Papers

We study the problem of maximizing a general and not necessarily monotone submodular function subject to a matroid independence constraint. This problem has a rich history, with multiple algorithms using both discrete and continuous methods. Recently, [Ganz-Rozenman, Kulik, Schwartz and Singh STOC `26] presented a novel hybrid approach based on a Poisson process that aims to combine the strengths of both discrete and continuous methods for the special case of the problem where the submodular function is monotone. Our main result is a new Poisson process based hybrid algorithm that works for both non-monotone and monotone submodular functions, achieving an approximation of $ \frac{1}{e}$ for the former and $1-\frac{1}{e}$ for the latter. The algorithm always maintains a feasible set and at random times governed by the Poisson process it performs a single element swap based on a best response set. The new idea is that our algorithm is spiteful as it can purposefully discard an element that is in both the current set and the best response set. Surprisingly, this spiteful step does not harm the approximation our algorithm achieves for monotone submodular functions but is necessary for the non-monotone case. As applications, we obtain fast approximation algorithms for maximizing non-monotone submodular function subject to a general matroid independence constraint as well as faster algorithms for a partition matroid.

We study exact simulation of diffusions via rejection sampling on path space using unbiased estimators of the density ratio obtained from Girsanov's theorem. When applied to the underdamped Langevin diffusion, it yields an algorithm for sampling from a strongly log-concave and log-smooth distribution with condition number $\kappa$, in dimension $d$, to accuracy $\varepsilon$ in Rényi divergence, in $\widetilde O(\kappa^{2/3} d^{1/3}\,\mathrm{polylog}(1/\varepsilon))$ queries. Under a third derivative bound, the dimension dependence improves to $d^{1/5}$. This improves substantially over the prior state-of-the-art complexity of $\widetilde O(\kappa d^{1/2}\,\mathrm{polylog}(1/\varepsilon))$ for the Metropolis-adjusted Langevin algorithm, and over the $d^{1/4}$ dimension dependence of Metropolized Hamiltonian Monte Carlo under the same third derivative bound. We also present applications to the mirror Langevin diffusion, and for obtaining Fisher information bounds in the non-log-concave case.

We prove a tight impossibility result for online vertex cover under edge arrivals. No randomized integral or fractional algorithm achieves a competitive ratio strictly below $2$ against an oblivious adversary, even on bipartite graphs. Since the standard algorithm that takes both endpoints of every uncovered edge is $2$-competitive, this settles the optimal ratio. Our proof is a direct reduction from the recent breakthrough blueprint framework of Assadi, Jiang, and Xiang.

Recent breakthroughs in Cluster Editing have motivated attempts to adapt these approaches to obtain better-than-$2$ approximations for Cluster Deletion. We rule out this possibility under the Unique Games Conjecture: Cluster Deletion is NP-hard to approximate within a factor of $2-\epsilon$ for every fixed $\epsilon>0$, matching the known $2$-approximation [Veldt et al., WWW 2018]. Our approximation-preserving reduction from Vertex Cover also implies NP-hardness of approximation within $\sqrt2-\epsilon$. We also show that better-than-$2$ approximations are possible in restricted settings. We close the paper with a brief discussion of the relationship between Cluster Editing and Bad Triangle Transversal. In particular, we give a $31$-vertex graph~$G$ for which the two optimal values differ, answering an open question of Adriaens and Tatti [ICML 2026].

The Graph Edit Distance (GED) is a widely used graph similarity measure asking for the minimum cost of a sequence of edits transforming one (labeled) graph into another. The considered edit operations are deletion, insertion, and relabeling of nodes and edges. Special cases include the Graph Isomorphism problem, as well as many other graph problems that ask for the existence or minimum cost of a certain substructure, like the Traveling Salesman or Maximum Clique problem. We present a novel exponential time algorithm to compute the exact GED and a corresponding edit sequence in $O^*(4 + \varepsilon)^n$ time and polynomial space, provided one of the two graphs admits strictly sublinear balanced separators. In particular, the claimed runtime holds if one of the graphs is $K_h$-minor free (e.g., planar), or has bounded treewidth, which is the case for many real-world applications (e.g., all instances in GEDLIB). This substantially improves the best known worst-case running time bounds of $O^*(n!)$ for these graph classes.

