Papers

A partial matrix has entries in $\{0,1,\star\}$, and a disambiguation replaces each $\star$ by $0$ or $1$. We construct partial matrices whose fully specified submatrices satisfy strong restrictions, yet every disambiguation contains every binary matrix of a prescribed size. Our first result answers a question of Alon, Hanneke, Holzman and Moran on the disambiguation of linear classifiers with margin. For $0<\varepsilon<\pi/2$, let $M_\varepsilon^d$ be the partial matrix indexed by points of the unit sphere $\mathbb S^d$, with entry $0$ for pairs at spherical distance at most $\varepsilon$, $1$ for pairs at distance at least $\pi-\varepsilon$, and $\star$ otherwise. Although these matrices have VC-dimension bounded independently of $d$, we prove that every disambiguation contains every binary $k\times k$ matrix once $d$ is sufficiently large. This also yields a partial concept class of Littlestone dimension $1$ with no disambiguation of finite VC-dimension. We also construct, for every $k$, a finite partial matrix whose fully specified $2\times2$ submatrices are all constant, while every disambiguation contains every binary $k\times k$ matrix. A symmetric analogue holds for partial graphs: for every $k$, there exists a partial graph of VC-dimension at most $1$ whose fully specified induced subgraphs are all cliques or stable sets, yet every disambiguation contains every $k$-vertex graph as an induced subgraph. A disambiguation can be viewed as a $2$-coloring of the unspecified entries, making Ramsey theory a natural framework for forcing prescribed patterns. Our proofs draw on two recent Ramsey theorems: the geometric argument uses Pálvölgyi's Dense Block theorem, while the combinatorial constructions rely on the girth Ramsey theorem of Reiher and Rödl, a suitable strengthening of the induced Ramsey theorem.

Locally decodable codes (LDCs) enable recovery of any message symbol by probing only a small number of positions in a possibly corrupted codeword. The central parameters of an LDC are its rate, locality, and error tolerance. Ideally, one would like all three parameters to be constant. However, classical lower bounds show that such codes cannot exist. A recent line of work introduced amortized locally decodable codes (aLDCs), in which the decoder is tasked with recovering an entire block of consecutive message symbols rather than a single symbol. While prior work obtained ideal aLDCs with constant rate, constant error tolerance, and constant amortized locality, those constructions relied on either shared randomness hidden from the channel or computational assumptions restricting the channel. Another well-studied relaxation is the notion of a relaxed locally decodable code (RLDC), in which the decoder may output a special failure symbol $\bot$ rather than risk decoding incorrectly. In this work, we introduce the notion of an amortized relaxed locally decodable code (aRLDC), combining amortized decoding with the relaxed decoding paradigm. Unlike prior ideal aLDC constructions, our model is fully information-theoretic and makes no assumptions about shared randomness or computational limitations of the adversarial channel. We construct the first aRLDC with constant rate, constant error tolerance, and constant amortized locality. Moreover, for any block of length $\Omega(\mathrm{polylog}(k))$, our decoder achieves amortized locality $1+\delta^{1 - o(1)}$, where $\delta$ is the error tolerance parameter. Thus, asymptotically, recovering a long block requires essentially less than two codeword probe per message symbol recovered. By contrast, without amortization no RLDC can simultaneously achieve constant rate, constant error tolerance, and constant locality.

We give a simple exposition of the Matrix Spencer theorem due to Akbas and Sra [AS26].

We construct a simple rank-$7$ oriented matroid on $24$ elements with no simplicial tope, disproving the Las Vergnas simplex conjecture from 1980. We then show that the mutation graph on uniform oriented matroids is disconnected for infinitely many values of rank $r$ and ground-set size $n$, disproving the Cordovil–Las Vergnas conjecture from 1988.

