Papers

Given $n$ points in a metric space, Max-Min diversification asks for a subset of $k$ points maximizing the minimum pairwise distance between the selected points. This is arguably the most fundamental notion of diversity with applications across a wide range of domains. We consider this problem under partition constraints, previously studied as Fair Max-Min Diversification (FMMD). Here, each point has a color in $[m]$, and a feasible solution must contain exactly $k_i$ points of color $i$, where $k_1,\ldots,k_m$ are prescribed parameters satisfying $\sum_i k_i=k$. We give the first constant factor approximation for the problem using local search, that runs in time $f(m)\cdot \operatorname{poly}(n)$, in which all constraints are satisfied exactly. All previously known algorithms either provided an $\widetilde \Theta(m)$ approximation factor, had running times exponential in the solution size $k$, or satisfied the fairness constraints only approximately or in expectation. We further generalize our result to the problem where each point may belong to an arbitrary subset of colors. Given lower and upper bounds $\ell_i$ and $u_i$ for every color $i$, the goal is to find $k$ points whose color counts satisfy all these bounds while maximizing their diversity.

The Unanimous Vote problem is to determine a fixed order in which to flip each of $n$ biased coins, where each coin can be flipped only once, such that the expected number of flips until seeing both a head and a tail (or flipping all coins) is minimized. Duman Keles et al. (arXiv:2510.16678 [cs.DS]) gave an $\mathcal{O}(n \log n)$-time algorithm for this problem. Extensions of the Unanimous Vote problem are a rich source of stochastic optimization problems. We focus on three: (1) a variant in which each coin can be flipped arbitrarily many times (a solution is thus an infinite sequence of coin choices), (2) a generalization with $d$-sided dice, that can each be rolled once, where dice must be rolled until two different outcomes are observed (or all dice have been rolled), and (3) a different generalization with $d$-sided dice, where dice must be rolled until all $d$ outcomes have been observed. For (1), we show that there is an optimal sequence which follows a simple greedy rule; the same rule only gives a 1-additive approximation for the original problem (arXiv:2510.16678 [cs.DS]). The rule also yields a correspondence between a particular optimal sequence and a related mechanical word, which we exploit to characterize the conditions under which this optimal sequence is periodic. We establish tight multiplicative and additive adaptivity gaps for this variant. For (2), we show that two different generalizations of the greedy rule from (arXiv:2510.16678 [cs.DS]) can be combined to obtain a PTAS. For (3), we give an $\mathcal{O}(\log d)$-approximation algorithm by reducing the problem to Submodular Ranking (arXiv:1007.2503 [cs.DS]); the same reduction technique can be used to yield approximation algorithms for other stochastic probing problems. Finally, we pose a number of related open questions.

The local Hamiltonian problem is the canonical $\mathsf{QMA}$-complete problem, and $O(2^n)$ time classical algorithms and $O(2^{n/2})$ time quantum algorithms are known to solve the problem in the worst case. It is not clear how to improve these brute force strategies for a broad class of the problem because ground states are highly entangled in general, and we cannot directly apply known strategies for classical CSPs. In this work, we present exponentially faster classical and quantum algorithms under two mild assumptions: (1) the Hamiltonian is frustration-free on YES instances, and (2) it is approximately regular, meaning that every qubit is acted upon by approximately the same number of constraints. We complement these upper bounds by showing that, assuming (Q)SETH, quantum 5-SAT admits no non-trivial worst-case speedup. Our lower bound further demonstrates that the dependence of our algorithms on regularity is in some sense nearly optimal. Specifically, quantum 5-SAT remains (Q)SETH-hard even for Hamiltonians in which all but $O(\sqrt{n})$ qubits participate in only constantly many constraints, while the remaining $O(\sqrt{n})$ qubits each participate in $O(\sqrt{n})$ constraints. By contrast, if either the size of this high-degree subset or the degrees of its qubits is reduced by a factor of $n^\delta$, for any $\delta>0$, our algorithm solves the problem in time $O(2^{(1- \varepsilon)n})$ for some $\varepsilon>0$. Together, our upper and lower bounds establish a fine-grained complexity dichotomy for quantum satisfiability.

In the designer-port routing-labeling problem, every vertex of a rooted tree receives a binary label and the child edges receive distinct port numbers. Given only the labels of a source and a destination, a decoder must return the first port on their path. Gawrychowski, Janczewski, and Lopuszanski gave labels of length $\log_2 n+O((\log_2\log_2 n)^2)$, whereas the previous lower bound was $\log_2 n+\Omega(\log_2\log_2 n)$. We prove that every scheme for all $n$-vertex trees needs a label of length $\log_2 n+\Omega((\log_2\log_2 n)^2)$ for every sufficiently large $n$. The result allows arbitrary port assignments and imposes no computational restriction on either the encoder or the decoder. Thus the second-order term in the optimal worst-case label length is determined up to constant factors.