Treewidth is a fundamental graph invariant that quantifies how tree-like a given graph is. It is extensively used with dynamic programming to design fixed-parameter tractable algorithms for many NP-hard graph combinatorial optimization problems. However, despite broad theoretical applicability, treewidth dynamic programming (TDP) does not scale in practice beyond graphs with very small treewidth. Rather than applying TDP as a standalone technique, in this paper, we demonstrate that TDP can serve as a broadly applicable enhancer for a wide range of graph combinatorial optimization algorithms. Our framework leverages the concept of treewidth modulators, which refer to vertex sets whose removal significantly reduces the treewidth. We further propose an empirically efficient procedure for generating such treewidth modulators. To enhance an algorithm $\textit{A}$, we use $\textit{A}$ to heuristically make decisions on the modulators vertices, after which the remaining decisions outside the treewidth modulators become scalable for TDP. To demonstrate the general applicability of our proposed framework. We experimented with three classic graph combinatorial optimization models: Maximum Independent Set, Minimum Vertex Cover, and Max Cut. We apply TDP to enhance algorithms across diverse paradigms, including evolutionary search, greedy heuristics, and graph-neural-network-based heuristics. For all combinations of optimization models and base algorithms, TDP significantly improves performance over the original methods. In many settings, TDP-enhanced greedy heuristics are competitive with, and sometimes clearly outperform, state-of-the-art commercial solvers.

We prove that the $\textsf{Greedy}$ binary search tree is $2^{O(\sqrt{\log\log n})}$-competitive. It is widely conjectured that $\textsf{Greedy}$ is $O(1)$-competitive, but before this work it was not known to be $f$-competitive, for any non-trivial $f(n)=o(\log n)$. Our analysis differs from prior analyses of binary search trees. It takes what might be called a "scaling" approach, where the cost at a refined scale is related to the cost at a coarser scale, and Wilber's interleave lower bound.

Budget-feasible mechanism design is a classic framework introduced by Singer, but there is still a wide gap between existing upper and lower bounds. In this paper, we significantly advance the state of the art. First, without computational constraints, we show that there exists a universally truthful budget-feasible mechanism with the following approximation ratios: - $3$ for monotone submodular valuations and $e+1$ for nonmonotone submodular valuations, improving over $3.798$ and $9.742$, respectively. - $e+1$ for XOS valuations, improving over $28$. In large markets, our approximation can be improved deterministically to $e$. - $2e+1$ for subadditive valuations, improving over $33$. In large markets, our approximation can be improved deterministically to $2e$. Moreover, for subadditive valuations, we obtain a constant-approximation mechanism that runs in polynomial time using demand queries. This improves over the previous best approximation of $O(\log \log n)$, resolving a long-standing open problem going back to Dobzinski, Papadimitriou, and Singer, who conjectured that a constant approximation requires exponentially many demand queries. We obtain these results through a simple and unifying framework based on non-truthful indirect mechanisms, recently coined compensation design. In particular, through a potential argument, we establish constant price-of-stability bounds for compensation design based on marginal-contribution payment rules, which we then translate into truthful direct mechanisms. For subadditive valuations, the core of the argument is a new smoothing lemma showing that every subadditive function can be approximated within a factor of $2$ by a self-bounding function. This is also of independent interest, readily addressing an open question in multiwinner elections by showing the existence of a $2e$-approximate core even under subadditive valuations.

Chechik, Kaplan, Thorup, Zamir, and Zwick (STACS 2016) claimed a simple deterministic linear-time comparison-based algorithm for solving deterministic two-player, turn-based, zero-sum terminal-payoff games, also known as deterministic graphical games (DGGs). We give a counterexample to their algorithm. We also give a deterministic linear-time reduction from the directed $s$-$t$ bottleneck path (BP) problem to the DGG problem. Consequently, a linear-time comparison-based algorithm for computing the value of a designated start vertex in a DGG would yield a linear-time comparison-based algorithm for directed $s$-$t$ BP. Whether directed $s$-$t$ BP admits such an algorithm has remained open since Gabow and Tarjan gave their $\mathcal{O}(m\log^* n)$-time algorithm. Thus, a positive resolution of the open question for DGGs would also resolve the corresponding open question for directed $s$-$t$ BP.

