Papers

We study the best separable state problem (BSS), which asks for the maximum acceptance probability of a quantum measurement over unentangled states. In classical terms, the goal is to maximize $\langle(x \otimes y), M (x \otimes y)\rangle$ over unit vectors $x,y$ where $0 \preceq M \preceq I$; we call this value $\mathrm{BSS}(M)$. We study $\mathrm{BSS}$ in the "perfect completeness" regime, where given $M$ such that $\mathrm{BSS}(M) = 1$ the goal is to find the best possible solution $x,y$ – this generalizes the problem of finding a rank-one matrix as close as possible to a given subspace of $\mathbb{R}^{n \times n}$ guaranteed to contain a rank-one matrix. The strongest known algorithmic guarantees for this problem are: (1) an algorithm which finds a solution with value $1-\varepsilon$ in time $\exp(\sqrt{n} (\log n)^{O(1)} / \varepsilon^2)$, due to Barak, Kothari, and Steurer, and (2) an algorithm which finds a solution with value $q/n$ in time roughly $n^{O(q)}$, due to Bhattiprolu, Ghosh, Guruswami, Lee, and Tulsiani. We give a much simpler approach to rounding the SoS relaxation, generalizing the canonical "global correlation rounding" technique, and obtain a better running time. Given $M$ with $\mathrm{BSS}(M) = 1$, our algorithm finds a solution with value $1-\epsilon$ in time $n^{O(\sqrt{n/\varepsilon})}$, and a solution of value $q/n$ in time $n^{O(\sqrt q)}$. Using the same techniques, we prove a new variant of the "pinning lemma", a measure-decomposition theorem widely used in LP/SDP rounding, high-dimensional probability, and statistical physics, which we believe is of independent interest.

A number of fundamental graph problems admit simple algorithms based on iterative peeling: repeatedly remove all vertices whose current degree is below a fixed threshold. This paradigm underlies algorithms for density-dependent edge orientation, density-dependent coloring, densest subgraph, and $k$-core decomposition. In this paper, we study these problems in the sub-linear MPC model and achieve the following round-approximation tradeoffs. For density-dependent edge orientation, given any integer $t > 0$, we compute an orientation with maximum out-degree at most $(2+\epsilon)(t+1)\alpha(G)$ in $O(\lg^{1/(t+2)} n \cdot \operatorname{poly}(\lg \lg n))$ rounds, where $\alpha(G)$ denotes the minimum possible maximum out-degree of an orientation of $G$. In the $\operatorname{poly}(\lg\lg n)$-round regime, this gives an $O(\lg\lg n/\lg\lg\lg n)$-approximation, improving the approximation factor of the recent work by Ghaffari and Grunau [PODC 2025]. We obtain a similar improvement for density-dependent coloring. For densest subgraph, we obtain a $(4+\epsilon)$-approximation in $\widetilde O(\lg^{1/3} n)$ MPC rounds and a $(6+\epsilon)$-approximation in $\widetilde O(\lg^{1/4} n)$ MPC rounds. This improves the $\widetilde O(\sqrt{\lg n})$ round complexity of Ghaffari, Lattanzi, and Mitrović [ICML 2019] with a slightly larger approximation factor. This is the first $O(1)$-approximate algorithm for densest subgraph to break the $\Theta(\sqrt{\lg n})$ round-complexity barrier in the sub-linear MPC model. For $k$-core decomposition, given any integer $t > 0$, we compute approximate coreness values within a factor of $(2+\epsilon)(t+1)$ in $O(\lg^{1/(t+2)} n \cdot \operatorname{poly}(\lg \lg n))$ MPC rounds for any integer $t > 0$. This improves the $\widetilde O(\sqrt{\lg n})$ round complexity of Ghaffari, Lattanzi, and Mitrović [ICML 2019], again giving a round-approximation tradeoff.