Analyses of dual lattice attacks have often assumed that the individual scores associated with short dual vectors are mutually independent. Laarhoven-Walter used this heuristic to derive explicit trade-offs between the target radius and query time for bounded distance decoding (BDD) with preprocessing. Ducas-Pulles subsequently demonstrated theoretical and experimental failures of this heuristic and proposed an alternative model conditioned on the target norm. In this note, we prove an explicit asymptotic trade-off for (decision-)BDD with preprocessing in the Haar-random lattice model, without heuristic assumptions. Using moment identities of Siegel and Rogers, we analyze cosine scores over complete dual balls and bound both error probabilities when distinguishing targets planted at a prescribed radius from uniform targets modulo the lattice. Optimizing the dual radius yields a trade-off between target radius and query time that matches the asymptotic prediction from the conditional model of Ducas-Pulles.

In spatial matching markets, a supply unit's flexibility is measured by its service radius, the maximum distance at which it can serve demand. In dimensions $k \geq 2$, we study how a platform should allocate service radii among the supply nodes subject to a budget on their sum. The platform makes this choice before observing supply and demand locations, with the objective of maximizing the expected fulfilled demand. We show that the shape of a preferred allocation depends on the total budget: under suitable conditions, large budgets favor allocations that are more uniform in the sense of majorization, while small budgets favor concentration. We also characterize a non-uniform allocation that is asymptotically optimal for a very-sparse regime, and show that the uniform allocation is suboptimal in this regime. Our results provide theoretical explanations for the radius allocation questions raised by the numerical experiments in [ASY26b].

In this work we initiate the study of competitive algorithms with arrival-time incentive compatibility for random-order online bipartite matching in settings where the users care only about receiving service (matched vs. unmatched) and not which offline resource serves them, while the platform's objective is to maximize total matching reward. This captures applications such as ride-sharing where the users primarily care about being matched to a ride while the platform internalizes the cost of dispatching a distant driver; dispatching homogeneous service requests to heterogeneous servers (cloud/edge routing); and assigning customer requests to a pool of providers with different flexibility (e.g., English-only vs. bilingual agents). Our main question is: Is constant-competitive matching possible for incentive-compatible, random-order edge-weighted matching on complete bipartite graphs? Motivated by the LP-based treatment of incentive compatibility in the classical secretary problem by Buchbinder et al., we impose a constraint that the ex ante probability of selection is equalized across all arrival positions. We answer our main question in the affirmative and propose the first constant-competitive algorithm for incentive compatible edge-weighted random-order online matching on complete bipartite graphs. The competitive ratio of our algorithm is parameterized by the imbalance factor $k := n/m$ – where $n$ and $m$ are the numbers of online and offline nodes, respectively, and $k$ is a positive integer. In particular, we obtain a competitive guarantee of the form $c_k - O(1/\sqrt{m})$ where $c_1 \approx 0.162$ and $c_k \to 0.02308\ldots$ as $k \to \infty$. We also present algorithms with strictly improved competitive ratio of $\approx 0.07 + O(1/m)$ for the binary-weighted case ($0$-$1$ rewards).

Influence maximization is a central algorithmic challenge in network analysis, aiming to identify a set of $k$ seed nodes in a graph with $n$ nodes and $m$ edges that maximizes the expected cascade of information under standard diffusion models. The seminal work of Borgs, Brautbar, Chayes, and Lucier (SODA'14) yielded a fundamental breakthrough\footnote{The conference version of their paper originally claimed a runtime of $\tilde O_{\epsilon}(n+m)$, but this was subsequently corrected to a runtime of $\tilde O_{\epsilon}((n+m)k)$ in an updated version of the paper that is available online. We validate the necessity of this additional factor $k$ in Section~\ref{sec:lowerbound} by demonstrating that if their algorithm is restricted to a runtime budget of $\tilde{O}_{\epsilon}(n+m)$, the approximation ratio deteriorates to $O(k^{-1/4})$.} for this problem by achieving an $\tilde O_{\epsilon}((n+m)k)$ time algorithm for approximating the solution within a factor of $1-1/e-\epsilon$. In the years since, numerous efforts have attempted to improve the runtime of this algorithm; however, these works have been successful in only shaving logarithmic factors or improving the dependence on $\epsilon$, leaving the existence of an almost linear-time algorithm as an open question. In this work, we resolve this long-standing open question. We present a novel algorithm that approximates the influence maximization problem within a factor of $1-1/e-\epsilon$ in time $\tilde{O}_{\epsilon}(n+m)$, effectively removing the multiplicative dependence on $k$ from the time complexity.