We present a new online algorithm for the well-known Multi-Level Aggregation Problem (MLAP) with arbitrary delay functions, achieving a $2D$-competitive ratio, where $D$ is the depth of the underlying tree. This result improves the current best-known competitive ratio of $O(D^2)$ and asymptotically matches the $D$-competitive bound previously known only for the deadline variant, thereby closing the asymptotic gap between the two settings. Our key technical contribution is a novel dual fitting framework that provides a unified analysis for both settings; in particular, it also establishes a $D$-competitive ratio for MLAP with deadlines. Our analysis is built upon two new ideas: a hindsight dual construction, which resolves the infeasibility issues in traditional online primal-dual methods, and a time-dependent dual packing that maintains feasibility over dynamic request sets.

The demand matching problem generalizes both the knapsack problem and the $b$-matching problem. In this problem, each edge of a graph has a demand and a weight, each vertex has a capacity, and the goal is to find a maximum weight subset of edges whose total incident demand at every vertex does not exceed its capacity. We study $(\alpha, \beta)$-bicriteria approximation algorithms, which return a solution of weight at least $1/\alpha$ times the optimum while allowing an additive capacity violation of at most $\beta$ times the maximum edge demand. We give an iterative relaxation algorithm for the demand matching problem that exploits a structural characterization of strictly fractional extreme points of the natural LP relaxation, which reduces the residual rounding problem to odd-cycle instances. Combined with a better-of-two rounding strategy, this yields $(7/6, 1)$- and $(1, 1)$-bicriteria approximation algorithms for general and bipartite graphs, respectively. We further generalize this approach to obtain a parametric family of algorithms, including a $(1, 4/3)$-bicriteria approximation. Separately, for the more general $k$-hypergraph demand matching problem, we give a greedy, combinatorial $(k, 1)$-bicriteria approximation algorithm. We complement these algorithmic results with matching lower bounds relative to the natural LP relaxation for $\beta = 0$ and all $\beta \geq 1$, completely characterizing the trade-off between weight approximation and additive capacity violation in this range.

The standard method of exponential moments for proving concentration bounds can often be replaced by an argument based on elementary symmetric polynomials. We introduce an additional element of randomness into this framework, which reduces the problem to bounding product moments over a uniformly sampled set of indices. We show that this approach gives useful bounds in three settings. For read-$\Delta$ families under limited independence, we obtain bounds governed by the degrees of randomly induced dependency subgraphs, improving the dependence on worst-case degrees. For random binary linear hashing with (semi-)random inputs, we derive fixed-bin and maximum-load bounds by controlling the rank defect of random tuples of input keys. Finally, for stochastic processes, we show how decay of product moments yields concentration bounds, recovering the spectral and mixing-time scales for finite-state Markov chains.

For parameters $\alpha,\beta \geq 1$, a spanning tree $T$ of a weighted graph $G$ rooted at a designated vertex $r$ is called an $(\alpha,\beta)$-shallow-light tree (SLT) if (i) for every vertex $v$, $d_T(r,v) \leq \alpha \cdot d_G(r,v)$ (root-stretch $\alpha$), and (ii) $w(T) \leq \beta \cdot w(\mathsf{MST})$ (lightness $\beta$). The pioneering work of Khuller, Raghavachari, and Young (SODA 1993) constructed $\left(1+\epsilon, \tfrac{2}{\epsilon}+1\right)$-SLTs for general weighted graphs, and proved that this tradeoff between root-stretch and lightness is tight even for series-parallel graphs. They further asked whether even a slight improvement, namely reducing the lightness to $\tfrac{2-c}{\epsilon}$ for any constant $c>0$, is possible in the Euclidean plane. We resolve this longstanding question in the affirmative. Specifically, we show that every Euclidean instance admits an SLT with root-stretch $1+\epsilon$ and lightness at most $\left(\frac{5}{3} + o_\epsilon(1)\right) \cdot \frac{1}{\epsilon}$, thereby significantly improving upon the longstanding $2/\epsilon$ barrier. As our second main result, we provide a construction of SLTs in the Euclidean plane, with root stretch $1+\epsilon$ and lightness at most $\left(\frac{2\pi}{\sqrt{4\pi^2+1}}+o_\epsilon(1)\right)\frac{1}{\epsilon} \approx (0.987+o_\epsilon(1))\frac{1}{\epsilon}$. Notably, this reduces the leading $2/\epsilon$ term in the lightness bound by more than a factor of two, and comes quite close to the lower bound of $\left(\frac{2\pi}{2\pi+1} +o_\epsilon(1))\right) \cdot \frac{1}{\epsilon} \approx (0.862 +o_\epsilon(1))\frac{1}{\epsilon}$ by Elkin and Solomon (FOCS 2011).

