Papers

We present our submission to the IJCAI 2025 'Counterfactual Routing Competition' (CRC 25). The goal of the competition is to find counterfactual explanations for the shortest path problem. This requires deciding what the minimal changes to a road network would make a route chosen by the user the optimal route. This enables explanations such as "Your suggested route would indeed have been optimal, if road X were not a bicycle path." Our solution models the problem as an integer program, iteratively incorporating constraints until an exact solution is found. In the final evaluation on held-out test instances, our method ranked fourth in solution quality and obtained its solution fastest on every instance, with an average runtime of 9.0 seconds compared to 118.8 seconds for the next-fastest submission.

We study non-adaptive selection of a feasible set $S$ that maximizes the expected sum of the $k$ largest realized values among independent nonnegative discrete random variables. The same objective arises in team hiring and as VCG welfare in an $\ell$-unit auction. The main setting is a fixed-dimensional nonnegative packing family, whose natural LP has $d=O(1)$ packing inequalities with binary coefficients. We give a PTAS for every $k\ge 1$ on every such family, including binary one- and two-dimensional knapsack, by approximating the occupancy functional $p\mapsto\mathbb{E}[\min(k,N(p))]$ and realizing the resulting signatures in the packing LP. As a generic guarantee the scheme is essentially optimal: there is no FPTAS that works for every such $\mathcal{F}$ unless $P=NP$, and no EPTAS unless $W[1]=FPT$. An incomparable sufficient condition is a query-weight exact-sum oracle (DAG paths, matchings), which likewise yields a PTAS for every $k$. The same signatures give a PTAS for $\min_{S\in\mathcal{F}}\mathbb{E}[\mathrm{Top}_k(S)]$ on every fixed-$d$ covering family; two-dimensional covering knapsack rules out a generic FPTAS on that class.

We study the basic problem of approximating the number of spanning trees of a graph. For a graph with $n$ vertices, $m$ edges, We propose an algorithm that approximates the number of spanning trees in $\widetilde O(m+n^{3/2}\eps^{-1})$ time. Our algorithm improves upon the previously best known $\widetilde O(m+n^{15/8}\eps^{-7/4})$ time algorithm by Chu, Gao, Peng, Sachdeva, Sawlani, and Wang [FOCS 2018] and the $\widetilde O(m^{3/2}\eps^{-1})$ time algorithm by Liu, Peng, and Yang [FOCS 2026]. Notably, our algorithm is based on the novel concept of $\ell_1$-regularized resistance. We propose simple and efficient algorithm for computing $\ell_1$-regularized resistance and we show that they can be used to approximate the number of spanning trees by combining with the determinant sparsifier framework of Durfee, Peebles, Peng, and Rao [FOCS 2017]. Our algorithm matches the best known size of the determinant sparsifiers.

We study assortment and procurement design for a digital content platform offering both ad-supported and subscription access. Users are heterogeneous in content preferences and ad tolerance and self-select between the two modes or an outside option. For a fixed common subscription price and ad load, the platform chooses assortment distributions specific to each user type and access mode, together with content-family-level buy-versus-rent decisions to maximize profit. Rental costs scale with realized consumption, whereas buying provides a reusable pool of titles whose cost depends on the largest induced requirement across user types and modes. We show that the resulting problem is NP-hard. We then develop a scalable approximation framework based on a candidate buy set, a relaxation of the procurement coupling, and a decomposition into linear programs with a single equality constraint. These subproblems are solved by dual bisection with cardinality-constrained assortment optimization, followed by restricted-master postprocessing to recover primal feasibility. The method yields computable optimality-gap bounds, an interpretable threshold-based procurement heuristic, and asymptotic optimality under proportional market scaling as market size and grid resolution increase. Numerical experiments show strong performance at moderate market scales and grid sizes.

We prove that fully independent SparseStack achieves the oblivious subspace embedding parameters conjectured by Nelson and Nguyen (FOCS 2013): $m=O((d+\log(1/\delta))/\varepsilon^2)$ rows and $s=O(\log(d/\delta)/\varepsilon)$ nonzero entries per column for distortion $\varepsilon$ and failure probability $\delta$ on any fixed $d$-dimensional subspace, with explicit constants. The proof turns random-matrix concentration into a problem in finite-dimensional linear algebra. A coupling first reduces the moment estimates to a model with independent finite-valued entries. We represent these variables by multiplication operators, so their matrix moments become exact matrix elements of a deterministic operator on a finite tensor product. The central estimate bounds the contribution of $\ell\ge1$ occupied sites sharing the external factor $\mathbb{R}^d$ by $d+\ell-1$ rather than $d\ell$, yielding additive dependence on the dimension and the moment order. This approach controls both spectral edges without Gaussian comparison. The main theorem has been formally verified in Lean 4.