Optimizing over the {0, 1/2} rank-1 Chvàtal-Gomory (CG) closure of a binary integer linear program is NP-hard. While polynomial-time approximation schemes (PTAS) are established for monotone packing and covering formulations, extending these guarantees to mixed-sign variants remains an open challenge. In this paper, we study the approximability of the CG closure for k-slack bounded (capacity-bounded) min-closed systems, a generalized packing formulation with mixed-sign constraints that extends weighted Boolean Horn logic. In these systems, a constant upper bound $k \ge 1$ bounds the ratio between the negative penalty coefficient $q$ and the right-hand side capacity $b$ of every constraint, enforcing $q \le k \cdot b$. We prove that a constant-degree sum-of-squares relaxation yields a PTAS for maximizing linear objectives over the first CG closure generated by multipliers in $\{0\}\cup[\tfrac1f,1]$, for any constant integer $f \ge 2$. This closure is contained in the {0, 1/2} rank-1 CG closure.Optimizing over the {0, 1/2} rank-1 CG closure of a binary integer linear program is NP-hard. While PTASes are established for monotone packing and covering formulations, extending these guarantees to mixed-sign variants remains an open challenge. In this paper, we study the approximability of the CG closure for \emph{$k$-slack bounded (capacity-bounded) min-closed systems}, a generalized packing formulation with mixed-sign constraints that extends weighted Boolean Horn logic. In these systems, a constant upper bound $k \ge 1$ bounds the ratio between the negative penalty coefficient $q$ and the right-hand side capacity $b$ of every constraint, enforcing $q \le k \cdot b$. We prove that a constant-degree sum-of-squares relaxation yields a PTAS for maximizing linear objectives over the first CG closure generated by multipliers in $\{0\}\cup[\tfrac1f,1]$, for any constant integer $f \ge 2$.

Residual-String Equality, denoted $\texttt{ResStringEq}$, is the property consisting of all pairs of strings over $\{0,1,*\}$ that are equal after deleting all `$*$' symbols from them. This property was first introduced by Fischer, Magniez, and Starikovskaya (SODA 2018), who used it to show a lower bound on testing the $\texttt{Dyck}$ languages, where $\texttt{Dyck}_m$ is the language consisting of balanced sequences of parentheses over $m$ parenthesis types. They showed that testing $\texttt{ResStringEq}$ on inputs of length $n$ requires $\Omega(n^{1/5})$ queries, and presented a reduction from testing $\texttt{ResStringEq}$ to testing $\texttt{Dyck}_m$ where $m \geq 2$. Furthermore, they showed that $\texttt{Dyck}_m$ can be tested with $O(n^{2/5+\delta})$ queries for every constant proximity parameter, where $\delta>0$ is an arbitrarily small constant. In this work, we nearly close the remaining gap, by showing that testing $\texttt{ResStringEq}$, and hence $\texttt{Dyck}_m$ where $m\geq 2$, requires $\Omega(n^{2/5})$ queries. We also show a stronger lower bound of $\Omega(\sqrt{n})$ for testers that make non-adaptive queries. We establish that the $\Omega(\sqrt{n})$ bound is nearly tight, by presenting a non-adaptive tester for $\texttt{ResStringEq}$ that uses $O(n^{1/2+\delta})$ queries for an arbitrarily small constant $\delta>0$. Furthermore, we extend this non-adaptive tester to the $\texttt{Dyck}$ languages, with the same query complexity. Finally, we improve the dependence on the proximity parameter $\epsilon$ in the tester of Fischer, Magniez, and Starikovskaya, reducing it from $O(1/\epsilon)^{\mathrm{poly}(1/\delta)}$ to $O(1/\epsilon)^{O(\log(1/\delta))}$.

This paper gives two parallel algorithms for $\Delta+1$ edge coloring, where $\Delta$ denotes the maximum degree of any vertex. The first is a deterministic parallel algorithm with $\tilde{O}(\Delta^3)$ span and $\tilde{O}(m \Delta^3)$ work. Our second algorithm and our main result is a more efficient randomized algorithm, achieving $\tilde{O}(\Delta^2)$ span and $\tilde{O}(m \Delta^2 )$ work both with high probability. These bounds substantially improve over the recent deterministic parallel algorithm of Elkin and Khuzman, which has $\tilde{O}(\Delta^4)$ span and $\tilde{O}(m \Delta^5)$ work. Our deterministic algorithm thus represents a $\tilde{O}(\Delta)$ improvement on span and $\tilde{O}(\Delta^2)$ on work compared to their algorithm, and our randomized algorithm improves the span and work by $\tilde{O}(\Delta^2)$ and $\tilde{O}(\Delta^3)$ factors, respectively. Moreover, our improvements do not come at the expense of larger logarithmic factors.

