describe the same hereditary graph classes. As a consequence of our equivalence theorem, we also show that every hereditary class $\mathcal C$ with strongly sublinear tree-independence number admits a subexponential-time algorithm that, given $G\in\mathcal C$, computes a tree decomposition of $G$ with strongly sublinear independence number.

In the vertex cover interdiction problem, we are given an undirected graph $G=(V,E)$, two integers $t$ and $k$ and a vertex subset $B\subseteq V$, and we are asked to find a set $X \subseteq B$ with $|X|\leq t$ such that $X$ hits (i.e., intersects) all the vertex covers of $G$ of size at most $k$. Recently, Grüne and Wulf proved that the problem is $\Sigma_2^p$-complete. However, their reduction relied on the fact that the vertex cover problem is NP-complete. This, in turn, means that we do not know the complexity status of the vertex cover interdiction problem when the input graph is restricted to a bipartite graph since the vertex cover problem can be solved in polynomial time for bipartite graphs. One of our main results shows that the vertex cover interdiction problem is NP-complete for bipartite graphs. In contrast, when $k$ is restricted to the minimum vertex cover size, i.e., we are only required to hit all the minimum vertex covers, we show that the vertex cover interdiction problem can be solved in polynomial time for bipartite graphs. This motivates us to study the parameterized complexity of the vertex cover interdiction problem for bipartite graphs when the difference of $k$ and the minimum vertex cover size is taken as a parameter. With this parameter, we show that the problem is $\mathrm{W}[1]$-hard, but can be solved in polynomial time when the parameter is constant (i.e., in XP time). We also show that the problem is fixed-parameter tractable when parameterized by $k$.

Over the past decade, a growing body of research has shown that $\gamma$-weak submodularity broadly arises in numerous subset selection tasks, including feature selection, neural network pruning, and video summarization. Despite its prevalence, maximizing a $\gamma$-weakly submodular function subject to a general matroid constraint remains challenging. To date, the only known approximation guarantee is the conservative $(1+1/\gamma)^{-2}$ factor established by \citet{chen2018weakly}. To improve upon this result, this paper proposes a novel algorithm called \MGPE, which repeatedly performs maximum-gain local exchanges through careful control of a non-homogeneous Poisson clock, and proves that this \MGPE\ can attain an approximation ratio arbitrarily close to $\rho_\gamma=1-\left(\gamma/(2-\gamma)\right)^{ \frac{\gamma^2}{2(1-\gamma)} }$. In sharp contrast to the previous guarantee, our obtained factor $\rho_\gamma$ not only strictly improves upon $(1+1/\gamma)^{-2}$ for every $\gamma\in(0,1]$, but also can asymptotically approach the optimal $(1-1/e)$-approximation for submodular maximization as $\gamma\to1$. Furthermore, we surprisingly find that when the matroid constraint reduces to a cardinality or the objective satisfies the stronger notion of $\alpha$-weak DR-submodularity, \MGPE\ can automatically recover the tight approximation ratios of $1-e^{-\gamma}$ and $1-e^{-\alpha}$, respectively. Here, $\alpha\in(0,1]$ denotes the DR ratio.