We consider efficient algorithms to learn multiband signals and Fourier-sparse signals. A mutliband signal has a Fourier transform supported by a bounded number of intervals, say $I_1 \cup I_2 \cdots \cup I_n$. There is a long line of research on multiband signals. In particular, Avron et al. showed an efficient reconstructing algorithm whose sample complexity is almost optimal. However, all previous algorithms for multiband signals consider the reconstructing problem in which the locations of $I_1,\ldots,I_n$ are given as a priori knowledge. On the other hand, although the problem of learning Fourier-sparse signals with $k$ arbitrary frequencies dates at least to Prony in 1795, designing efficient and robust learning algorithms is still an open problem. The state-of-the-art is an efficient algorithm of $\tilde{O}(k^3)$ samples and $\tilde{O}(k^{3 \omega})$ time from the very recent work by Cai et al., while the statistical upper bound is $\tilde{O}(k^2)$ samples. Let $[-1,1]$ be the time window in which the noise is $\ell_2$ bounded. 1. We show an efficient algorithm to recover the locations of the bands $I_1,\ldots,I_n$ in $\hat{x}$ within $\tilde{O}(n+\sum_i |I_i|)$ samples and $\tilde{O}(n+\sum_i |I_i|)$ time. Furthermore, combining this with the reconstructing algorithm by Avron et al. provides an efficient interpolation algorithm within $\tilde{O}(n+\sum_i |I_i|)$ samples. 2. We show that every $k$-Fourier-sparse signal $x$ admits a multiband approximation $z$ whose Fourier transform is of support size $|\mathrm{supp}(\hat{z})|=\tilde{O}(k^2)$. Furthermore, we show an interpolation algorithm for $k$-Fourier-sparse signals with $\tilde{O}(k^2)$ samples and $\tilde{O}(k^5)$ time.

Temporal graphs model dynamic networks by assigning positive integer time labels to the edges, while information propagates along temporal paths, whose edge labels are traversed in nondecreasing order. A temporal $\alpha$-spanner of a temporal graph with $n$ vertices is a temporal subgraph that approximates the minimum-hop temporal distance between every pair of vertices within a factor of $\alpha$. While general temporal graphs may not admit sparse temporal $\alpha$-spanners for any value of $\alpha$, temporal cliques are known to admit temporal $(2k-1)$-spanners of size $\widetilde{\mathcal{O}}(kn^{1+1/k})$ for every positive integer $k$. We present a simple recursive algorithm that computes, for every temporal clique on $n$ vertices, a temporal $3$-spanner of size $n^{1+2/\sqrt{\ln n}}=n^{1+o(1)}$, thereby improving the previous best upper bound of $\widetilde{\mathcal{O}}(n^{3/2})$. We also show that a modified version of our algorithm computes temporal $3$-spanners of size $\mathcal{O}(nL)$ when the lifetime is bounded by $L$, i.e., all time labels are in $\{1,\ldots,L\}$, thus improving the previous bound of $\mathcal{O}(2^Ln\log n)$. Both results are particularly striking in light of the known lower bound of $\Omega(n^2)$ on the size of temporal $2$-spanners, which already holds for temporal cliques of lifetime $L\geq 3$. Both algorithms rely on a new simple recursive decomposition that certifies temporal connectivity for a large collection of source-target pairs using only $\mathcal{O}(n)$ carefully selected edges and recursively processes only the remaining pairs. Besides yielding substantially improved upper bounds, this approach is significantly simpler than previous constructions.

This paper presents a new randomized algorithm for solving the exact shortest vector problem. For the $n$-dimensional lattice $\mathcal L$, our algorithm runs in time and space $2^{n/2+o(n)}$. Our algorithm can be viewed as a $q$-ary analogue of the midpoint Hessian for an odd prime $q$; more precisely, we use the fact that, for a shortest vector $v$, the gradient (rather than Hessian) of the periodic Gaussian function at $v/q$ is nearly proportional to $v$ (up to sign), even after aggregation over a relatively large random affine coset. We compute the relevant coset gradient along a chain of intermediate lattices using a combinatorial procedure inspired by Wagner's generalized birthday algorithm, yielding the $2^{n/2+o(n)}$ time and space complexity. A variant of the algorithm solves the exact closest vector problem on every input $(y,\mathcal L)$ with a distance guarantee $\operatorname{dist}(y,\mathcal L)\le 1.039\lambda_1(\mathcal L)$ within the same time and space complexity. This guarantee holds for a random target and a random lattice drawn according to the Haar-Siegel measure. Thus, this algorithm solves a closest vector problem on such random instances in time and space $2^{n/2+o(n)}$.