We determine the optimal query complexity of ground-state preparation to trace-distance error $\varepsilon$ when an energy threshold in the spectral gap is known. Let $U_H$ be an $\alpha$-block-encoding of a Hamiltonian with unique ground state $|\psi_0\rangle$, and suppose $|\langle\psi_0|U_I|0\rangle|\ge\gamma$ for a state-preparation oracle $U_I$. The threshold lies at least $\Delta/2$ above the ground-state energy and at least $\Delta/2$ below every excited-state energy. We give two algorithms that prepare a state within trace distance $\varepsilon$ of the ground state. One uses $O((\alpha/\Delta)(\gamma^{-1}+\log(1/\varepsilon)))$ calls to $U_H$ in expectation; the other uses $O((\alpha/(\gamma\Delta))\log(1/\varepsilon))$ calls to $U_H$ in the worst case. We prove a lower bound matching the expected query count; the corresponding worst-case lower bound follows from Somma and de Wolf [SdW26]. The respective bounds on calls to $U_I$ are $O(1/\gamma)$ in expectation and $O(\gamma^{-1}\log(1/\varepsilon))$ in the worst case. On $(N+1)$-dimensional systems, these $U_I$ bounds are also optimal when the expected or worst-case count of $U_H$ calls, respectively, is $o((\alpha/\Delta)\sqrt N)$. Both algorithms use a constant-accuracy spectral filter to construct a purifier, which we then sequentially compose during amplitude amplification to prepare a state with constant overlap with the ground state. The expected-query algorithm repeats the preparation followed by one high-accuracy spectral filter until success. The worst-case algorithm uses filters of increasing accuracy and limits the total number of queries.

We study the problem of learning a sparse quantum state, an $n$-qubit quantum state whose density matrix has at most $s$ nonzero matrix entries in an unknown product basis. While such states admit compact classical descriptions, they can carry long-range entanglement that prevents reconstruction from local reduced density matrices alone. Therefore, previous learning approaches addressed such long-range-entangled states using many entangling gates to extract the necessary information. In this work, we show that sparse states can nevertheless be efficiently learned using only single-qubit measurements. Specifically, when the sparsity $s$ is constant, our algorithm can learn sparse states from single-qubit measurements with polynomial sample complexity and classical computational complexity. When $s$ grows polynomially with $n$, sparse states can still be learned from single-qubit measurements with polynomial sample complexity, although efficient classical computation is not guaranteed in general. In this regime, however, the classical computational complexity becomes quasipolynomial when the state is sparse in an unknown basis that is a product of a known fixed finite set of single-qubit bases (e.g., eigenbases of Pauli operators). These results establish efficient learning of sparse states with long-range entanglement without entangling gates, and the single-qubit measurement requirements make our algorithms compatible with current quantum devices.

Correlation clustering has been extensively studied over the last two decades in many different computational models owing to its wide-ranging practical applications. The goal is to compute a partition of the vertex set such that the total number of disagreements, i.e. the sum of edges between clusters, and non-edges within clusters, is minimized. In this paper, we study correlation $k$-clustering, in which the total number of clusters is restricted to $k$. Correlation $k$-clustering is NP-hard, and while previous work has presented a polynomial time approximation scheme for constant $k$, there are no known results for general $k$. Our first result is a polynomial-time constant-factor approximation algorithm for correlation $k$-clustering for general $k$. The main focus of this work is in the more challenging online setting. Noting that the best competitive ratio under adversarial arrivals is known to be $\Omega(n)$, we concentrate on the well-studied random-order model, where vertices arrive in random order and on arrival of a vertex, edges to its earlier-arrived neighbors are revealed. We prove a surprising lower bound of $\Omega(\log k)$-competitiveness for any online algorithm, which can be extended to $\Omega(\log n)$ when $k = \text{poly}(n)$. Finally, the main result of this paper is a polylogarithmic upper bound on the competitive ratio for correlation $k$-clustering, using an algorithm inspired by the classic Pivot algorithm for correlation clustering.