In the classical Art Gallery Problem (AGP), guards are placed in a polygon so that together they see every point. The Witness Set Problem (WSP), introduced by Amit, Mitchell, and Packer, is a natural dual to the AGP. In this paper, we study the WSP in weak visibility polygons (WVPs), the simple polygons in which every point is seen from some point of one fixed edge. A witness set is a set of points whose visibility regions are pairwise disjoint, so that no single guard sees two of them. A maximum witness set, therefore, lower-bounds the guard number. Exact polynomial-time algorithms for the WSP are known only for monotone mountains, a proper subclass of WVPs. We give the first exact polynomial-time algorithms for the WSP in WVPs, in two settings. In the Discrete Witness Set Problem (DiscWSP), the witnesses come from a given set of $m$ points, and we find a maximum witness subset in $O(n + m \log(n+m))$ time on an $n$-vertex polygon. The algorithm rests on a structural fact: the visibility intersection graph of a WVP, in which two points are adjacent if their visibility regions intersect, is a trapezoid graph, that is, an intersection graph of trapezoids between two parallel lines. Moreover, the class of these graphs properly contains the interval graphs and the permutation graphs, which may be of independent interest in graph theory. We also prove an $\Omega(n \log n)$ lower bound in the algebraic decision-tree model for instances with $m = \Theta(n)$, so our algorithm for DiscWSP is optimum. In the Continuous Witness Set Problem (ContWSP), a witness may be any point of the polygon, and we give an exact algorithm running in $O(n \log n + \rho^{2}(n + \rho^{2}))$ time, where $\rho$ is the number of reflex vertices.

We study deterministic relative approximation of the total variation distance between high-dimensional distributions given by succinct descriptions. We develop an abstract deterministic approximation framework based on representing the total variation distance as a support function of a low-dimensional zonotope. As applications, we obtain FPTASs for several models. Given two mixtures of product distributions over $[q]^n$ with a total of $K$ component distributions, our algorithm approximates their TV-distance within a factor of $1+\varepsilon$ in time $\widetilde O_K(nq(n/\varepsilon)^{2K})$. We also give an FPTAS for mixtures of $n$-step Markov chains over $[q]^n$ with a total of $K$ component distributions, with running time $\widetilde O_K(nq^2(n/\varepsilon)^{2K})$. Finally, for two latent-tree Ising models with the same underlying tree topology, we give an FPTAS for the TV-distance between their leaf marginals in time $O(|V|^{13}\varepsilon^{-12})$.

Asynchronous Byzantine Reliable Broadcast (BRB) is a fundamental primitive that guarantees agreement and validity in distributed systems subject to Byzantine faults, but it lacks ordering guarantees. In this paper, we address Byzantine Causal Reliable Broadcast (BCRB), which builds on BRB to enforce causal message ordering. We present a novel BCRB protocol that decouples causal ordering from the BRB layer, achieving constant-size $\mathcal{O}(1)$ message metadata overhead and $\mathcal{O}(n^2)$ communication word complexity as against $\mathcal{O}(n^3)$ communication word complexity of existing protocols; here $n$ is the number of processes. We present two variants of our protocol: a cryptographic version using a threshold encryption scheme and sequence gating, and its non-cryptographic version. In the cryptographic version, senders broadcast ciphertexts immediately, and decryption shares are piggybacked on out-of-band ACKs, preventing early decryption and front-running. In both versions, causal safety is achieved probabilistically. We evaluate the probability of causal safety violations using a random variable path analysis under independent exponential link delay distributions. We show that both variants satisfy liveness and the probability of weak safety violation is bounded by $\mathcal{O}(f^{-3}\cdot\ln^3 f)$, where $f$ is the upper bound on the number of Byzantine processes, and $f < n/3$ and $f=\mathcal{O}(n)$. Further, for the crypto version, we show that the probability of strong safety violation is bounded by $\mathcal{O}(f^{-1} \cdot \ln^2 f)$. We also show how to modify our two protocols to guarantee 100% weak safety keeping $\mathcal{O}(1)$ message space overhead but with $\mathcal{O}(n^3)$ messages and $\mathcal{O}(n^3)$ communication word complexity.