Navigability is a fundamental property of graph-based search structures and plays an important role in the analysis of nearest-neighbor algorithms. Informally, a graph is navigable if, from any current point and toward any desired target, there is always an outgoing edge that moves strictly closer to that target. While this property provides a strong guarantee for greedy search, it can be inherently expensive: in the worst case, navigable graphs require $\Omega(n^{3/2})$ edges, where $n$ is the size of the dataset. Recently, Avi and Musco introduced $(1-\epsilon)$-almost navigability, a natural relaxation in which, from every current point, such a progress-making edge is required for almost all targets, while an $\epsilon$ fraction of targets may fail this condition \cite{avimusco2026almost}. They showed that every dataset admits such a graph with $O(n/\epsilon)$ edges. In this work, we prove a matching lower bound, establishing the optimality of their construction and giving a complete characterization of the sparsity achievable by almost-navigable graphs. For most values of $\epsilon$, our hard instances lie in Euclidean spaces of polylogarithmic dimension, across the full worst-case range of $\epsilon $, dimension $d=O(\sqrt{n}\log^{3/2} n)$ suffices. The same construction also sharpens our understanding of ordinary navigability: in the latter dimension, we exhibit datasets for which every navigable graph has $\Omega(n^{3/2})$ edges. Our proof reveals a surprising connection between almost navigability and the classical Zarankiewicz problem of constructing dense graphs with limited pairwise neighborhood overlap. This connection lets us translate extremal graph constructions into hard geometric instances for navigation, linking two seemingly different notions of graph sparsity.

We study the low-degree Steiner forest decomposition. Given a graph $G=(V,E)$ and a terminal set $U\subseteq V$, the standard decomposition returns a set $X\subseteq V$ of size at most $|U|/2$ and a forest $T\subseteq G-X$ of maximum degree $\Delta$ such that, for every connected component $C$ of $G-X$, some connected component of $T$ contains all terminals in $U\cap V(C)$. This is the central decomposition behind several connectivity oracles under vertex failures [DP20, LS22, LW24]. The state-of-the-art algorithms either take $O(mn\log n)$ time with degree bound $4$ [DP20], or take $m^{1+o(1)}$ time with the weaker degree bound $O(\log^{2}n)$ [LW24]. We show that if $T$ is allowed to contain vertices of $X$, then a degree-$4$ decomposition can be computed by a very simple algorithm in $O(m\alpha(n))$ time. Further, we show that this relaxed decomposition is equally useful for constructing connectivity oracles under vertex failures. As a consequence, we obtain a deterministic connectivity oracle under $d$ vertex failures with $\tilde{O}(m)$ space, $\tilde{O}(md_\star)$ preprocessing time ($d_\star$ is an upper bound on the number of failed vertices), $\tilde{O}(d^{2})$ update time, and $O(d)$ query time. Up to polylogarithmic factors, this oracle strictly improves all known oracles; in particular, it removes the $n^{o(1)}$ factors from the preprocessing and update times of [LS22, LW24].

Let $G\in\mathbb{R}^{M\times N}$ have independent standard Gaussian entries. For a fixed margin $\kappa\in\mathbb{R}$, the asymmetric binary perceptron asks for $\sigma\in\{\pm1\}^N$ such that $G\sigma/\sqrt{N}\ge\kappa\mathbf{1}_M$. We study the online version of this problem, in which the columns of $G$ arrive sequentially and each sign must be chosen irrevocably before future columns are revealed. We determine the exact threshold $\alpha_{\mathrm{on}}(\kappa)$ for every fixed $\kappa$: for $M/N\to\alpha$ with $\alpha<\alpha_{\mathrm{on}}(\kappa)$, there is a deterministic online algorithm, using $O(MN)$ arithmetic operations and polynomial bit complexity, that succeeds with high probability, while for $\alpha>\alpha_{\mathrm{on}}(\kappa)$, no online algorithm succeeds with high probability. The threshold is characterized by a one-dimensional stochastic control problem for Brownian motion. The main difficulty is to upgrade a single-coordinate Brownian limit to simultaneous feasibility of all $M=\Theta(N)$ constraints, which we do with half-line monotonicity and a short final correction block. At zero margin, we give a computer-assisted proof that $0.32747<\alpha_{\mathrm{on}}(0)<0.36664$. In particular, every density below $0.32747$ is achievable online by such an algorithm, more than tripling the best density previously proved attainable by any polynomial-time algorithm, online or offline (the previous bound was $\alpha\le0.1$, due to Li, Schramm, and Zhou). As $\kappa\to+\infty$, the online threshold agrees to first order with the offline storage capacity. As $\kappa\to-\infty$, it has the same asymptotic scale as the best known offline polynomial-time guarantee, while the storage capacity is larger by a factor of order $\kappa^2$.