The Frieze-Jerrum rounding is a standard tool for rounding SDP relaxations of graph partitioning and clustering problems, assigning nodes to at most $k$ clusters using $k$ independent Gaussian vectors. Its analysis hinges on the collision probability $P_k(\rho)$ that two nodes whose SDP vectors have inner product $\rho$ are assigned to the same cluster. No tractable closed form for $P_k$ is known for $k\geq 4$, making it difficult to certify approximation guarantees and hindering the systematic search for better algorithms. We develop a Hermite-coefficient certification framework to derive accurate and tractable bounds on $P_k$. Using the Hermite expansion of Gaussian noise stability, we express $P_k$ as a power series with nonnegative coefficients, reduce these coefficients to one-dimensional Gaussian integrals, and certify finitely many of them, yielding rigorous bounds on $P_k$ over the entire correlation range. Our framework yields strengthened polynomial-time approximations for several clustering problems. For MaxAgree Correlation Clustering, we derive a $0.7818$-approximation, the first improvement in two decades over the $0.7666$ ratio of Swamy (2004). On the hardness side, we show that the integrality ratio of the standard SDP relaxation is at most $0.802$, and that approximation beyond that is Unique Games-hard. We also improve the best known ratios for the variant with at most $K$ clusters, MaxAgree$[K]$ (e.g., from $0.77$ to $0.8151$ for $K=3$). For Max $K$-Cut we resolve, via a structural property of the Hermite expansion, a conjecture of de Klerk et al. (2004) characterizing the Frieze–Jerrum approximation ratio for every $K\ge3$; we show that this ratio is tight, and determine it to within $10^{-6}$ accuracy for $K\le16$. Finally, we reduce the additive approximation error for modularity maximization from $0.42084$ (Kawase et al., 2021) to $0.3790$.

Let $\lambda_k(G)$ be the $k$th eigenvalue of the normalized Laplacian of a finite undirected weighted graph $G$ with positive degrees, where $k$ is an integer satisfying $1\le k\le |V(G)|$. Let $\varphi_k(G)$ be the minimum possible maximum conductance of $k$ disjoint nonempty vertex sets. We prove $\varphi_k(G)\le C\sqrt{\lambda_k(G)\log(k+1)}$ for an absolute constant $C$. The number of sets and the spectral index are both $k$, and conductance is measured in the original graph. The logarithmic dependence is optimal up to an absolute constant. The proof combines geometric partitioning of the spectral embedding with minimum-cut improvement and adaptive projections in coefficient space. A single conductance threshold is used throughout the construction. The resulting maps have disjoint supports, and the sum of their Gram matrices is bounded below by an absolute positive multiple of the identity. A dyadic maximal estimate bounds the sum of their internal energies uniformly over unit coefficient vectors. A dimension argument using local eigenvalues then yields exactly $k$ disjoint sparse cuts.

Controlling tail risk is an important objective in online optimization, and recently it has been studied in the context of competitive analysis. Continuing this line of research, we investigate pure tail constraints, which capture the tradeoff between expected and worst-case competitiveness. For two fundamental search problems, online bidding and line search, we derive the Pareto-optimal frontiers of this tradeoff. We then investigate another classic problem, TCP acknowledgment, which has structure similar to the iterated ski rental problem. There, we construct an algorithm whose tradeoff coincides with the known Pareto-optimal tradeoff for ski rental. The lower bounds for this problem are substantially more involved as the problem exhibits adaptive structure: an online algorithm observes requests of the adversary (packet arrivals) and may adaptively adjust its actions (acknowledgments) on this basis. We emphasize that all previous work on tail risk in the context of competitive analysis was restricted to non-adaptive problems, where the feedback given to an algorithm was essentially limited to a binary indicator of whether the algorithm has succeeded or not. Nonetheless, we identify a set of constraints implied by tail bounds in this adaptive setting, and show that they imply a nontrivial lower bound on the TCP acknowledgment problem.

We study truthful-in-expectation (TIE) mechanisms for allocating indivisible chores alongside ex-post maximin share (MMS) guarantees. For goods, Bu and Tao (FOCS 2025) established a (1/n)-approximation for TIE mechanisms, and this was substantially improved by Babaioff, Feige, and Manaker Morag (FOCS 2026), who established an \Omega(1/\log n) approximation, where n is the number of agents. The corresponding problem for chores has received less attention. The best-known result is due to Aziz, Li, and Wu (MAPR 2024), who gave a TIE mechanism with an O(\sqrt{\log n}) MMS approximation guarantee that holds only in expectation. They also established a 6/5 lower bound for TIE mechanisms for two agents. In this paper, we present the first TIE mechanism for chores that achieves a constant ex-post MMS approximation guarantee. Specifically, our mechanism guarantees an ex-post ratio of 1.97 for any number of agents n, which improves to 4/3 for n=2 and 3/2 for n=3. On the hardness side, we tighten the two-agent lower bound to 4/3, showing that our mechanism is optimal for n=2. More generally, we establish a lower bound of 13/12 on the ex-post MMS approximation ratio achievable by TIE mechanisms for every n\ge 3.