We study approximate counting and sampling algorithms for the hard-core model on $\Delta$-regular bipartite graphs under a spectral expansion condition. Let $M_G$ be the biadjacency matrix of $G$. For every fixed $\xi\in(0,1)$, we give an FPRAS for the hard-core partition function and an efficient approximate sampler whenever \[ \lambda\leq \frac{1-\xi}{\sigma_2(M_G)}. \] The main idea is to introduce a family of quadratic tilts in the left-right occupation imbalance and show that each tilted measure can be sampled efficiently using Glauber dynamics. A discrete Gaussian identity expresses the original hard-core model as an exact positive mixture of these tilted measures; truncation and simulated annealing then yield efficient counting and sampling algorithms. For the complementary high-fugacity regime, we refine the polymer-model approach and show that the required phase-dominance and cluster expansion conditions follow from the singular-spectrum bound alone. Combining the two regimes, we obtain efficient approximate counting and sampling at every fugacity $\lambda>0$ whenever \[ \sigma_2(M_G)\leq c\left(\frac{\Delta^2}{\log(\mathrm e\Delta)}\right)^{1/3} \] for an absolute constant $c>0$. In particular, this recovers all-fugacity algorithms for random $\Delta$-regular bipartite graphs for all sufficiently large $\Delta$, while providing an efficiently verifiable certificate of their success on a given instance.

We study the parameterized complexity of Induced Subgraph Isomorphism (ISI) and Maximum Common Induced Subgraph (MCIS) with respect to the cluster vertex deletion number $k$. For ISI, we give a randomized $O^*(k^{O(k)})$-time algorithm, showing that ISI is fixed-parameter tractable under this parameter and resolving an open question of Hanaka et al. [WALCOM 2026]. Our algorithm is optimal under the Exponential Time Hypothesis (ETH), and is based on a reduction to Exact Multicolored Matching solvable via algebraic techniques. For MCIS, we present a randomized $O^*(2^{O(k^2)})$-time algorithm via a reduction to a weighted variant of Exact Multicolored Matching, and we prove a matching ETH-based lower bound by showing that a $k$-by-$k$ binary matrix feasibility problem with list-constrained rows and columns admits no $O^*(2^{o(k^2)})$-time algorithm, which may be of independent interest. These results reveal that, in this setting, MCIS is strictly harder than ISI. Finally, for the three-graph variant 3-MCIS, we show that it becomes NP-hard already when each input graph has cluster vertex deletion number 2.

Worst-case-optimal (wco) join algorithms have demonstrated their power – in both theory and practice – to efficiently solve complex Basic Graph Patterns (BGPs). Modern graph query languages, such as SPARQL and GQL, have BGPs at their core, but also have a wide range of other features, including filters (aka.\ selections). Such conditions are typically handled via pre- or post-filtering, before or after processing the BGPs. In this paper we show how to uplift wco join algorithms so as to incorporate such filtering natively, improving efficiency. We demonstrate the superiority of this approach by extending the Ring – a compact index that provides wco resolution of BGPs within almost no extra space on top of the graph – so as to handle property graphs using our new techniques while retaining compactness. We implement this extension and experimentally show that it outperforms various baseline systems.