For an $n\times n$ nonnegative matrix $A$, the Bethe permanent, which is computable in deterministic polynomial time, satisfies the tight universal comparison \[\operatorname{Bethe}(A) \leq \operatorname{per}(A) \leq 2^{n/2}\operatorname{Bethe}(A).\] The lower bound, due to Gurvits, is attained on forests. The upper bound, due to Anari and Rezaei, is attained by the adjacency matrix of a disjoint union of $4$-cycles. Confirming a conjecture of Anari, we provide an optimal girth-dependent refinement of the above comparison. More precisely, we show that if the bipartite support graph of $A$ has girth at least an even integer $g \geq 4$, then \[\operatorname{Bethe}(A) \leq \operatorname{per}(A) \leq 2^{2n/g}\operatorname{Bethe}(A).\] The upper bound is attained by the adjacency matrix of a disjoint union of $g$-cycles.

A layout of a graph G is an injective function $f : V(G) \rightarrow Z$, and the bandwidth of a layout f is $bw(G,f) = max_{uv \in E(G)} |f(u) - f(v)|$. The bandwidth bw(G) of G is the minimum bandwidth of a layout of G. Computing the bandwidth of a graph is a notoriously hard problem: assuming P != NP, there is no polynomial time algorithm, even on very restricted classes of trees [Monien, SIAM Journal on Algebraic Discrete Methods, 1986], and no constant factor approximation, even on trees [Dubey et al., JCSS 2011]. Assuming the Exponential Time Hypothesis, there is no algorithm with running time $f(k)n^{o(k)}$ to determine whether an input graph has bandwidth at most k, even on very restricted classes of trees [Dregi and Lokshtanov, ICALP 2014]. In this paper we show that Bandwidth on general graphs is FPT-approximable. In particular we give an algorithm that takes as input a graph G and an integer k, runs in time $2^{O(9^k)}n^{O(1)}$, and outputs a subtree T of G such that $bw(T) \geq k$ or a layout of G of bandwidth at most $(10^{85} k^{28})^{4^k}$. This resolves in the affirmative an open problem of Chung and Seymour [Discrete Mathematics, 1989], who asked whether the bandwidth of every graph G is upper bounded in terms of the maximum bandwidth of a subtree of G. Our theorem leads to a forbidden subgraph characterization for graphs of bounded bandwidth, and can be seen as an analog for bandwidth of the classic grid minor theorem for treewidth, the forbidden subtree theorem for pathwidth, and the forbidden subpath theorem for treedepth.

The shortest common superstring problem is to find the shortest string that contains every string in a given set as a substring. It is conjectured that the greedy algorithm that repeatedly selects a pair of strings with maximum overlap and merges them is a $2$-approximation algorithm, and this conjecture had remained open for nearly four decades. In this paper, we disprove this conjecture and show that the approximation ratio of this algorithm is at least $9/4$.