Weighted ultrametric embedding with outliers asks for a minimum weight set of points whose deletion makes the remaining metric an ultrametric, equivalently, leaves no triple with a unique largest distance. We give a deterministic $5/2$-approximation using $O(n^5)$ arithmetic and comparison operations, improving the previous factor $3$ for arbitrary nonnegative vertex weights. Approximating the problem within any factor strictly below $2$ is UGC-hard.

Given an undirected weighted graph $G$, a $\gamma$-approximate low-stretch spanning tree (LSST) $T \subseteq G$ is a tree that approximates the distance metric of $G$ up to a $\gamma$-factor in expectation. Currently, existing algorithms to find a provably good LSST carefully construct an approximate shortest-path tree from an arbitrary source. The resulting algorithms are intricate. In contrast, practitioners observed that a much simpler heuristic performs surprisingly well: choose an arbitrary root, run Dijkstra's algorithm, and use the resulting shortest-path tree as an LSST. In this paper, we give a smoothed analysis of shortest-path tree algorithms, such as Dijkstra's algorithm, that explains this behavior. We show that adding a small perturbation to the weights of the input graph suffices to turn the shortest path tree rooted at an arbitrary node in the resulting graph into an $\tilde{O}(1)$-approximate LSST. We further show that the set of perturbations can be computed efficiently from few low-diameter decompositions (LDDs). Thus, our proof is also constructive in the sense of giving a novel approach to computing LSSTs.

The development of round elimination into a general-purpose technique [PODC 2019] marked a turning point in our understanding of the hardness of many graph problems in the distributed setting and led to several breakthrough results. However, the round elimination technique seems unable to yield randomized lower bounds of $\omega(\log \log n)$ rounds as a function of the number $n$ of nodes. Very recently, Khoury and Schild [FOCS 2025] introduced a new technique called round elimination via self-reduction, which bypasses the limitations of classical round elimination. Using this approach, the authors show that any randomized algorithm for maximal matching requires $\Omega(\sqrt{\log n})$ rounds in the LOCAL model. Their elegant technique is, in some respects, similar to classical round elimination while being fundamentally different in others. However, it is tailored specifically to maximal matching rather than being applicable to a broad class of problems. In this paper, we show that self-reduction is, in fact, a special case of classical round elimination, thereby turning it into a general-purpose approach. In particular, we introduce a new way to measure the error of an algorithm and show that, under this new measure, classical round elimination can indeed yield $\omega(\log \log n)$ randomized lower bounds. More specifically, we identify a large class of problems for which this improvement is entirely black-box: once a problem is shown to belong to the class, stronger randomized lower bounds follow automatically from the classical round-elimination framework. As an application, we prove $\Omega(\sqrt{\log n})$ randomized lower bounds for a range of graph problems, namely, maximal matching on regular $2$-colored graphs, $\frac{1}{k}$-integral matching, and maximal $H$-packing.

Karlin, Klein, and Oveis Gharan introduced a randomized better-than-$3/2$ approximation algorithm for metric TSP [KKO21] and subsequently established the corresponding improvement in the integrality gap of the subtour-elimination LP [KKO22], with an explicit constant $\varepsilon>1.00000\cdot10^{-36}$. Gurvits, Klein, and Leake subsequently improved the certified saving to $2.18000\cdot10^{-34}$ [GKL24]. In this paper, we obtain a randomized polynomial-time $(3/2-\varepsilon)$-approximation for every fixed $0<\varepsilon<\varepsilon_\star$, where $\varepsilon_\star>2.78621\cdot10^{-18}$, and consequently the subtour-elimination LP has integrality gap at most $3/2-\varepsilon_\star$. The classical worst-case integrality-gap lower bound is $4/3$ [Wil90].

Low-energy estimation and state preparation for general $k$-local Hamiltonians are fundamental challenges in quantum complexity theory. Buhrman et al.~ [BGLGST, PRL 2025] recently broke the natural Grover bound $O^\ast(2^{n/2})$ for both problems, with the improvement depending on the relative accuracy $\varepsilon$ and the locality $k$. In this work, we present faster exponential quantum algorithms for these problems, where the binary entropy function governs the runtime exponent. For sufficiently small $\varepsilon/k$, our algorithms improve the exponent by a factor of $\log(k/\varepsilon)$ over [BGLGST, PRL 2025]. Our main technical result is an entropy-governed lower bound on the dimension of the Hamiltonian's low-energy subspace, obtained by depolarizing its ground state. For fixed $k$, this bound is optimal up to constant factors in the exponent. The same framework yields tighter bounds for Heisenberg, $XY$, and Ising models on arbitrary interaction graphs.