We study restricted-link augmentation to $2$-vertex-connectivity. An instance consists of a graph $G$, possibly disconnected, a set $L$ of admissible links on its vertices, integer link costs in $\{1,\dots,W\}$, and an integer $k$; the task is to add at most $k$ links of minimum total cost so that the resulting multigraph is $2$-vertex-connected. Recent work gives $O^*(k^{O(k)})$-time algorithms for unweighted $\lambda$-vertex-connectivity augmentation for every $\lambda\leq 4$ [Carmesin and Ramanujan, SODA 2026], and an $O^*((k+\lambda)^{O(k)})$-time algorithm for arbitrary $\lambda$ [Korhonen and Thorup, arXiv 2026]. We give a deterministic algorithm with running time $O^*(36^kW)$. Thus, for $\lambda=2$, the unweighted running time improves from $O^*(k^{O(k)})$ to $O^*(36^k)$, and the algorithm also handles link costs with pseudo-polynomial dependence on $W$. We reduce the problem to a boundary-pair variant of $2$-vertex-connected spanning subgraph, where each vertex is assigned a pair of incident edges with an associated pair cost. We solve this variant using a cancellation identity, inspired by Cut&Count [Cygan et al., TALG 2022], obtained by applying Möbius inversion to decompositions along cut vertices: the identity cancels every connected spanning graph with more than one block and keeps exactly the $2$-vertex-connected spanning graphs.

A b-coloring is a proper vertex coloring such that every color class contains a vertex, a so-called b-vertex, which sees all colors in its closed neighborhood. This type of coloring has been intensively studied from both structural and algorithmic point of view. Recently, Zaker [DAM 2025] introduced the notion of a b*-coloring, which is a b-coloring in which there is a vertex that sees a b-vertex of every color in its closed neighborhood. The b*-chromatic number is the maximum integer k such that there is a b*-coloring with k colors. We partially answer a question posed by Zaker and prove that graphs of girth at least 7 are b*-monotonic, which means that the b*-chromatic number does not increase by taking an induced subgraph. In addition, we discover a class of d-regular graphs of girth at least 5 with b*-chromatic number d+1, which strengthens a result about b-colorings by Dettlaff, Furmańczyk, Peterin, Roux, and Ziemann [AMC 2024]. We also study the parameterized complexity of finding b*-colorings, and show that for many structural parameters, the complexity coincides with that of finding b-colorings. In particular, the b*-chromatic number can be computed in polynomial time on any class of bounded clique-width. For most parameters, the translation from b-colorings is straightforward but for the feedback edge number, the FPT algorithm for b*-colorings is actually much simpler than that for b-colorings by Balabán [MFCS 2026].

Maximum Coverage and Partial Set Cover are fundamental parameterized covering problems. The former fixes a budget $k$ and maximizes coverage; the latter meets a target with as few sets as possible. Badanidiyuru, Kleinberg, and Lee (SoCG 2012) give an EPAS for the former on bounded-VC set systems, while Jain et al. (SODA 2023) show that on $K_{d,d}$-free incidence graphs, $k+1$ sets suffice whenever $k$ sets meet the target. We ask whether this guarantee extends to all bounded-VC set systems. Our first result is negative. Unless FPT = W[1], Partial Set Cover admits no parameterized $(2-\delta)$-approximation even at VC-dimension seven. Under ETH, it has no parameterized approximation scheme there and no $2^{o(d)}$-approximation at VC-dimension $d$. On the positive side, bounded semi-ladder index restores this guarantee. It is stronger than bounded VC-dimension but strictly generalizes the $K_{d,d}$-free setting. For Weighted Partial Set Cover, if $k$ sets cover weight $W$, we find $k+1$ sets covering weight $W$ in $2^{O(\Gamma k\log k)}N$ time, where $\Gamma$ is the downward intersection complexity and $N$ is the input size. The framework supports per-class targets and matroid independence, with applications to partial dominating set and geometric and bounded-size covering. Finally, we give a deterministic FPT reduction from Weighted CC-MaxSAT to a bounded family of Weighted Maximum Coverage instances, preserving incidence structure and approximation schemes with constant-factor accuracy loss. This gives an EPAS at bounded semi-ladder index. We improve the deterministic BKL bounded-VC implementation; combined with our reduction, it yields a $2^{\widetilde{O}(kd/\varepsilon)}N^{O(1)}$-time EPAS for bounded-VC Weighted CC-MaxSAT.