Sensitivity oracles preprocess a graph so that queries can be answered after any $f$ edge insertions and deletions, without recomputing from scratch. For structural optimization problems the known landscape is limited: for flows and cuts, all known compact oracles handle only $f\le2$ failures; existing oracles for $s$- and global min-cut apply only to undirected graphs; and for matchings, arborescence and spanning-tree packings, and arboricity, no efficient oracle is known for $f>1$. We present a unified algebraic framework based on sensitivity oracles for matroid packing, covering, and parity of sparse linear matroids, yielding the first oracles supporting an arbitrary number $f$ of updates across all of these problems (all constructions randomized Monte-Carlo). Concretely, we obtain efficient oracles for exact $(s,t)$-max-flow/min-cut, resolving an open problem of Baswana, Bhanja, and Pandey (ICALP'22) with near-optimal space; for all-pairs $k$-bounded flow, generalizing the near-optimal reachability oracle of Brand and Saranurak (FOCS'19, the case $k=1$); the first oracles for any $f$ for directed $s$- and global min-cut; oracles for $k$-disjoint arborescences, $k$-disjoint spanning trees, colorful spanning trees, and arboricity; and oracles for the existence of an $\alpha$-factor, with perfect matching as the case $\alpha=1$. We further introduce the subset sensitivity model, in which updates are confined to a susceptible edge set of size $\sigma$ fixed during preprocessing. Here we decouple updates from the matroid representation and eliminate the dependence on $k$ and the matroid density altogether: all of the above are supported with $\widetilde O(f^\omega)$ query time and $O(f\sigma^2)$ space. We also prove a matching $\Omega(\min\{\sigma^2,n^2\})$-bit lower bound when $f\ge2$, establishing optimality.

The \emph{$s$-Club Cluster Edge Deletion} problem asks whether, given a graph $G$ and an integer $k$, one can delete at most $k$ edges so that every remaining connected component has diameter at most~$s$. This generalizes the classical Cluster Edge Deletion problem by permitting components of bounded diameter instead of requiring cliques. On general graphs, $2$-Club Cluster Edge Deletion is known to be fixed-parameter tractable when parameterized by $k$, but it remains open whether it admits a polynomial kernel, as posed in~\cite{ABUKHZAM2023113864}. Motivated by this question, we study the problem on interval graphs and obtain a polynomial vertex kernel of size $\mathcal{O}(k^{5})$. As a complementary result, we also show that the \emph{$s$-Club Cluster Edge Deletion} problem is polynomial time solvable on unit interval graphs. We also show that $2$-Club Cluster Edge Deletion is NP-hard even on split graphs.

We study path diversification in trusted-node networks, where sensitive material is relayed through intermediate nodes, some of which may be compromised. Our randomized routing rule assigns each vertex an independent random priority and repeatedly expands the highest-priority vertex on the global frontier of the explored region. Let $G$ have $n$ vertices, let $s,t$ be honest endpoints, and let $C$ be a set of $c$ compromised intermediate vertices, called a cartel, whose deletion leaves $s$ and $t$ connected. For every fixed $c$ and every fixed target probability $q\in(0,1)$, we prove that $\Theta(n^c)$ independent executions are sufficient in the worst case for some route to avoid $C$ with probability at least $q$.

An LLM application often sells or internally allocates several service products: a small or premium model, a short or long token cap, and possibly multiple posted prices. The operational decision is not merely which model answers a prompt. A price changes purchase probability, a token cap changes both user value and the tail of resource consumption, and accepted requests compete for shared compute and premium-model capacity. Demand and output length are initially uncertain, while an offline model may provide useful but imperfect predictions. We formulate sequential pricing and admission with stochastic resource consumption. Each arriving request belongs to an observable segment. The platform chooses a product–price pair or makes no offer; purchase, revenue, and resource use are then random. An offline predictor supplies a uniform, validated error radius for every segment–product cell. We propose Prediction-Clipped UCB (PCUCB), which intersects the offline prediction interval with an online confidence interval, evaluates products using resource shadow prices, and reserves a sample-path envelope before commitment. The prior gives a fast start when accurate, while online learning protects the platform when predictions are coarse. The analysis is modular. On a simultaneous confidence event, regret against a buffered fluid benchmark is bounded by a pacing term plus the cumulative diameter of the intersected intervals. For $J$ segment-product cells and prediction radius $\varepsilon$, this yields \[ \widetilde O\left( \sqrt{T}+(1+\bar\Lambda) \min\{T\varepsilon,\sqrt{JT}\} \right), \] where $\bar\Lambda$ bounds operational shadow prices. Thus the algorithm smoothly interpolates between an almost full-information regime and learning from scratch. Hard feasibility holds on every sample path through reservation envelopes.

Daviaud, Guillon, and Merlet proved that comparison of max-plus automata is undecidable under a fixed state bound of 553 and explicitly left the range from 2 to 552 states open. We resolve the two-state endpoint. More strongly, given an arbitrary finite max-plus automaton $A$ and a max-plus automaton $B$ with at most two states, it is decidable whether $[\![A]\!](w)\leq [\![B]\!](w)$ for every word $w$. The structural reason is a one-dimensional projective normal form for two-state dynamics. Outside an effective bounded region, a transition has one of three tail behaviors: it propagates the unbounded projective gap with gap-independent height increment, forgets the gap with gap-independent height increment, or reads the gap magnitude into the height increment and then forgets it. In particular, any transition whose output depends on the unbounded gap necessarily destroys that gap. This yields an exact one-counter realization of $B$. Effective semilinearity of context-free Parikh images then reduces comparison to Presburger arithmetic. As a consequence, two-state max-plus comparison, equivalence, and positivity are decidable.