We study the temporal exploration problem on temporal digraphs. We prove that a lifetime of $O(n^2)$ suffices to guarantee the existence of a temporal exploration on always-unilateral temporal digraphs. We complement this with a $\Omega(n^2)$ lower bound, even in the case where each snapshot has maximum undirected degree 2; for always-strong temporal digraphs, the lower bound still holds even if the maximum undirected degree is 3. This stands in stark contrast with the undirected setting. For the large minimum degree setting, we show that a lifetime of $4n/3 - 1$ is sufficient and necessary for guaranteeing the existence of a temporal exploration on temporal digraphs where each snapshot is semicomplete. For always-strong temporal digraphs where each snapshot has minimum undirected degree at least $n - c - 1$, we prove that a lifetime of $O(cn)$ guarantees the existence of a temporal exploration, and we also prove that this is asymptotically tight. From a computational perspective, our results for temporal semicomplete digraphs also yield a polynomial-time, factor-$4/3$ algorithm for deciding if a temporal semicomplete digraph admits a temporal exploration within the first $\ell$ snapshots. We complement this showing that no polynomial-time, factor-$(4/3 - \epsilon)$ approximation algorithm exists, even if every snapshot is a tournament, unless P$=$NP.

Given $n$ points in a metric space, partitioned into groups, $X_1,\dots,X_m$, and integer quotas, $k_1,\dots,k_m$, summing to $k$, the Fair Max-Min Diversification problem asks for a set of $k$ points, exactly $k_i$ from each group $X_i$, maximizing the minimum pairwise distance. Addanki et al. (ICDT 2022) described a ball LP for this problem and rounding algorithms that yield a factor 2 approximation whose fairness holds only in expectation and a factor 6 approximation with relaxed fairness guarantees, as well as an $(m+1)$-approximation with exact fairness. We introduce two new algorithms. The first method refines the rounding of Addanki et al., yielding a 2-approximate solution that is $\varepsilon$-fair with high probability, meaning that from every group $X_i$, at least $(1-\varepsilon) k_i$ points are chosen. The second method in polynomial time returns a 4-approximation to the optimal value of the exact fairness version. In time $n^{O(1)} 2^{O(k)}$, which is fixed-parameter tractable in $k$, we achieve an exactly fair 4-approximation. Our method adapts the augmenting procedure behind Haxell's theorem (Graphs Combin., 1995). This approximation factor does not depend on $m$. Moreover, we show that no rounding of the ball LP achieves a smaller factor with exact fairness.

In classical fixed-interval scheduling, each job of a given set must be processed during a prescribed time interval. The goal is to assign each job to exactly one machine such that no two jobs assigned to the same machine overlap in their interiors, and the number of machines used is minimized. Without further constraints this is interval graph coloring and is solvable in polynomial time through greedy approaches. We study a variant in which every used machine must remain idle during a contiguous break of prescribed length $x$ somewhere in the scheduling horizon. We show that this additional constraint makes the problem hard, except when $x\le1$. First, for every fixed $x\ge2$, deciding whether $k$ machines suffice for the assignment of a given set of jobs is NP-complete, even when all coordinates are bounded linearly in the number of jobs, so machine minimization is strongly NP-hard. Second, for $x=1$, we give a polynomial-time algorithm that solves the problem exactly. Finally, we give a deterministic polynomial-time algorithm that, on every feasible instance, outputs a schedule using at most $\OPT+1$ machines, where $\OPT$ is the true minimum. Unless $\mathrm{P}=\mathrm{NP}$, no polynomial-time algorithm guarantees $\OPT$ machines for any fixed $x\ge2$, so the additive guarantee of one is best possible.

Online resource allocation is a fundamental problem in revenue management, sponsored search, and platform operations. Most prior work assumes nonincreasing assignment rewards, capturing diminishing returns. We instead study increasing returns, where assigning more customers to the same product can unlock larger value through scale, visibility, or network effects. We consider capacity-limited products assigned to sequentially arriving customers, where the reward of each product depends on the total number of customers assigned to it. We focus on full compatibility, where every product can be assigned to every customer. The number of customers is unknown to the online algorithm. We show that representative classical approaches, such as online greedy algorithm and LP-based independent rounding, can perform arbitrarily bad in this setting, even when products share a common bonus function. For homogeneous bonus functions, we give a simple polynomial-time algorithm that achieves a tight $1/2$-competitive ratio. For arbitrary heterogeneous bonus functions, we show that no constant competitive ratio independent of the number of products is possible: for $m$ products, the optimal competitive can be as small as $\Theta\left({1}/{\sqrt m}\right)$. This shows that heterogeneous delayed rewards can force any online algorithm to guess the realized arrival counts. We then identify a structured heterogeneous regime that restores a constant guarantee. When each bonus function is nonnegative, nondecreasing, and discrete concave, we design an intermediate target repair algorithm with a deterministic pathwise guarantee of $0.2466$. The algorithm repeatedly computes an offline target for a larger demand level and repairs the current allocation toward it using a prefix-robust assignment order. This coordinates buildup while remaining robust to early stopping and yields a distribution-free guarantee.