In this note, we introduce the notion of ranked spreadness, a strengthening of the usual spread condition in which the elements of each member can be ordered so that their one-coordinate marginals decay geometrically with their rank. This additional structure removes the dependence on the maximum set size in random-containment estimates. We prove width-free hitting and weighted-concentration theorems for ranked-spread set systems, together with an elementary kernel-extraction theorem showing that ranked spreadness arises naturally in arbitrary distributions on small sets. Our main application is to the simulation of nonadaptive property testers by sample-based testers. If a one-sided tester has average query complexity $d$ and rejects every far input with probability at least $\delta$, then, for every integer $c>d/\delta$, it admits a one-sided sample-based simulation with expected sample complexity $O_{d,\delta,|\Sigma|}\bigl(n^{1-1/c}\bigr)$. More generally, if positive inputs are rejected with probability at most $\gamma$ and far inputs with probability at least $\delta>\gamma$, the same conclusion holds for every $c>d/(\delta-\gamma)$. In particular, for constant-query nonadaptive testers we obtain an exponent $1-\Theta(1/q)$, matching, up to the dependence on the rejection gap, the exponent conjectured by Fischer, Lachish, and Vasudev.

Let $\{0, 1\}^n$ be the Boolean cube, endowed with the probability product measure, where ${\Bbb P}(1)=p$ and ${\Bbb P}(0)=q$ with $0 < p \leq q$ and $p+q=1$. For $i=1, \ldots, m$, let $\phi_i: \{0, 1\}^n \longrightarrow {\Bbb C}$ be $L_i$-Lipschitz functions in the Hamming metric, such that each $\phi_i$ depends on at most $r$ coordinates of $x \in \{0, 1\}^n$, where $rp \geq 12$. For $j=1, \ldots, n$, let $I_j $ be the set of indices $i$ such that $\phi_i$ depends on the $j$-th coordinate. We prove that $E \exp\left\{ \sum_{i=1}^m \phi_i \right\} \ne 0$ provided $\sum_{i \in I_j} L_i \leq {1 \over 10 \sqrt{rp}}$ for all $j$. This translates into a regime for $\pm 1$ spin systems, where a linear increase in the energy of multi-spin interactions requires only a logarithmic increase of the external field to keep the partition function zero-free and the system away from the phase transition. As a corollary, we obtain efficient deterministic algorithms to approximate the partition function in the zero-free region.

We present an average case model of classical problems in combinatorial optimization where there are color constraints. In all cases we seek some (spanning) sub-structure of a complete graph of minimum cost. The edges are randomly colored either red or blue. We bias against the red edges by placing a bound on the number of them that are allowed in our structure. This bound will be lower w.h.p. than what would occur without discrimination. We examine the effect of this bias on the minimum cost of a desired structure. We consider minimum cost spanning trees, shortest paths, minimum cost perfect matchings and the asymmetric traveling salesperson problem.

In recent work, Marcussen, Rubinfeld, and Sudan introduced the notion of quality control problems, which aim to capture the task of determining if a given input is truly random. Formally, their goal is to accept typical inputs from the specified distribution while rejecting every input whose value of a specified statistic is far from the distributional baseline. This captures the empirical practice of using specified statistics as a proxy for the quality of randomness. Empirical algorithms, however, have not exploited the asymmetry in the definition of quality control problems, which require soundness guarantees in the worst-case while only seeking average-case completeness. Their work abstracted a problem definition emphasizing this asymmetry and used it to give efficient quality control algorithms for assessing the randomness of graphs. In this work, we introduce and study quality control problems over sequences, where the goal is to distinguish a sequence of i.i.d. characters from sequences where some specified pattern appears too often (or too infrequently) as a subsequence. We consider this problem in both the finite-alphabet setting and for real-valued sequences. We refer to the former setting as the pattern counting problem. In the latter case, the natural notion of a pattern is to consider the relative ordering of the characters in the subsequence, and we refer to this as the permutation pattern counting problem. Algorithms to approximately count (permutation) patterns of length $k$ in a worst-case sequence of length $n$ can provably require exponential in $k$ queries into the sequence. In contrast, we show that by taking advantage of the asymmetry in the definition of quality control, we give algorithms that run in poly$(k)$ time to solve these problems. We also prove that any quality control algorithm (over some natural distributions) requires superlinear queries in $k$.