We study deterministic approximation algorithms for the permanent of a nonnegative matrix through the Bethe permanent, an approximation computable in polynomial time. The tight analysis of Anari and Rezaei gives a universal comparison between the permanent and the Bethe permanent within a factor $(\sqrt 2)^n$. The simple example of the unweighted $4$-cycle $C_4$ (or a union of disjoint $C_4$'s) shows that this bound is tight. We show that such $4$-cycle obstructions can be identified and exploited algorithmically. Given a Bethe optimizer, our algorithm identifies nearly isolated weighted $2\times2$ blocks and peels off a vertex-disjoint family of them. If the total weighted correction is large, we can improve the Bethe approximation; if it is small, we show that the Bethe permanent is within a factor of $(\sqrt2 - \varepsilon)^n$ of the truth. Combining these facts, we obtain a deterministic polynomial time $(\sqrt2-\varepsilon)^n$-approximation algorithm for the permanent of an arbitrary nonnegative $n\times n$ matrix, where $\varepsilon>0$ is some absolute constant.

We give the first polynomial-time algorithm, after half a century, for the celebrated Győri-Lovász theorem, which resolved a conjecture of Frank (1975). The theorem, one of the simplest existential theorems to explain, states that every $k$-connected graph can be partitioned into $k$ disjoint connected subgraphs of arbitrary prescribed positive sizes. This is a fundamental structural result with broad applications, such as flexible allocation of connected subnetworks of prescribed sizes in sufficiently connected cloud infrastructures. While Lovász (1977) gave a highly non-constructive proof for a stronger directed version using algebraic topology, Győri's original constructive proof (1976) requires exponential time. Despite more than 50 years of effort, no polynomial-time algorithm was known even for $k>4$. Determining the computational complexity of the Győri-Lovász theorem—whether it admits even a sub-exponential-time algorithm or is computationally hard (in particular, PLS-complete or PPAD)—has remained one of the central open problems in algorithmic graph theory. In this paper, we finally resolve this long-standing problem by a fundamentally new proof of the existential theorem via introducing the novel concept of flow-essential assignment, which genuinely marries matching and cut structures and yields the first polynomial-time constructive algorithm for the Győri-Lovász theorem. In fact, we obtain a polynomial-time algorithm for Lovász's stronger directed version, whose proof was non-constructive even for DAGs; for DAGs, we further obtain a near-linear-time algorithm. We also develop polynomial-time algorithms for weighted generalizations where the seminal work of Chen, Kleinberg, Lovász, Rajaraman, Sundaram, and Vetta (JACM'07) on confluent flows established only existential non-constructive results.

A machine faces many jobs whose lengths are hidden. Spending one unit of time to inspect a job may reveal a short job that should be finished now, or it may reveal nothing useful while every other job waits. When should the machine keep looking, and when should it start working? We study natural variants of this question and provide optimal algorithms in both the worst-case and instance-optimal frameworks. The resulting algorithms are often quite simple, which may make them useful in practice.

A finite connected graph is rarely a Cayley graph. We measure how far it is from being one: given $G$ with $n$ vertices and $m$ edges, how few edges must be added, or added and deleted, before the result is a Cayley graph of an abelian group of order $n$ on the same vertex set? This defines two invariants, the completion number $\gamma^{+}$ (additions only) and the Cayley edit distance $\gamma_{\triangle}$ (both), each normalized by $m$. We show that deciding the edit version is NP-complete already for a fixed cyclic host, by a reduction from Hamiltonian Cycle in which the edit cost of a labeling is $n+m-2k$ when it realizes a longest path with $k$ edges; the optimal cost is $m-n+2pp(G)$, bounded in polynomial time by the matching number. We prove that irregularity alone forces $\gamma^{+}(G)\ge n\Delta^{*}/(2m)-1$, where $\Delta^{*}$ is the least $d\ge\Delta$ with $nd$ even, computable in linear time from the degree sequence; we characterize equality exactly. It is attained on the star, where $\gamma^{+}(K_{1,q})=(q-1)/2$ and the star maximizes $\gamma^{+}$, while $\gamma_{\triangle}$ stays bounded by an absolute constant. We determine paths and grids exactly, $\gamma^{+}(P_n)=\gamma^{+}(P_n\,\square\,P_n)=1/(n-1)$, and show $\gamma_{\triangle}(K_{1,q})\to 2$, not the $3/2$ suggested by the additive case. We report an exhaustive certified census of all $995$ connected graphs on at most seven vertices. The degree bound is attained on $89.4\%$ and the two invariants separate strictly on $84.7\%$, though both rates vary sharply with order: attainment $100\%,100\%,84.8\%,89.7\%$ and separation $0\%,61.9\%,73.2\%,87.7\%$ for $n=4,5,6,7$, dominated by the $853$ graphs on seven vertices. The star uniquely maximizes both. Edit count and the bi-Lipschitz distortion of the completed host are independent, moving oppositely on stars and paths.Data and certificates at doi:10.5281/zenodo.21852006.