We present a deterministic CRT-based sparse Fourier architecture for exact recovery of noiseless, on-grid, at-most-\(k\)-sparse spectra with an engineered transform length. The transform length \(N\) is selected as an exact product of small pairwise-coprime stage primes, allowing each stage to use transforms whose size scales with the sparsity rather than the ambient dimension. Three co-prime-increment time shifts provide a one-sided singleton-consistency screen: every genuine singleton passes, while passing collision bins remain provisional candidates. Each candidate bin is mapped by a total label-emission rule to at most one global frequency label, and per-stage label sets are intersected, yielding a deterministic \(O(k)\) candidate bound for every input. Candidate construction uses \(O(k\log N)\) samples. In a comparison real-RAM model with exact arithmetic and exact phase evaluation but without unit-cost root-index decoding, it requires \(O(k\log^2 N/\log k)\) arithmetic operations; in a stronger root-index-oracle model the arithmetic cost is \(O(k\log N)\). Under all-stage genuine-singleton survival, the candidate set contains the true support. Exactness, however, does not depend on that survival condition: a consecutive-sample residual verifier accepts a sparse reconstruction only when it is exact, and all failures route to a dense FFT. The resulting hybrid algorithm therefore returns the exact spectrum for every input in the stated noiseless, on-grid, at-most-\(k\)-sparse model, with \(O(N\log N)\) worst-case runtime. The verified sparse path additionally incurs an \(O(k^2)\) verifier cost in the root-index-oracle model. Neither computational model is a bit-complexity model.

We simplify algorithms for Knapsack, output-sensitive Subset Sum, and near-convex min-plus convolution. For bounded Knapsack, we give a deterministic algorithm using $O(N+W^2\log^3(W+2))$ arithmetic and comparison operations, where $W$ is the maximum item weight and $N$ counts input records with binary-encoded multiplicities. Following Bringmann's approach, we partition the items and bound the weight added or removed within each part when correcting a greedy solution to an optimum. These bounds keep the dynamic-programming tables small. The same analysis gives the corresponding bound with maximum profit in place of weight. For multiple-choice Knapsack, we give a randomized $\widetilde O(N+w^2\min\{r,w\})$ algorithm, where $N$ counts alternatives, $r$ counts classes, and $w$ is the maximum within-class weight range. Randomly grouping classes exploits cancellation between positive and negative weight changes. For Subset Sum of $n$ nonnegative integer vectors in any fixed dimension $d$, we obtain expected time $\widetilde O(n+s\sqrt n)$, where $s$ counts attainable sums in the target box. With high probability, all sums are returned within the same bound. This improves the $\widetilde O(n+s n^{d/(d+1)})$ bound of Bringmann, Fischer, and Nakos for $d>1$. The key step computes a sumset inside a box without generating the potentially much larger unrestricted sumset. Finally, we simplify the $\widetilde O(N(D+1))$ algorithm for min-plus convolution of integer arrays of total length $N$, where $D$ is the sum of their maximum deviations above convex arrays. The deviations can change the minimizing pairs substantially, but restrict relevant candidate values to short intervals.

We present a new class of near-linear algorithms for efficiently integrating general tensor fields defined on trees with distance dependent kernels, the Structure-Adaptive Tree Field Integrators (STAD-TFIs). STAD-TFIs exploit the tree's underlying structure through decompositions built around path backbones and single vertex separators, and use two-dimensional fast Fourier transforms to compute interactions jointly. By exploiting this structural information, STAD-TFIs achieve more computationally efficient integration than their regular efficient tree field integrators (TFI) counterparts. We provide a detailed theoretical analysis of our proposed approach and complement it with an exhaustive empirical evaluation, ranging from speed tests on synthetic trees, through accelerated Sinkhorn-based relaxations of the Optimal Transport algorithms on real meshes, to Topological Attention Transformers for vision tasks. To the best of our knowledge, we provide some of the first results showing that efficient to compute and accurate relaxations of the geodesic Sinkhorn-based solutions of the Optimal Transport problem can be derived by applying fast TFI methods.