Ultra-log-concave distributions are ubiquitous in probability, combinatorics, and statistical mechanics and have been studied extensively. In this paper, we introduce a quantitative high-dimensional extension of this notion, called $\delta$-ultra-log-concavity, for probability measures on $\mathbb{N}^d$ with downward closed support. When $\delta = 1$, this notion coincides with the class studied by Gurvits (2009) via strongly log-concave generating functions, and with the class defined by Anari, Oveis Gharan, and Vinzant (2021) via completely log-concave generating functions; in one dimension, it reduces to classical ultra-log-concavity. We establish several functional inequalities, including a weighted Poincaré inequality, a discrete Brascamp–Lieb inequality, and a weighted Wu-type modified log-Sobolev inequality. Our approach combines integrated Bakry–Émery calculus for a canonical birth-death chain with Poisson stochastic localization, which arises as the time reversal of coordinatewise binomial thinning. We further establish concentration of measure, maximum-entropy principles, and several closure properties for ultra-log-concave measures, and develop applications to queueing models, polymatroids, antiferromagnetic Potts models, and hardcore models. Finally, a lattice scaling limit of the discrete theory yields Poincaré and Brascamp–Lieb inequalities for Laguerre diffusions.

In this paper, we determine the precise asymptotics of the number of parts of stable regularity equipartitions in terms of the Littlestone dimension: every graph $G$ of Littlestone dimension $\operatorname{Lit}(G)\leq\ell$ has a regular equipartition into excellent sets with $(1 + o_{\epsilon\to 0,\ell}(1))\cdot\epsilon^{-\ell-1}$ parts and in the other direction, for every $\ell\in\mathbb{N}_+$, there is an infinite family of graphs, all of Littlestone dimension $\ell$, whose equipartitions into good sets must have size at least $(1 + o_{\epsilon\to 0,\ell}(1))\cdot\epsilon^{-\ell-1}$. Dropping the equitability condition, we determine the asymptotics of non-equitable partitions up to a multiplicative $\log(1/\epsilon)$: every graph $G$ with $\operatorname{Lit}(G)\leq\ell$ has a regular partition into excellent sets with $(1 + o_{\epsilon\to 0,\ell}(1))\cdot\epsilon^{-\ell}\cdot\ln(1/\epsilon)$ parts and in the other direction, for every $\ell\in\mathbb{N}_+$, there is an infinite family of graphs, all of Littlestone dimension $\ell$, whose partitions into good sets must have size at least $(1 + o_{\epsilon\to 0,\ell}(1))\cdot\epsilon^{-\ell}$. We also show that such partition can be obtained algorithmically efficiently in an approximation scheme fashion: replacing the $o_{\epsilon\to 0,\ell}(1)$ term above by a constant $c > 0$, we obtain randomized $O_{c,\epsilon,\ell}(n\cdot\log(n))$-time algorithms for partitions/equipartitions into good sets, a deterministic $O_{c,\epsilon,\ell}(n^2)$-time algorithm for partitions into good sets, a deterministic $O_{c,\epsilon,\ell}(n^6)$-time algorithm for equipartitions into good sets, a deterministic $O_{c,\ell,\epsilon}(1)\cdot n^{O(\ell\cdot 2^{2\cdot\ell+4})}$-time algorithm for partitions/equipartitions into excellent sets, and $O_{c,\epsilon,\ell}(\log(n+1))$-space algorithms for partitions/equipartitions into good/excellent sets.

We study Greedy for online metric matching with $n$ servers and $n$ requests sampled independently and uniformly from $[0,1]^d$. Servers are available initially, and Greedy irrevocably matches each arriving request to its closest available server, incurring a cost of their distance. We prove that Greedy has competitive ratio $O(1)$ for every fixed $d\ne2$, and $\Theta(\sqrt{\log n})$ for $d=2$. Previously, constant competitiveness was shown for $d = 1$ [BFP23], and no non-trivial results for this setting were known for higher dimensions. Our proof first analyzes Greedy on the flat torus and then transfers the estimates back to the cube.