We study online discrepancy minimization: vectors $v_1,\ldots,v_T\in\mathbb{R}^n$ arrive sequentially, and each must immediately be assigned a sign $x_t\in\{\pm1\}$, with the aim of minimizing $\|\sum_{t=1}^T x_t v_t\|_\infty$. We give a polynomial-time potential-based algorithm combining a regularization of the $\ell_\infty$-norm with restriction to an adaptively chosen coordinate set. For i.i.d. inputs with independent, symmetric, centered, unit-variance sub-Gaussian coordinates of sub-Gaussian norm at most $\sigma$, the algorithm achieves terminal discrepancy $O(\sigma^8\sqrt{n})$ with probability at least $1-\exp(-\Omega(\sigma^3\sqrt{n}))$. If the coordinates are independently masked by Bernoulli variables with mean $k/n$, where $k\gtrsim(\log n)^2$, the bound improves to $O(\sigma^8\sqrt{k})$, with failure probability $\exp(-\Omega(\sigma^3\sqrt{k}))$. Both guarantees hold for every prescribed finite horizon $T$, with no dependence on $T$. The dense result substantially generalizes a theorem of Bansal and Spencer (2020) for Rademacher inputs and gives an efficient $O(\sqrt{n})$ bound for Gaussian inputs, as conjectured by Gamarnik et al. (2022). When $T$ is polynomially larger than $n$, this is conditionally close to optimal: under worst-case hardness assumptions for standard approximate lattice problems, Vafa and Vaikuntanathan (2025) showed that no polynomial-time algorithm, even offline, can improve the $\sqrt{n}$ scale by a fixed polynomial factor in $T/n$.

For a constant $\eta \geq 0$, Pathwidth-$\eta$ Deletion is the problem of deciding whether, for a given graph $G$ and integer $k$, there is a set $S \subseteq V(G)$ of size at most $k$ such that the pathwidth of $G - S$ is at most $\eta$. The problems Treewidth-$\eta$ Deletion and Treedepth-$\eta$ Deletion are defined similarly for the parameters treewidth and treedepth, respectively. A landmark result of Fomin et al. [FOCS, 2012] shows that, for any constant $\eta$, all three problems admit a kernel on $O(k^{c(\eta)})$ vertices, where $c(\eta)$ is a constant depending on $\eta$. Giannopoulou et al. [ACM TALG, 2017] show that, in some sense, this result is optimal for Treewidth-$\eta$ Deletion: for $\eta \geq 2$ and even when parameterizing by the size of a vertex cover $M$ of the input graph, there is no kernel of size $O(|M|^{\frac{\eta+1}{2}-\varepsilon})$, for any $\varepsilon > 0$. Contrasting this result, they prove that Treedepth-$\eta$ Deletion admits a uniform polynomial kernel, that is, a kernel of size $O(k^c)$ for a constant $c$ that is independent of $\eta$. In comparison, the question whether Pathwidth-$\eta$ Deletion admits a uniform polynomial kernel has been neglected in the literature. As treewidth and pathwidth tend to behave similarly, it is natural to expect that no uniform kernel exists when parameterizing by the size of a vertex cover. Surprisingly, we show this not to be the case. More concretely, we prove the existence of a uniform polynomial kernel for Pathwidth-$\eta$ Deletion when parameterizing by (1) the solution size $k$ plus the size of a set $M$ such that $G - M$ has bounded treedepth; (2) the (vertex-deletion) distance to pathwidth-$1$ graphs; (3) the distance to the class of graphs with treedepth at most $\eta + 1$. This leads us to conjecture that Pathwidth-$\eta$ Deletion admits a uniform kernel when parameterizing by the solution size $k$.

We study the self-intersection time of the non-backtracking random walk on connected undirected graphs. For every fixed $\Delta \geq 3$ we show that the expected self-intersection time is $O(\sqrt{n} \log n)$ on $n$-vertex graphs with minimum degree at least $3$ and maximum degree at most $\Delta$. For regular graphs with a uniform spectral gap, we improve this to $O(\sqrt{n})$. We also show an $\Omega(\sqrt{n})$ lower bound on a class of regular expanders. Our upper bound on the expected self-intersection time implies an improved mixing time bound on Glauber dynamics for the Ising model on $\Delta$-regular graphs at the tree uniqueness threshold.