The Lempel-Ziv (LZ) factorization is a fundamental method for compressing repetitive strings. Its multiplicative sensitivity to an operation measures how much that operation can increase the number of phrases, as the maximum ratio of the phrase counts after and before the operation. While its sensitivity to single-character edits is known, its sensitivity to more global operations such as prefix deletion and string reversal remains unknown. Further open questions concern its size relationships with other compressed representations and whether it can be converted into an LZ-like encoding of polylogarithmic height without asymptotically increasing its size. In this paper, We determine the multiplicative sensitivity of the LZ factorization to prefix deletion, cyclic rotation, and string reversal by establishing an $\Omega(\log n)$ lower bound for each operation that matches the known upper bound, where $n$ is the input length. Using the result for prefix deletion, we show that the combined size of the relative LZ factorization and the LZ encoding of its reference string can be asymptotically smaller than that of the LZ factorization. We also obtain asymptotically tight worst-case ratios of $\Theta(\log n)$ for the size of the LZ factorization divided by the minimum size of a collage system and for the size of the lex-parse divided by that of the LZ factorization. We give lower and upper bounds on the worst-case ratio of the minimum size of an LZ-like encoding of height at most $h$ to the size of the LZ factorization, with matching bounds for $h \geq \lg n$. In particular, we show that the minimum size of an LZ-like encoding with height at most $h \in O({\rm poly}\,\log n)$ can be larger than the size of the LZ factorization by a factor of $\Omega(\log n / \log\log n)$. All our lower bounds follow from a common string construction based on the bit-reversal permutation.

Aggarwal, Dadush, Regev, and Stephens-Davidowitz (ADRS; STOC 2015) sample $2^{n/2}$ discrete Gaussians at an arbitrary parameter in $2^{n+o(n)}$ time, and above smoothing in $2^{n/2+o(n)}$ time. They ask whether the latter bound suffices for one sample at an arbitrary parameter. We answer this question affirmatively: for every rank-$n$ lattice $L\subseteq\R^n$ specified by a rational basis and every rational $s^2>0$, we produce one sample from $D_{L,s}$ within statistical distance $\exp(-\Omega(n^3))$ in expected $2^{n/2+o(n)}$ time and $2^{n/2+o(n)}$ space on every execution. The algorithm samples from random superlattices that are smooth at the required scale with constant probability and outputs the first point in $L$; a Gaussian-mass comparison shows that the $2^{n/2}$ samples produced by one ADRS call contain a point of $L$ with inverse-polynomial probability. The factor $2^{n/2}$ is tight in this Gaussian-mass comparison. For every fixed rational $\alpha<1.4697$, the same comparison gives a sub-$2^n$ algorithm for exact CVP on targets satisfying $\dist(y,L)\le\alpha\lambda_1(L)$, without a uniqueness assumption, and an exact-SVP algorithm in $2^{0.7315n+o(n)}$ time.

This paper presents two randomized proper-coloring algorithms that control color frequencies in the synchronous CONGEST model without paying a diameter-dependent coordination cost. Let $\lambda \geq 1$ denote the desired failure exponent. For every fixed $\delta > 0$, the first algorithm uses $\chi = \lceil (2+\delta)\Delta \rceil$ colors and, with probability at least $1 - n^{-\lambda}$, outputs a proper coloring that bounds the deviation of every color frequency from $n/\chi$ by $O_\delta(\sqrt{(\lambda+1)(n/\chi)\lg n} + (\lambda+1)\lg n)$. Under an explicit load condition, this additive guarantee yields two-sided relative balance. The second algorithm works with every $\chi > \Delta$ and gives a one-sided frequency cap controlled by the palette slack $\chi - \Delta$. In particular, it uses $\Delta + \lceil (\Delta+1)/\lceil \ln n \rceil \rceil$ colors and caps every used color class by $O((\lambda+1)(\sigma \lg^2 n + \lg n))$, where $\sigma = n/(\Delta+1)$. Both algorithms run in $O((\lambda+1)\lg n)$ rounds, with no dependence on the network diameter; for the first algorithm, the multiplicative constant in the time bound depends on $\delta$.