We prove two main algorithmic results in spectral discrepancy. First, we give a deterministic polynomial-time rounding theorem for rational positive semidefinite matrices of arbitrary rank. The algorithm starts from any rational fractional signing and assigns one sign per original matrix. Its discrepancy is less than $3.37\,\|\sum_i \mathrm{Tr}(A_i)A_i\|^{1/2}$. This yields Kadison–Singer half-partitions with error below $1.69\sqrt{\varepsilon}$, as well as deterministic graph signings that control signed adjacency and signed degrees simultaneously. The proof builds on the spectral-potential method of Ezeunala and Jiang (2026) and introduces a new way to choose rounding directions. We prove polynomial bit complexity for the rounding procedure. Second, we give a Las Vegas algorithm for the Bilu–Linial signing problem on an arbitrary prescribed graph. If $G$ has $n$ vertices and maximum degree $\Delta\ge3$, the algorithm terminates almost surely. It uses fewer than $100n^3$ insertion attempts in expectation and returns a signing with $\|A_s\|<2\sqrt{2(\Delta-1)}$. For bipartite graphs its one-sided form gives the sharp universal bound $\|A_s\|<2\sqrt{\Delta-1}$. The algorithm builds the signing by inserting vertices and recursively deleting and restoring neighbors after rejected insertions. In the analysis, the $\sqrt2$ gap to the Bilu–Linial conjecture comes from a factor of two in the bound for vertex deletions in the two-sided case. On a $d$-regular bipartite Ramanujan base the same signing produces a Ramanujan $2$-lift of that prescribed base.

The first almost-linear time maximum and minimum cost flow algorithm of Chen-Kyng-Liu-Peng-Probst Gutenberg-Sachdeva (FOCS 2022), reduced these flow objectives to a sequence of min-ratio cycle problems. Solving this core primitive requires approximately minimizing the ratio of a linear gradient term and an undirected length term. In Chen-Kyng-Liu-Peng-Probst Gutenberg-Sachdeva (FOCS 2022) and the subsequent work of Chen-Kyng-Liu-Meierhans-Probst Gutenberg (STOC 2024), intricate data structures were given to solve the min-ratio problem. We show that such a cycle can be extracted directly from the dynamic distance oracle of Kyng-Meierhans-Probst Gutenberg (STOC 2024) using linearity. This simplifies previous algorithms that relied on multiple additional steps to extract the cycle, and can be seen as evidence that solving the min-ratio cycle problem really is all about distances. As a result, we obtain a faster primal maxflow and min-cost flow solver that also extends to incremental graphs.

We give a fixed parameter tractable (FPT) algorithm with running time $f(k,\Delta)\cdot {|I|}^{O(1)}$ for integer linear programs with 4-block structure, parameterized by the maximum block dimension $k$ and the largest absolute matrix entry $\Delta$. This result resolves a long-standing open question in parameterized complexity, and answers a conjecture by Eisenbrand and Rothvoss (2026) in the positive. This result has several implications for other block-structured integer programming models, including 3-block, mixed fracture number, and special cases of 4-block programs with large entries outside of the diagonal. Important tools for this algorithm are structural properties of generalized $n$-fold integer programs shown by Ligthart~(2026) and an algorithm by Veselov et al.~(2020) for optimizing discrete convic functions.

The A* search is a fundamental path-finding algorithm in artificial intelligence. While admissible and consistent heuristics guarantee efficient performance by expanding each state at most once, modern search applications frequently employ powerful but inconsistent heuristics derived from machine learning, randomized evaluations, etc. A long-standing theoretical barrier to using these inconsistent heuristics is the risk of catastrophic node re-expansion, which yields a worst-case exponential time complexity of $\Omega(2^n)$. However, empirical observations contradict this pessimistic bound, demonstrating that inconsistent A* operates highly efficiently in practice. To bridge this significant gap between theory and practice, this paper presents the first smoothed analysis of the A* algorithm using inconsistent heuristics. We model typical real-world noise by applying slight random perturbations to the edge weights of worst-case search graphs. Our main result proves that the expected smoothed time complexity of inconsistent A* is bounded by a polynomial, specifically a total iteration number of $O(n^2 m \kappa)$, where $n$ is the number of nodes, $m$ is the number of edges, and $\kappa$ controls the scale of random perturbations. Furthermore, we also show that this result naturally extends to the functionally equivalent problem of Dijkstra's algorithm on negative-weight graphs.