In the Ultrametric Violation Distance problem, we are given a set of distances between $n$ points, and the goal is to modify the minimum number of distances so that the resulting set forms a valid ultrametric. In other words, the task is to fit an ultrametric to the given data, where the quality of the fit is measured by the $\ell_0$-norm of the error. While variants of this problem under the $\ell_\infty$ and $\ell_1$-norms have been well studied, the complexity of Ultrametric Violation Distance under the $\ell_0$-norm remained largely unexplored until recently. This changed with the work of Cohen-Addad, Fan, Lee, and Mesmay [FOCS 2022], who introduced a constant-factor approximation algorithm. Significant further progress on approximation algorithms was made in subsequent work by Charikar and Gao [SODA 2024], and by An, Kao, Lee, and Lee [FOCS 2025]. In this paper, we initiate a systematic study of Ultrametric Violation Distance from the perspectives of kernelization and fixed-parameter tractability (FPT). By the work of Fan, Gilbert, Raichel, Sonthalia, and Van Buskirk [SWAT 2020], the problem is known to be FPT when parameterized by the number of violated distances $k$. We show that the problem admits a kernel with $\mathcal{O}(k^2)$ points. Additionally, we present a single-exponential-time algorithm with running time $9^k \cdot n^{\mathcal{O}(1)}$, which is asymptotically tight.

We study constructive discrete multimarginal optimal transport (MOT) among $m$ distributions on $n$ points, where the cost tensor $C$ has $N=n^m$ entries. For additive accuracy $\varepsilon$, let $\kappa=\max\{1,(\max C-\min C)/\varepsilon\}$. Classically, we give two algorithms that return exactly feasible additive-$\varepsilon$ couplings. A deterministic box–simplex method with rounding runs in $O(m^2N\kappa\log N)=\widetilde O(m^3N\kappa)$ time, while an exact reduction to positive packing followed by rank-one completion runs in randomized time $\widetilde O(m^2N\kappa)$. At fixed $\kappa$, both match the $\Omega(N)$ cost of writing a dense coupling, up to factors in $m$ and logarithms. Quantumly, in the general entry-access model, tensor Sinkhorn followed by sparse recovery returns an exactly feasible additive-$\varepsilon$ coupling as a classical list of $\widetilde O(m^2n\kappa^2)$ atoms, using $\widetilde O(m^4\sqrt{Nn}\,\kappa^3)$ coherent cost queries without materializing the tensor. Finally, for fixed $m$ and constant normalized accuracy, we prove sparse-output lower bounds of $\widetilde\Omega(N)$ randomized classical queries, even with unrestricted output size, and $\widetilde\Omega(\sqrt{Nn})$ quantum queries for outputs with at most $n\operatorname{polylog}(n)$ atoms. Hence, for fixed $m$ and normalized accuracy between $(\log n)^{-O(1)}$ and a sufficiently small constant, explicit sparse MOT construction has query complexity $\widetilde\Theta(N)$ classically and $\widetilde\Theta(\sqrt{Nn})$ quantumly.

Persistent efforts in recent years have been devoted to devising distributed algorithms for fundamental optimization problems in planar graphs. In particular, for Single-Source Shortest-Paths, there is an $\tilde O(D^2)$-rounds exact algorithm [Li, Parter STOC'19] for directed planar graphs, and an $\tilde {O}(D)$-rounds $(1+o(1))$-approximation algorithm [Rozhon, Grunau, Haeupler, Zuzic, Li STOC'22] for undirected planar graphs (where $D$ is the graph's hop-diameter). Recently [Abd-Elhaleem, Dory, Parter, Weimann PODC'25], a matching bound for the exact case was obtained for the Maximum $st$-Flow problem. Namely, an $\tilde O(D^2)$-rounds exact algorithm for directed planar graphs. However, for the approximate case, they give a $D\cdot n^{o(1)}$-rounds $(1-o(1))$-approximation algorithm for undirected planar graphs that works only for the special case where both $s$ and $t$ lie on the same face. In this paper, we remove the restriction that both $s$ and $t$ must lie on the same face (we also eliminate the $n^{o(1)}$ factor). Namely, we present the first distributed near-optimal $\tilde{O}(D)$-rounds $(1-o(1))$-approximation algorithm for Maximum $st$-Flow in general undirected planar graphs. Our main technical contribution is a distributed implementation of the classical Reif's [SICOMP'83] centralized algorithm. This is achieved by a careful recursive incision procedure on the planar dual $G^*$ of the graph $G$. It is challenging, because we need to simulate dynamic changes (incisions) over the dual graph $G^*$, while we can only communicate over the input graph $G$.