Bilu and Linial conjectured that every finite $d$-regular graph with $d \geq 2$ admits an edge signing $\sigma$ whose signed adjacency matrix $A_\sigma$ has spectral radius at most $2\sqrt{d-1}$. We prove that every signing meeting the mixed-root condition $r_{A_\sigma}\leq\sqrt{2(d-1)}$ satisfies \[ \rho(A_\sigma) < \frac{3+\sqrt5}{2}\sqrt{d-1}, \] where $r_{A_\sigma}$ is the largest root of the mixed determinantal polynomial $\chi[A_\sigma,-A_\sigma]$. The interlacing theorem of Ravichandran and Srivastava guarantees a signing satisfying the mixed-root condition, so our result improves the coefficient $2\sqrt2$ in their two-sided spectral bound. In the proof, we construct a positive matrix-valued probability measure supported on the roots of $\chi[A_{\sigma},-A_{\sigma}]$. The second moment gives a simple matrix inequality $A_{\sigma}^2 + dI \preceq 4r_{A_\sigma}^2I$, which yields a preliminary coefficient $\sqrt{7}$. Estimates for the fourth moment use information about short walks to obtain the coefficient $(3+\sqrt{5})/2$. For triangle-free graphs, the coefficient improves to $\sqrt6$, and for graphs with girth at least five, it improves to $\sqrt{35/6}$. As a result of independent interest, we extend the construction to $\chi[A_1,\ldots,A_k]$ for Hermitian matrices $A_1,\ldots,A_k$ with zero diagonal, and compute the first two moments explicitly. Finally, an explicit signing of $K_8$ shows that the mixed-root condition alone cannot guarantee a coefficient below $(4+\sqrt5)/\sqrt6$.

Let $\sigma_{n,d}$ denote the elementary symmetric Boolean function of $n$ variables and degree $d$. Previous work determined $C(\sigma_{n,d})$ when $d$ is odd or a power of two, but the general even non-power-of-two case remained open. We determine the certificate complexity for every degree $1\le d\le n$, thereby completing the classification for elementary symmetric Boolean functions. Writing $d=2^t m$ with $m$ odd, we obtain an explicit formula in which the possible deficit from the maximal value $n$ is controlled by $2^t$, while the exact value is determined by a binary containment condition involving $m$. In particular, \[ n-2^{\nu_2(d)}+1\le C(\sigma_{n,d})\le n, \] and we characterize when the upper bound is attained. Moreover, we determine the least positive period of the certificate-complexity deficit $\Delta_d(n)=n-C(\sigma_{n,d})$: it is $1$ for odd $d$, equals $d$ when $d$ is a power of two, and equals $2^{\lfloor\log_2 d\rfloor+1}$ for even non-power-of-two $d$. This least-period problem is distinct from the classical periodicity of the underlying value sequence $\binom{j}{d}\bmod 2$. The known odd-degree and power-of-two formulas are recovered as special cases.

We give an algorithm counting the permutations that avoid a fixed pattern of the following form: the direct sum of an increasing pattern and 231. The first members of the family are 1342 and 12453. For each member the algorithm uses polynomially many operations and stored integers, with degrees that grow linearly in the length of the pattern. It comes from a recurrence that reads a permutation from left to right and records the constraints that the letters read so far impose on those still unread. This recurrence has exponentially many states, but part of each state is protected: later steps carry it along unchanged and do not depend on it, and factoring the protected part out leaves a dynamic program of polynomial size. For 12453 a translation symmetry sharpens the bounds to degree seven for the operations and degree four for the storage. We also compute the number of 12453-avoiding permutations of every length up to 150. The previously published series, due to Biers-Ariel (2019), reached length 38. We also give a sampler of uniformly random avoiders. A floating-point implementation of it, proved to be within total variation distance $3.5\cdot10^{-5}$ of uniform for ideal random bits, draws the one million 12453-avoiding permutations of length 300 shown in a heatmap. The literal and kernel recurrences for 1342 and 12453 are verified in the Lean 4 proof assistant.

We show that every planar graph with no $t \times t$ grid minor has treewidth at most $4t +4$. This improves on the previously best known bound of $\frac{9}{2}t - \frac{11}{2}$, due to Gu and Tamaki (2012), and is within a factor $2$ of optimal. A key step in the proof is showing the following result, which might be of independent interest: Every $2$-connected plane graph $G$ with radius $d$ and faces of size at most $k$ has a tree-decomposition of width at most $\max\{3d+ k+5, 2d+2k+1\}$ such that the vertex set of every face of $G$ is contained in some bag.

We construct a finite connected simple cubic graph $F$ such that every signing of its edges yields a signed adjacency matrix with an eigenvalue outside $[-2\sqrt2,2\sqrt2]$. This disproves the Bilu–Linial signing conjecture for general regular graphs. The graph $F$ is not Ramanujan, and the conjecture restricted to Ramanujan base graphs remains open.