For a given geometric graph-convexity on a graph $G$ equipped with a weight function on the vertices with value in $\mathbb{Z}$, the Max Weight Convex Set problem consists in determining the convex set $S$ with maximum weight (sum of the weight of the vertices in $S$). Although the problem is NP-complete in general, it remains polynomial for particular cases. After a survey of known results, our main contribution uses a generalisation of the maximum subsequence problem to laminar trees. Then we derive a linear algorithm for proper interval graphs and a quadratic one for interval graphs. Both improve the state of the art.

In this paper, we study the Pattern Avoidance problem of determining whether a given graph $G$ admits a linear vertex order which avoids a given pattern $P$, i.e., a vertex sequence with some forced and forbidden edges, on every suborder. Such patterns form a natural ordered counterpart to induced subgraphs in the order-invariant setting, and it is known that Pattern Avoidance captures a broad variety of graph problems including Bandwidth, Vertex Coloring, Queue Number, and extends to vertex-deletion problems such as Odd Cycle Transversal. We show that Pattern Avoidance is $\Sigma_2^{\textsf{P}}$-complete and furthermore remains intractable (in both the classical and parameterized sense) even under a variety of severe restrictions to both the pattern $P$ and the graph $G$. As our main contributions, we complement these lower bounds with the following tractability results, which provide a unifying framework for recognizing pattern-definable graph classes: - a fixed-parameter algorithm w.r.t. the vertex integrity of $G$ plus $|V(P)|$, - a fixed-parameter algorithm w.r.t. the neighborhood diversity of $G$ plus $|E(P)|$, and - a polynomial algorithm for Pattern Avoidance on forests for almost all constant-sized patterns.

We consider the Boolean function ${\rm SMul}_{n-1}^n(\boldsymbol{x},\boldsymbol{y})$, which computes the middle bit of the multiplication of two natural numbers represented as $n$-bit binary strings $\boldsymbol{x}$ and $\boldsymbol{y}$, drawn from a restricted domain. We investigate the width of OBDDs computing ${\rm SMul}_{n-1}^n$. We introduce a combinatorially defined function $s_*(n)$ and show that the width of such OBDDs is $\Theta(2^{s_*(n)})$.

The query-optimal algorithm of [CGWZ26] for general time-dependent Hamiltonian simulation uses $$ q = O\left( \alpha T + \frac{\log(1/\varepsilon)}{\log\left(e + \log(1/\varepsilon)/(\alpha T) \right)} \right) $$ queries to $\mathrm{HAM\mbox{-}T}$ within $\varepsilon$ error for a Lipschitz-continuous time-dependent Hamiltonian $H(t)$ on $[0,T]$ satisfying $\left\lVert H(t)\right\rVert\leq\alpha$. However, its direct circuit implementation incurs a substantially larger gate overhead. In this note, we give an implementation of the same algorithm that retains its optimal query complexity and uses $$ O\left[ q \left( a + \log\left(1 + \frac{T(\alpha + \beta T)}{\varepsilon} \right) \right) \right] $$ one- and two-qubit gates, where $a$ is the number of block-encoding ancilla qubits and $\beta$ is the Lipschitz constant of $H$. The main ingredient is an exact dyadic factorization of the ordered update product in the underlying one-query transducer.