We introduce the Safe Bicycle Network with Bounded Detours (SBNBD) problem, motivated by upgrading rural road networks for bicycle traffic. Given an undirected graph with safe and unsafe edges, edge lengths, upgrade costs, terminal pairs, a budget, and a detour factor $\alpha$, the task is to upgrade unsafe edges so that each terminal pair is connected by a safe path of length at most $\alpha$ times its shortest-path distance in the original network. We study SBNBD from a parameterized perspective. We prove strong NP-hardness on restricted graph classes, including planar graphs of treewidth two, graphs with feedback vertex set number one, and graphs of maximum degree three, and complement these lower bounds with polynomial-time algorithms for trees and graphs of maximum degree two. We show fixed-parameter tractability for the number of unsafe edges and prove matching SETH-based lower bounds, a polynomial-kernel lower bound, and W-hardness for natural parameters. Our main structural result maps any instance to an equivalent instance with $O(\mathrm{fes}+p)$ vertices and edges, where $\mathrm{fes}$ is the feedback edge number and $p$ the number of terminal pairs; this yields fixed-parameter tractability for $\mathrm{fes}+p$. Finally, we evaluate ILP-based algorithms on OpenStreetMap road networks for small German municipalities and their surroundings. The instances have small treewidth upper bounds and moderate feedback edge structure. Preprocessing based on the $\mathrm{fes}+p$ reduction and tree-decomposition-based cut generation both improve exact solving, especially on harder instances. Experiments with different detour factors show that increasing $\alpha$ can reduce the upgraded-edge length, revealing trade-offs between upgrade cost and allowed relative detours. Overall, structural graph parameters provide a useful algorithmic lens for safe bicycle-network design.

We prove that no randomized integral or fractional algorithm for online vertex cover under general vertex arrivals achieves a competitive ratio strictly below $1+\sqrt{e}/2\approx1.824360635$, even on bipartite graphs and against an oblivious adversary. This improves the previous lower bound of approximately $1.753$. Our proof extends the complete-bipartite alternating construction of Wang and Wong to an arbitrary number of alternations. The resulting adversary is described by a monotone integral recurrence. If the recurrence never violates the competitive budget, its iterates converge to an integrable fixed point; classifying all such fixed points forces the excess ratio to be at least $\sqrt{e}/2$. A truncated discrete recurrence and a Riemann-sum argument convert every strict continuous violation into a finite, algorithm-dependent but realization-oblivious input. We also exhibit a critical fixed point showing that $1+\sqrt{e}/2$ is the exact limit of this homogeneous complete-bipartite recurrence, rather than a numerical artifact.

We study the stretch–size tradeoff for geometric spanners in high-dimensional $\ell_p$ spaces. Our main contribution is a simple proof of a lower bound shown by Har-Peled, Indyk, and Sidiropoulos [SODA 2013]: Every $2$-hop $t$-spanner of the pointset $\{0,1\}^d$ under $\ell_2$ norm has at least $(2^d)^{1+\Omega(1/t^2)}$ edges. Our proof further extends this result to spanners with Steiner vertices. In addition, we establish a connection between bounded-hop spanners and general spanners, as follows. If every subset $Y$ of an $n$-point metric has a $t$-spanner with at most $\mu|Y|$ edges, then the metric has an $O(t)$-hop $O(t)$-spanner of size $O(n(\mu+\log n))$. Consequently, hop-restricted spanner lower bounds for a metric imply lower bounds without hop restriction for one of its subsets.

The single-selection prophet inequality is a canonical Bayesian online selection problem in which independent nonnegative values arrive sequentially and the decision-maker must irrevocably select at most one. Classical single-threshold guarantees are tight in the worst case, but the hard instances that prove tightness are highly irregular: the prophet's advantage is driven by rare, very large realizations of the maximum. We refine this worst-case picture by imposing a bound on the relative variance of the prophet's value, $\mathrm{Var}(\max_{i\in[n]}X_i)/\mathbb E[\max_{i\in[n]}X_i]^2$. This yields a nonparametric complexity measure that interpolates between deterministic instances, where the full prophet value can be recovered, and the unrestricted worst-case regime. Our main technical contribution is a general kernel method for single-threshold prophet inequalities. The method represents an instance by the quantile function of the maximum and rewrites the payoff of a threshold as a linear kernel functional of this quantile. This turns the worst-case analysis into an infinite-dimensional convex program, restores strong minimax duality in quantile space, and reduces the bounded-variance adversary's problem to a one-parameter variational family. Applying this framework, we obtain an exact characterization of the IID bounded-variance curve and asymptotically optimal finite-horizon thresholds, a closed-form expression for the fixed-order non-identical model, and a prophet-secretary lower-bound program together with a strict separation from the IID benchmark at every positive finite variance constraint. As a further application of the same kernel viewpoint, we derive an exact formula for IID random horizons under a convexity condition on the horizon pgf, which includes monotone-hazard-rate horizons, highlighting the broad applicability of this new technique for single threshold settings.