Consider a constraint satisfaction problem $\mathbf{C}$ on finitely many independent random variables with dependency graph $G$. Let $p_a$ be the violation probability of a constraint $a\in \mathbf{C}$ and $N_G^2 (a)$ the set of constraints at distance one or two from $a$ in $G$. Suppose that, there exists $x\in (0,1)^{\mathbf{C}}$ such that, for a sufficiently small universal constant $c > 0$, and for all $a \in \mathbf{C}$, \[ p_a \leq c \cdot x_a \prod_{b\in N_G^2(a)}(1-x_b). \] Under the above analog of the asymmetric Lovász Local Lemma, we give an FPRAS for the probability that all constraints are satisfied, and an approximate sampler, running in polynomial expected time, for the product distribution conditioned on this event. The degree of the polynomial in the running time is independent of the domain sizes, constraint sizes, or degree of the dependency graph. Up to the choice of the constant $c$, our condition on $p_a$ matches known hardness results. Our work builds on the method of Liu, Wang, Yin, Zhang, and Zhou, who obtained an FPRAS for the probability of satisfaction in the setting of the symmetric Lovász Local Lemma. Our sampling result is new even in this special case.

The Directed Feedback Vertex Set problem (DFVS) asks whether a digraph can be made acyclic by deleting at most $k$ vertices. Whether DFVS admits a polynomial kernel parameterized by $k$ is a major open problem in kernelization, even for planar digraphs. We resolve the planar case by giving a deterministic kernel with $O(k^{66}\log^2 k)$ vertices and arcs. Our algorithm proceeds in three stages. First, we apply structural reduction rules to the input digraph, bounding the number of directed faces and some special vertices. Second, we pass to the planar dual, where vertex deletion corresponds to adding groups of reverse arcs to make each weakly connected component strongly connected. The structural bounds in the first stage yield a small retained vertex set in the dual. We then compress the dual instance by identifying vertices with the same distance records from this retained vertex set. The main technical contribution is a directed-cut argument showing that this identification preserves feasibility. Finally, we transform the polynomial-size dual instance back into an instance of Planar Directed Feedback Vertex Set via a $3$-CNF encoding and a planar graph construction.

Influence maximization asks for $k$ seed vertices that maximize the expected spread of a diffusion process in a network. Standard near-optimal-time algorithms based on reverse-reachable sampling achieve a $(1-1/e-\varepsilon)$ approximation, but their expected running-time bounds grow linearly with the seed budget $k$. We remove this multiplicative dependence: for the independent cascade model, our algorithm succeeds with probability at least $1-\delta$ in $O((m+n)\varepsilon^{-3}\log(2n/\delta))$ expected time. The result extends to triggering models with explicitly charged local sampling costs. We reserve $O(\varepsilon k)$ seed positions for cost-weighted random vertices, allowing reverse-reachable searches to stop as soon as they encounter a reserved seed. An independent sample-count estimation phase uses a statistic that also controls the expected search cost. Matching these quantities eliminates the multiplicative dependence on $k$ while preserving the approximation guarantee.

We present a spectral signing algorithm solving the Komlós problem with a constant discrepancy in polynomial time. Given a matrix $A\in\mathbb{R}^{m\times n}$ whose columns have Euclidean norm at most $1$, the algorithm finds a vector $\varepsilon\in\{-1,1\}^n$ satisfying $\|A\varepsilon\|_\infty\le C$, where $C$ is an absolute constant. By minimizing a cubic spectral potential, our spectral signing algorithm updates the fractional coloring toward Boolean signs with time complexity $O((mn^9+n^{10})\log(2+m+n))$.