In many real-life matching problems, waiting agents might abandon before being matched, such as patients deceasing before receiving organs, passengers/drivers cancelling ride requests, or raw materials/intermediary products degrading in production lines. This poses the need for incorporating reneging in stochastic matching models. In this work, we consider matching models on hypergraphs with batch arrivals and general-weight matchings. Since our model allows fractional weights, we may not be able to talk about individual items, and thus reneging is not required to be independent between items of the same class. For the simplicity sake's, we assume items arrive at discrete time. We show that stability depends on the exact nature of reneging, in stark contrast with the non-reneging case where it depends on the arrivals only through the arrival rates. To our best knowledge, this is the first such sensitivity result in stochastic matching. We uncover a new stabilising mechanism which exists neither in the non-reneging case nor in the graph case, which explains why incorporating reneging in stochastic matching is not straightforward. Together with balancing mechanism as hinted in the online assignment framework for the non-reneging case, it gives a criterion necessary and sufficient for stability. Finally, whilst verifying stability is a hard problem, we give a family of MaxWeight-type policies parameterised by $\varepsilon > 0$, which are maximally stabilising for all $\varepsilon$ sufficiently small. Unfortunately there is no effective bound for $\varepsilon$, but we show how to adjust its value during implementation.

The Art Gallery Problem (AGP) asks for the fewest guards that see all of a simple polygon. It is $\exists\mathbb{R}$-complete, hence NP-hard. We show that for a particular class of polygons, confining guards to a single edge makes AGP exactly and efficiently solvable. We call this the Strait Guarding Problem (SGP). Its input is a weak visibility polygon (WVP): a simple polygon where every point is seen from some point of one fixed edge, the base. SGP places the fewest guards on the base that jointly see the whole polygon. First, a structural fact: guards on the base edge that cover the boundary already cover the entire interior, turning a two-dimensional covering problem into a one-dimensional one. Our main result is the Witness-Guard Algorithm, which solves SGP exactly in $O((n + \mathrm{OPT} \cdot \rho)(\log n + \log \mathrm{OPT}))$ time, where $\rho$ is the number of reflex vertices in the WVP and OPT is the minimum number of guards. It is output-sensitive and certifies optimality by a witness set of size OPT derived from its output. We also study the guarding-the-vertex version and prove a tight $\Theta(n \log n)$ bound, with the lower bound following from Sorting. As a corollary of SGP, we obtain two results for altitude terrain guarding (ATG), a special case that SGP generalizes. We give a linear-time perfect-guarding algorithm, improving the previous $O(n^2 \log n)$ bound of Daescu, Friedrichs, Malik, Polishchuk and Schmidt. We also resolve their problem on the minimum guarding altitude, in $O(nk + k^2 \log k)$ time, improving on the $O(k^2 \lambda_{k-1}(n) \log n)$ bound of Kang, Kim and Ahn.

We revisit reconfiguration in the fair allocation of indivisible goods, where the goal is to transform one fair allocation into another through a sequence of exchanges while preserving fairness at every step. Our focus is on the hierarchy of envy-freeness up to $k$ goods (EF$k$). We show that for any fixed $k$, two EF1 allocations with the same size vector need not admit a reconfiguration path whose intermediate allocations satisfy EF$k$. This impossibility persists even when the two allocations arise from standard EF1 approaches: the envy cycle elimination algorithm or the maximum Nash welfare solution. In contrast, we prove that allocations with the same size vector produced by recursively balanced picking sequences, including round-robin, are always connected via a path that maintains EF2. We also show that deciding whether an EF1 reconfiguration path exists is NP-hard for any fixed number of agents. Furthermore, we complement these exchange-based results by studying a more permissive model that also allows transfers, establishing additional connectivity guarantees.

The Komlós conjecture is a classic problem in discrepancy theory; it asks whether an absolute constant $K$ exists such that given any $n$ vectors $a_1,\ldots,a_n$ inside the $m$-dimensional Euclidean ball, regardless of how large $m,n$ are, there is always a selection of signs $\varepsilon_1,\ldots,\varepsilon_n$ guaranteeing $$\|\varepsilon_1a_1+\ldots+\varepsilon_na_n\|_\infty \leq K.$$ We show that if the $\varepsilon_i$'s are allowed to take not just the values of $\pm 1$ but any unit modulus complex number, which we refer to as complex discrepancy, then the above inequality holds for a finite, explicit constant $K_{\mathbb{C}}$. Here, the $\ell^\infty$ norm of the resulting vector in $\mathbb{C}^m$ is the largest modulus of its entries, and thus the complex discrepancy of real vectors is equivalent to their rank-$2$ vector discrepancy. This quantity provides an upper bound (modulo uniform constant prefactor) on Gaussian discrepancy – a discrepancy measure introduced by Chewi, Gerber, Rigollet and Turner. Thus, we also resolve the Komlós conjecture for Gaussian discrepancy. Our paper builds upon the recent work of Bansal and Jiang on the Beck-Fiala and Komlós conjectures, which we approach from the formalism of Burkholder and the Bellman function method from probability and harmonic analysis. Our work was in part motivated by the realization that the complex discrepancy of the columns of any unitary matrix is equal to 1, a fact that follows from a straightforward calculation based on Idel and Wolf's generalization of the Sinkhorn normal form for unitary matrices.