We study online metric matching with per-request action predictions. On the real line, every deterministic $(1+\varepsilon)$-consistent algorithm has robustness at least $1+\sum_{j=1}^{k-1}2^{j+1}/\varepsilon^j$, and we give a deterministic algorithm for arbitrary metrics with the same leading term. Thus, for every fixed $k$, $\lim_{\varepsilon\downarrow 0}\varepsilon^{k-1}R_k^{\mathbb{R}}(1+\varepsilon)=\lim_{\varepsilon\downarrow 0}\varepsilon^{k-1}R_k(1+\varepsilon)=2^k$. The comparison is uniform up to an absolute constant for $0<\varepsilon\le 1/(k-1)$. We determine the two-server trade-off in both settings and the real-line three-server value $1+4/\varepsilon+8/\varepsilon^2$ for $0<\varepsilon\le\sqrt{13}-3$. When the predicted labels are distinct, the algorithm pays at most $(1+\varepsilon)$ times the cost of the predicted matching. For randomised algorithms, the fixed-$k$ dependence remains $\Theta_k(1/\varepsilon^{k-1})$. Uniformly in $k$, robustness is at most $M_0(\varepsilon)\rho_k^0$, where $\rho_k^0$ is the optimal strict prediction-free randomised ratio on the real line and $M_0(\varepsilon)=(2e+o(1))e^{2/\varepsilon}$. For every $\eta>0$, a lower bound $\exp((2-\eta)/\varepsilon)$ holds once $k\ge C_\eta/\varepsilon$. The randomised upper bound follows from a comparison theorem for two online algorithms whose states can be coupled at a cost bounded by their cumulative costs. For every fixed $c>1$, the least comparison factor $M^*(c,\varepsilon)$ under these assumptions satisfies $\lim_{\varepsilon\downarrow 0}\varepsilon\log M^*(c,\varepsilon)=2$. The guarantee is strictly multiplicative and has no diameter-dependent additive term. Irrevocable metric matching and metrical task systems satisfy the assumptions, and the exponent $2$ is optimal under them.

We establish counting and sampling analogues of the Lovász Local Lemma: we give efficient algorithms for approximately counting and exactly sampling satisfying assignments of general constraint satisfaction problems (CSPs) in the local lemma regime $$4 \mathrm{e} p (D+1)^2\leq 1, $$ where $p$ is the maximum constraint violation probability and $D$ is the maximum dependency degree. This condition is tight up to constant factors under $\mathbf{NP}\neq\mathbf{RP}$, matching known hardness bounds for counting and sampling in natural subclasses of CSPs. Our key ingredient is a novel $2$-tree expansion for constraint marginal probabilities that exhibits exponential decay of correlations throughout this regime. This expansion yields deterministic polynomial-time approximate counting for fixed local parameters, randomized approximate counting with quadratic cost, and exact sampling in expected near-linear time when the local parameters are fixed.

The Learned Count-Min Sketch (LCMS) is a learned data structure that estimates element frequencies in a multiset and has been experimentally shown to outperform classical data structures in the capacity-accuracy trade-off. However, its performance depends heavily on parameter selection. Because systematic optimization has not been adequately discussed, previous approaches relied on inefficient brute-force methods. In this study, we propose a method to rapidly optimize the parameters of the original LCMS. We experimentally confirmed that when the machine learning model performs well enough, using a single hash function is sufficient to optimize the weighted error metric. Based on this, we introduce a ternary search approach to efficiently find the optimal proportion of Unique Buckets. Our method achieves the same performance as brute-force approaches while speeding up parameter optimization by $216$-$729$ times when the machine learning model's performance is sufficient. Furthermore, even when the model's performance is suboptimal, our approach still determines appropriate parameters $221$-$740$ times faster than the brute-force approach.