Triangle counting is one of the most fundamental problems in network analysis. Given the massive sizes of real-world graphs, there is a long history of small-space streaming algorithms providing accurate estimates for this problem. However, most of the results focus on estimating the total triangle count or the number of triangles incident to individual nodes. In practice, one often wants fine-grained information to understand how triangles are distributed, as captured by clustering coefficients. In particular, a standard network analysis task requires computing the binned degree-wise clustering coefficient distribution, which provides a rich and informative summary of the structure of the graph. In this work we present BOLIDE, the first efficient and practical algorithm for estimating binned degree-wise clustering coefficients in streaming. Our algorithm makes a single pass over the edge stream, and is allowed to store only a small fraction of the total number of edges. BOLIDE carefully combines different sampling strategies to efficiently gather degree and triangle information across sets of nodes. As a result, our algorithm provably approximates the binned degree-wise clustering coefficients, and provides guarantees on the amount of memory used. Our experimental evaluation shows that BOLIDE accurately estimates clustering coefficient distributions while efficiently processing large datasets with billions of edges and triangles.

For a fixed integer $r\geq4$, the $K_r-e$-free graph sandwich problem asks whether, given graphs $G_1\subseteq G_2$ on the same vertex set, there is an induced-$K_r-e$-free graph $H$ between them. We give a deterministic algorithm taking $O(n+\alpha(G_2)^{r-3}m_2)$ time and space, where $m_2=|E(G_2)|$ and $\alpha(G_2)$ is the arboricity of $G_2$. In particular, the diamond-free case takes $O(n+\alpha(G_2)m_2)$ time. This improves the direct $O(n^r m_2)$ implementation of the previously known forced-edge closure. Our implementation maintains components of common neighborhoods indexed by $(r-3)$-cliques. A filtered frontier supports their merges within the clique-listing bound, while completion events avoid repeatedly searching for affected cliques. On feasible instances the output is contained in every feasible sandwich, independently of processing order. Applying the closure to $(G,K_n)$ gives an $O(n^{r-1})$-time bound for partitioned and nonpartitioned probe $K_r-e$-free recognition, improving the $O(n^{r+2})$ bound obtained from the direct sandwich closure. We also describe a direct static recognizer based on the same local characterization.

The Lyndon array stores, at each position of a word, the length of the longest Lyndon factor starting at that position and plays an important role in combinatorics on words, for example, in the construction of fundamental data structures such as the suffix array. In this paper, we introduce the Inverse Lyndon array, the analogous structure for inverse Lyndon words, namely words that are lexicographically greater than all their proper nonempty suffixes. Unlike standard Lyndon words, inverse Lyndon words may have non-trivial borders, which introduces a genuine theoretical difficulty. We show that the Inverse Lyndon array can be characterized in terms of the next greater suffix array together with a border-correction term, and we prove that this correction coincides with a longest common extension (LCE) value. Building on this characterization, we adapt the nearest-suffix framework underlying Ellert's linear-time construction of the Lyndon array to the inverse setting, obtaining an O(n)-time algorithm for general ordered alphabets. Finally, we show that the Inverse Lyndon array can also be used to reconstruct the canonical inverse Lyndon factorization in linear time.

We study the Busy Machine Time with Preemption and Migration and One Resource Requirement problem, motivated by energy minimization in cloud data centers. Given unlimited identical-capacity machines and jobs with release times, deadlines, processing times, and resource requirements, we allow free preemption and migration at integer times and seek to minimize total machine busy time. The problem is NP-hard, and previous results consist of a 2-approximation, 2-competitive algorithm for the case of uniform heights. We obtain a 22/9 < 2.445-approximation algorithm and a 2.5-competitive online algorithm, both running in O(n^2 log n) time. Our methods are based on new non-asymptotic performance bounds for the First Fit Decreasing algorithm for Bin Packing, and a new generalization of Span Minimization, the Huge-Tiny Busy Time problem, for which we present an exact offline algorithm and an optimal (3/2)-competitive deterministic online algorithm.