An explicit graph is given with left size $N$, left degree $\widetilde O(\log^2 N)$, right size $(1+o(1))K$ and $1$-expansion up to~$K$, meaning that every left subset of size $K' \le K$ has at least $K'$ neighbors. Let $\C(x)$ be the minimal length of a program that prints~$x$ (i.e., the central concept in Kolmogorov complexity). The $1$-expander is used to obtain an algorithm that on input $x$ computes in time $\poly(|x|)$ a list with $\widetilde O(|x|^3)$ programs such that at least 1 program prints $x$ and has length $\C(x) + O(1)$. This improves on the $O(|x|^{6+\eps})$ upper bound in~\cite{zim:c:shortlistshortproof} and is close to the $\Omega(|x|^2)$ lower bound from~\cite[theorem 4]{bmvz:j:shortlist}. In the companion paper “Online matching games in bipartite expanders: applications to data structures," the $1$-expander is used to obtain dynamic dictionaries in which the query operation has non-adaptive memory access.

We classify all smallest sets of edge deletions that destroy every nonnested perfect matching in a complete ordered graph. The vertices have a fixed linear order. A perfect matching selects edges that use each vertex exactly once; it is nonnested when no selected edge has both endpoints strictly between those of another. On $2k$ vertices, exactly $k$ deletions are necessary, and we describe all $2^k+k-2$ minimum deletion sets for every $k\ge2$. Allowing a matching to leave vertices unused changes the extremal problem. Barát, Freschi and Tóth proposed that an $n$-vertex ordered graph avoiding a nonnested $k$-edge matching can have at most $(k-1)n$ edges. We construct counterexamples for every $k\ge5$ and $n\ge2k+1$. For $n\ge3k$, our constructions exceed the proposed value by a number of edges proportional to $k^2$, matching the order of the known upper bound on this excess. The classification follows from cuts between consecutive intervals; the larger constructions coordinate deletions at the two ends of the vertex order. The exact extremal value remains open in general.

Single-machine asymmetric weighted earliness–tardiness (AWET) scheduling asks how to sequence jobs around a common synchronization date when early and late completion incur unrelated job-dependent penalties. At the boundary nonrestrictive date $d=\sum_jp_j$, a compact V-shaped schedule reduces the continuous-time problem to a quadratic choice of a nonempty early set. We establish four complementary results for this model. First, the positive-integer problem is strongly NP-complete by a unary-polynomial reduction from Restricted Exact Cover by 3-Sets. Second, unrestricted AWET admits a polynomial-time $(3+2\sqrt2+\varepsilon)$-approximation based on an anchored semidefinite relaxation and deterministic marginal thresholding. Third, when the earliness and tardiness ratio orders are strict reversals, the problem is weakly NP-complete but has an exact two-resource pseudopolynomial dynamic program. Fourth, for fixed total refinements whose ratio permutation is separable, an exact separating-tree recurrence and coordinated geometric trimming yield an FPTAS. The proofs use different manifestations of the same canonical objective: scale-separated prefix penalties, positive-semidefinite minimum-kernel covariance, a dominant completed load square, and a bounded four-coordinate decomposition interface. Together, the results show that the decisive issue is not merely whether the two ratio orders agree, but whether their interaction can be controlled by a global certificate or compressed into a bounded constructive interface.

We give a polynomial-size linear programming bound for $q$-ary all-symbol locally recoverable codes with locality parameters $(r,\delta)$, without assuming linearity. The key idea is to keep, for every ordered pair of codewords and every selected recovery view, the joint Hamming weight on the helper set, the recovered coordinate, and the rest of the code – rather than collapsing this triple into a single distance, as earlier formulations do. Averaging this three-block distribution over recovery views of the same length yields exact identities linking it to the global distance distribution, together with nonnegative product-Krawtchouk constraints that encode locality and spectral positivity simultaneously. The resulting LP has polynomially many variables, its optimum dominates the ordinary Delsarte bound, and an earlier outside-distance formulation, the convex-hull bound of Li–Wei–Xiong, and the dual-based bound of Gruica–Jany–Ravagnani all arise from it as coarser marginals. Exact rational certificates over $q=2,3,4$ show the bound is strictly stronger than the best of these prior LPs in thirteen of fifteen tested cases, pinning down seven exact maximum code sizes and twelve exact maximum linear dimensions.