Let $a$ and $b$ be distinct real weights. An $\{a,b\}$-edge-weighting of a tree assigns one of these weights to each edge and is proper if adjacent vertices have different sums of incident edge weights. For every such pair, we give an explicit structural characterization of the trees that do not admit a proper $\{a,b\}$-edge-weighting. If $ab(a+b)\neq0$, then $K_2$ is the only tree without such a weighting. If $a+b=0$, then a tree has no proper $\{a,b\}$-edge-weighting exactly when every vertex has degree $1$ or $3$ and the subgraph induced by the degree-$3$ vertices has a perfect matching. For the remaining case $ab=0$, form the spanning forest consisting of the edges whose deletion leaves two odd-order components. A tree $T$ has no proper $\{a,b\}$-edge-weighting exactly when both bipartition classes have odd order and every component of this forest satisfies two conditions. First, every component satisfies the preceding degree-and-matching condition. Second, within each component, the degree of a vertex $v$ in the forest plus twice the number of incident edges $e$ outside the forest for which the component of $T-e$ not containing $v$ has an odd number of vertices from each bipartition class is independent of $v$. For every fixed pair of distinct real weights, the proofs yield a linear-time algorithm that decides whether a proper $\{a,b\}$-edge-weighting exists and constructs one when it does.

We study the online Busy Time scheduling model on a single machine of unbounded capacity, with non-preemptive jobs. In our setting, flexible jobs arrive online with a processing time and deadline, both of which become known to the algorithm at the job's arrival time. The goal is to schedule jobs on the machine to finish all jobs by their deadlines, so that the total time when the machine is turned on (busy time, also called span in this setting) is minimized. We present a randomized online algorithm with an exact competitive ratio of exactly (1 + e) < 3.72, where e is Euler's number, against an oblivious adversary, and show that no randomized algorithm can have a competitive ratio better than $e$ against an oblivious adversary. This lower bound holds even for algorithms that are allowed to restart jobs and are given lookahead. Previous work either dealt with special cases, or with deterministic algorithms, for which the known upper bound on the competitive ratio is 5 and the known lower bound is 4. Our findings offer fresh insights into randomization in online energy-aware scheduling. In the setting where jobs have uniform processing times, and a job that is started by the algorithm must be finished, we show that no deterministic algorithm can do better than 2, even when the jobs are agreeable. This deterministic lower bound also holds for uniform processing times and restarts in the scenario where a job's deadline is only revealed at its starting deadline. For Capacitated Busy Time with p_max-lookahead, we obtain a deterministic online algorithm with competitive ratio at most 8 and a randomized online algorithm with competitive ratio at most (4+ e) < 6.72, against an oblivious adversary.

We consider standard top-down recursive Mergesort, where we do a single comparison before calling merge to check if the two recursively sorted subproblems happens to already be correctly ordered. (If so, we can skip the merging step). We show for any input $A[0..n)$ of elements consisting of $r$ runs (maximal increasing contiguous subranges in $A$) of respective lengths $L_1,\ldots, L_r$, the mergecost $M$ (the sum of output sizes of all merges) satisfies $M \le (\mathcal H+3)n$ for $\mathcal H = \sum_{i=1}^r (L_i / n) \log_2(n/L_i)$ the runlength entropy.

We show that set cover on a universe of size $n$ and with sets of size at most $k$ can be solved in time $2^{(1-1/k+O(1/k^{3/2}))n}$. This improves on a $2^{(1-0.929/k)n}$-time algorithm of Björklund (STACS 2010) for all sufficiently large $k$.

We give deterministic parallel algorithms that compute a $(\Delta+1)$-coloring and a maximal independent set for a simple graph with $n$ vertices and $m$ edges in $O(n+m)$ work and $O(\mathrm{poly}\log n)$ depth.

By using a simple textbook reduction from vertex cover, we show that the following three problems are all UG-hard to approximate with constant-factor smaller than two; minimum weakness strong triadic closure, cluster deletion and constrained correlation clustering. Additionally, we analyze the well-known low-diameter decomposition by Calinescu, Karloff and Raban applied to the standard LP relaxation semi-metric for constrained correlation clustering. As opposed to traditional pivot-based approaches, a CKR partition elegantly handles must-link and cannot-link constraints. It guarantees a 3-approximation in expectation, which matches the original approximation ratio by van Zuylen and Williamson. We conjecture that it in fact achieves a strictly better than 3-approximation, yet this remains an open problem.

We give a randomised algorithm for Equal Subset Sum that, on $n$ arbitrary integers of at most $m\le2^n$ bits, runs in time $O\big((5/3)^n\mathrm{poly}(n)+n^2m\big)$, never outputs a non-solution, and outputs a solution with probability $1-2^{-\Omega(n)}$ whenever one exists.