We present a new distributed algorithm for computing a minimum spanning tree (MST) in the \textsf{CONGEST-KT$_{1}$} model, where messages are limited to $O(\log n)$ bits and each vertex initially knows the identifiers of its neighbors. Our algorithm exposes a two-parameter time–message trade-off: for any $0 \leq \lambda \leq \kappa \leq 1/2$, it runs in $\tilde{O}(n^{\lambda}D_G + n^{1 - \kappa - \lambda} + n^{1 - 2\kappa + \lambda} + n^{1/2})$ rounds and uses $\tilde{O}(\min\{m, n^{1 + \kappa}\})$ messages, where $n$, $m$, and $D_G$ are the number of vertices, edges, and thenetwork diameter, respectively. In particular, setting $(\kappa, \lambda) = (1/3, 1/6)$ yields an MST algorithm running in $\tilde{O}(n^{1/2} + n^{1/6}D_G)$ rounds with only $\tilde{O}(n^{4/3})$ messages. Under the mild assumption $D_G = O(n^{1/3})$, this is round-optimal while improving the best known message bound of $\tilde{O}(n^{3/2})$. More broadly, our algorithm breaks the quadratic time–message trade-off barrier $\mathrm{\# rounds} \cdot \mathrm{\# messages} = \tilde{\Omega}(n^2)$, which no previous MST algorithm in the \textsf{CONGEST-KT$_{1}$} model has been able to overcome, and it does so for almost the entire range of the diameter $D_G$. As a byproduct, we also obtain new low-message broadcast, spanning-tree, and leader-election algorithms.

Classical lower bounds show that multiplying two degree-three polynomials over $\mathbb F_2$ requires nine scalar products in bilinear or quadratic models. They do not settle unrestricted Boolean multiplicative complexity: an XOR–AND circuit may reuse nonlinear intermediate wires, and Boolean equality is taken modulo $x_i^2=x_i$, so a multiplication can lower algebraic degree. Let $\operatorname{Mul}_4:\mathbb F_2^8\to\mathbb F_2^7$ output the seven coefficients of the product of two four-term binary polynomials. We prove that its unrestricted XOR–AND multiplicative complexity is exactly nine. This resolves, for a natural vector-valued quadratic function, the Boyar–Find question of whether a quadratic-circuit lower bound can persist against unrestricted nonlinear reuse. The proof is structural rather than exhaustive. A useful purely quadratic prefix is forced onto the three rational places of $\mathbb P^1(\mathbb F_2)$. In a hypothetical eight-AND circuit, the unique non-useful gate must carry a cubic high part. Any useful continuation then forces a rational tangent and exposes a first Hasse jet, while exterior jet separation together with Boolean idempotence prevents the same defect from exposing the second Hasse jet. The required useful suffix therefore cannot exist. A complete Lean 4 formalization verifies the Boolean-ANF semantics, the unrestricted circuit model, and the exact theorem; it uses no project-specific axiom or native decision procedure. The same zero-defect flag argument gives multiplicative complexity six for three-term multiplication, and the method isolates the multi-defect obstruction for five terms.

Under individual smoothness, known algorithms for nonconvex finite-sum optimization use $O(n+\sqrt n\Delta L_{\max}/\varepsilon^2)$ incremental first-order oracle (IFO) calls, while existing lower bounds miss the factor $\sqrt n$ in the second term. We prove the matching lower bound $\Omega(n+\sqrt n\Delta L_{\max}/\varepsilon^2)$ for randomized IFO algorithms, allowing component indices and query points to depend on the complete preceding history. Thus PAGE and SPIDER are minimax optimal up to universal constants under individual and mean-squared smoothness. Under the global Polyak-Lojasiewicz (PL) condition, prior lower bounds did not capture the correct dependence on $n$ under individual smoothness, and the regime of small $\kappa$ remained open for randomized IFO algorithms. We prove an $\Omega(n+\kappa_{\max}\sqrt n\log(\Delta/\varepsilon))$ lower bound for $\kappa_{\max}\ge\sqrt n$, and an $\Omega(n+n\log(\Delta/\varepsilon)/(1+\log(\sqrt n/\kappa_{\max})))$ lower bound for $3\le\kappa_{\max}\le\sqrt n$. Restarted PAGE attains matching upper bounds under mean-squared smoothness in terms of $\kappa_{\rm ms}=L_{\rm ms}/\mu$. Our lower bounds use dense weak hiding. Each hidden direction is spread across all components, making individual queries weakly informative while preserving the direction in the average. A sufficiently large dimension makes premature activation of future stages unlikely within the query budget. Balancing the cost of revealing a stage against the number of stages permitted by individual smoothness yields the missing $\sqrt n$ factor.