For an $n$-vertex graph of maximum degree $\Delta$ and diameter $D$, an equitable $(\Delta+1)$-coloring is a vertex coloring where the frequency of each color (namely, the number of vertices it colors) are all equal to $\sigma=n/(\Delta+1)$ (up to rounding). The Hajnal-Szemerédi Theorem guarantees the existence of such a coloring for every graph, and an $O(n^2\Delta)$ time sequential algorithm is known for computing such a coloring. Here, we study near-equitable graph coloring in distributed networks. The main question of interest is how close one can remain to the desired palette size of $\Delta+1$ while computing, in few distributed rounds, a coloring whose frequencies are close to $\sigma$. It appears that these two conflicting parameters exhibit a tradeoff, which we attempt to explore. We present a suite of fast randomized distributed algorithms representing varying points on this tradeoff, analyze their properties, and study their time complexity in the sequential, CONGEST and Congested Clique (CC) models.

In [AS21], Axiotis and Sviridenko conjectured that the linear dependence on the restricted condition number in sparse convex optimization cannot be improved by a polynomial-time algorithm. We establish their conjectured lower bound for least-squares objectives, conditional on the randomized exact-volume Small-Set Expansion Hypothesis in the weighted regular-graph formulation of Raghavendra, Steurer, and Tulsiani [RST12]. Concretely, for every fixed $\gamma\in(0,1]$, there is no randomized polynomial-time algorithm that, with probability at least $2/3$, returns a vector $x$ such that, writing $s=\lVert x\rVert_0$, \[ \lVert Ax-b\rVert_2^2 \leq \min_{\lVert z\rVert_0\leq k}\lVert Az-b\rVert_2^2+\varepsilon \quad\text{and}\quad s=O\!\left(k\,\kappa_{s+k}^{\,1-\gamma}\right), \] where $\kappa_r$ is the restricted condition number at sparsity level $r$. The result holds even on rational instances with $A$ of full column rank. The proof was first obtained using a fully automated Gemini-based agentic system developed internally at Google. The authors have verified the proof and edited it for clarity of presentation.

Quantizing high-dimensional vectors is fundamental to similarity search, distributed learning, and model compression. Feng, Indyk, Kapralov, Krachun, and Prokhorov established sharp guarantees for an unbiased dithered quantizer based on a randomized Hadamard transform [FIK+26]. Their $1/d$-scale inner-product estimator, however, uses a second randomized transform and residual quantization, increasing both communication and the leading constant in the proved bound. We show that this extra stage is unnecessary: pairwise-independent dithers across Hadamard coordinates suffice. The resulting unbiased single-stage estimator uses $b$ bits per coordinate and achieves \[ \mathbb{E}\!\left[ \left|\left\langle y,\widehat{x}-x\right\rangle\right|^2 \right] \leq \left(\frac{3\pi\sqrt{3}}{2}+o(1)\right) \frac{\lVert y\rVert_2^2}{d\,4^b}, \] as $b\to\infty$, with a dimension-free $o(1)$ term uniform over unit inputs and fixed queries. Compared with the two-stage construction of Feng et al., it eliminates the residual-stage $O(d)$-bit payload and reduces the leading upper-bound constant by a factor of approximately $5.93$. The proof was first obtained using a fully automated Gemini-based agentic system developed internally at Google. The authors have verified the proof and edited it for clarity of presentation.

We present a fully polynomial-time randomised approximation scheme (FPRAS) for the two-terminal reliability problem on general graphs, both directed and undirected. We also show that the complementary unreliability question is \BIS-hard. The key idea of the algorithm was discovered by GPT-5.6 Sol Ultra.