In open addressed hashing, quadratic probing is attractive for striking a nice balance between having a high locality of reference and a low number of probes per search. However, these are empirical observations, not theoretical guarantees. Indeed, until recently, it was not known whether quadratic probing had constant expected insertion cost under any positive load factor $\alpha > 0$, even with uniformly random hash functions. In a recent breakthrough—albeit a numerically understated breakthrough—Kuszmaul and Xi (2024) proved that any fixed offset sequence (including quadratic probing) does, in fact, have constant expected insertion cost for load factors $\alpha \leq 8.9\%$. This is well below what we would like to prove, that quadratic probing has constant insertion cost for any load factor $\alpha < 1-\epsilon$ bounded away from 1. In this paper, we prove that open addressed hashing with any fixed offset sequence has constant expected insertion cost for load factors up to $35.74\%$, and that for quadratic probing in particular, we can increase the load factor to $37.61\%$. Our main innovation is a new type of witness forest for recording collisions among the probe sequences.

We study per-colour independent (k)-rainbow domination and its connection with independent domination in generalized prisms. Building on the known prism identity and the trivial regime above the maximum degree, we focus on the boundary case where the number of colours equals the maximum degree. For every fixed (k\ge 3), we prove that the decision problem remains NP-complete even on a highly restricted class of graphs: (C_4)-free, bipartite, ((k,2))-biregular subdivision graphs arising from simple (k)-regular graphs. The reduction gives an exact correspondence between optimal rainbow-independent dominating functions on the subdivision graph and proper (k)-edge-colourings of the original graph. We also introduce an excess parameter measuring how far the domination number lies above its natural lower bound. For cubic graphs, this excess coincides with the classical edge-colouring degree and therefore with standard resistance parameters for subcubic graphs. These results reveal a sharp one-unit threshold: above the maximum degree the problem becomes trivial for every graph, while at the boundary NP-hard instances already occur within a very narrow structural family.

Web and social graphs are analyzed repeatedly for community structure, yet many of their edges are redundant for this purpose, which motivates sparsification. Existing sparsifiers preserve spectral quantities, cuts, local similarity, or a single clustering, but none preserves the thresholded connectivity structure of an edge-relevance filtration, the multiscale pattern by which groups form at high relevance and merge through weaker bridges. We study persistent-connectivity-preserving sparsification: given a graph, an edge-relevance filtration, and a proxy partition computed once during preprocessing, select a budgeted subgraph that preserves the labeled component partition at every threshold, and hence the zero-dimensional persistence diagram, while retaining community evidence for later analyses. Our method, TopoBudget, first extracts a tie-aware persistence backbone that enforces this constraint, then allocates the residual edge budget by greedily maximizing a backbone-conditioned submodular objective that rewards balanced recovery of proxy-internal degree. We prove exact preservation of the component partition at every threshold, and that the conditioned objective is monotone and submodular, so greedy attains a (1-1/e) guarantee for the fixed-backbone residual problem. On held-out synthetic benchmarks and six real Web and social graphs at equal budget, TopoBudget gives the strongest community preservation among topology-preserving methods under Louvain, remains competitive under Infomap, incurs zero topology mismatch, and runs substantially faster than an effective-resistance baseline. A no-backbone ablation shows that, on the real graphs, the mandatory backbone improves average quality while providing the exact guarantee. TopoBudget thus couples exact multiscale connectivity with budgeted, reusable community preservation.

In this note, we show that the approximation algorithm for Boolean Max $k$-CSP presented in [Makarychev and Makarychev 2014] yields a $(1-o_k(1))k/2^k$ approximation, as conjectured in [Makarychev and Makarychev 2017]. This improves the previous guarantee of $(0.626612-o_k(1))k/2^k$ from [Makarychev and Makarychev 2014] and asymptotically matches the known hardness results. The result is a short corollary of the Gaussian stochastic domination theorem of Mulgund.

The model of bulk-synchronous parallel (BSP) computation is an emerging paradigm of general-purpose parallel computing. Bruhat decomposition is an important method in numerical linear algebra, generalising ordinary LU decomposition while providing a symmetric structured way of expessing pivoting. Block-recursive algorithms have been developed in the past for various numerical linear algebra problems, including Bruhat decomposition; however, expressing recursive algorithms in the BSP model still remains a challenge. In this paper, we consider the communication and synchronisation complexity of Bruhat decomposition in the BSP model. We develop a new block-recursive, communication-efficient BSP algorithm for Bruhat decomposition.

Real distance data rarely cooperate: measurements are noisy, observations are missing, and the numbers that result seldom satisfy the triangle inequality. A family of methods exists to correct them, and every one of those methods rests on the same hope — that analysis run on the corrected data is a more faithful surrogate for the truth than analysis run on the raw data. We are not aware of anyone having tested that hope. We test it through the problem of Metric Repair, which asks for the fewest edges whose reweighting restores the triangle inequality. We implement a suite of algorithms, covering the literature and new methods, both with theoretical guarantees and heuristics, and evaluate their performance on real and synthetic data, both inherently non metric and corrupted. We demonstrate that the algorithms' performance is determined predominantly by the type and fraction of corruption, rather than the corruption's magnitude or graph size. We further test the effect of repair on downstream tasks, namely MDS and $k$NN, and ask if the repair got the result closer to the truth compared to a corrupted instance. In most cases it did not, and we identify the culprit. A small set of edges is not enough. Finding the correct set of edges, be it an injected corruption or a natural non-metricity, is critical. Moreover, deciding on a weight rule impacts performance: on data instances with available metric ground truth, a metric repair algorithm can pull the graph further from the truth, while an oracle access to the true weights helps. Surprisingly, the opposite can be true as well. Setting the weights is not an implementation detail; it is half the problem.

In this paper, we study the perfect fair-triangle packing problem (abbreviated as PFTP), which incorporates the fairness criterion from fair clustering into the maximum-weight triangle packing problem. Specifically, the input is an edge-weighted complete graph $G = (V, E)$ with $|V| = 3n$, where each vertex is colored red or blue. A fair triangle is a triangle containing vertices of both colors. PFTP asks for a partition of $V$ into $n$ fair triangles such that the total edge weight is maximized. To the best of our knowledge, this is the first paper to study PFTP. PFTP is NP-hard. Our main contributions are a deterministic $\frac 13$-approximation algorithm running in $O(n^3)$ time and an improved randomized $(\frac {16}{47}-\epsilon)$-approximation algorithm running in $O(n^4)$ time, where $\epsilon > 0$ is a fixed small constant. The deterministic algorithm is matching-based whereas the randomized algorithm employs several additional techniques, including maximum-weight $[1, 2]$-factor, a random cycle-breaking procedure, and maximum-weight matchings. Keywords: Triangle packing; fairness; approximation algorithms; randomized algorithms

We study an online variant of the Traveling Salesperson Problem (TSP) in which $n$ points arrive sequentially and must be inserted into an evolving tour. In the classical setting where arbitrary insertions are allowed, an $O(\log n)$-competitive algorithm has been known since the 1970s (Rosenkrantz, Stearns and Lewis 1977, Imase and Waxman 1991). Recently, Abrahamsen, Bercea, Beretta, Klausen, and Kozma [ESA 2024] introduced online metric TSP, a stricter model in which each arriving point must be assigned to a distinct cell of an array of size $m \ge n$, with the final tour order induced by the non-empty cells; the parameter $m$ captures the space usage of the algorithm. When $m = 2^{n}$, this model recovers arbitrary insertions and therefore admits an $O(\log n)$-competitive algorithm. In contrast, when $m = n$, i.e., when each point's position is fixed on arrival, Bertram [ESA 2025] recently showed that the competitive ratio is $\Theta(\sqrt{n})$. We investigate the tradeoff between space usage and competitiveness between these extremes. We note that this tradeoff was previously explored by the authors [SODA 2026] for the online sorting problem, which is the special case of online metric TSP on a line metric. Our main result is a deterministic online metric TSP algorithm using $m = (1+\epsilon) n$ space that achieves a competitive ratio of $O(\log^{3} n / \epsilon)$, for any $\epsilon \le 1$. In particular, increasing the space from $n$ to $2n$ improves the competitive ratio from $\Theta(\sqrt{n})$ to $O(\log^{3} n)$. We complement this with a lower bound showing that for $m = n^{1+\epsilon}$, any deterministic algorithm has a competitive ratio $\Omega(1/\epsilon)$, for all $\epsilon \ge \Omega(\log \log n / \log n)$. Consequently, even with $m = O(n \cdot \mathrm{polylog}(n))$, deterministic algorithms cannot achieve a constant competitive ratio.