Blocking flow-type maximum matching algorithms are based on finding maximal sets of shortest augmenting paths. They run in $O(m\sqrt{n})$ time, on both bipartite [HK73, Din70, Kar73a, Kar73a] and nonbipartite graphs [GT91, Gab17, Vaz24], but this time bound can be improved if the input is constrained. In this paper we consider $d$-regular bipartite and nonbipartite graphs. Previous algorithms show that a perfect matching in $d$-regular bipartite graphs can be computed in near-linear time deterministically [COS01] or sublinear time with high probability [GKK13]. On $d$-regular non-bipartite graphs, a $(1-1/(d+1))$-approximation can be computed in sublinear time $O(n \log n)$ with high probability [DH25], and hence a maximum matching can be computed in $O(n^2)$ time, w.h.p., which is slightly faster than the best deterministic algorithm for regular graphs [Yus13], running in O(n^2 log n) time. We prove that any blocking flow-type maximum matching algorithm based on finding shortest augmenting paths runs in $O(n^2)$ time on d-regular graphs, both bipartite and nonbipartite. On nonbipartite graphs this is an asymptotic improvement over $O(n^2 \log n)$ [Yus13] and an improvement over $O(m\sqrt{n})$ [GT91, Gab17, Vaz24] when $d = \omega(\sqrt{n})$. It also improves [DH25] by making its $O(n^2)$ bound deterministic. However, the main take-away message is that no new algorithms are needed: the "classic" matching algorithms automatically outperform [Yus13, DH25]. We also consider extensions of our results to graphs that are only "nearly regular," meaning that their degrees all lie within a specified range, $[d, \Delta]$.

This paper investigates a two-machine permutation flow shop in which the second machine is unavailable during one fixed interval $[s,t]$. We consider the non-resumable setting: an operation interrupted by the interval must restart from the beginning after the machine becomes available. The objective is to minimize the makespan. We establish three results. First, we give a polynomial-time $10/7$-approximation algorithm. Second, we develop a pseudopolynomial-time exact dynamic program. Third, we prove that the problem does not admit a fully polynomial-time approximation scheme (FPTAS) unless $\mathrm{P}=\mathrm{NP}$, even when the non-availability interval has unit length. Together, these results characterize a distinctive complexity profile: exact optimization is possible in pseudopolynomial time, whereas the usual route from such an algorithm to an FPTAS is impossible unless $\mathrm{P}=\mathrm{NP}$. They also reveal an approximability separation from the corresponding non-resumable problem with the interval on the first machine.

A tournament T is k-inducible if there are k linear orders on its vertex set such that, for every arc $i \to j$ of T, a majority of the orders rank i above j. For odd k, let N(k) be the least order at which some tournament is not k-inducible. Only N(3) = 8 is known exactly; for N(5) the best bounds were $12 \le N(5) \le 38$, from our previous paper [2], which also gave the first explicit example of moderate order, the Paley tournament $P_{43}$. Results. A bespoke search algorithm improves both ends: $13 \le N(5) \le 23$. The upper bound comes from proving that $P_{23}$ is not 5-inducible, the case Bachmeier et al. [1] reported they could not decide, their SAT solver not having terminated within a cumulative six weeks; ours takes 22 hours on one laptop. The lower bound comes from an analysis at order 12. We also show that $P_{31}$ is not 5-inducible, while $P_{19}$ is 5-inducible but not with unit margin, that is, not by a profile in which every arc is carried by exactly three voters against two. Both $P_{19}$ and $P_{23}$ are arc-critical for their respective properties, whereas $P_{31}$ and $P_{43}$ are not vertex-critical: deleting a vertex leaves a tournament that is still not 5-inducible. Method. The search places one vertex at a time, always choosing the vertex with the fewest options left, and propagates the consequences. Together with the automorphisms of the tournament, this decides on a single laptop instances that neither integer programming nor a general-purpose SAT solver can settle. The refutations for $P_{19}$ and $P_{23}$ are certified as well: the search is split into independent subproblems, a SAT solver emits a machine-checkable proof for each, and a separate program rechecks every proof. All results, subject to two human-checked lemmas, are reproducible from https://github.com/Leonardini/TournamentsBeyond5Voters.

Symmetric rendezvous on the line is a search problem in which two agents, initially placed at distance $2d$, must follow the same randomized strategy to meet as quickly as possible. In the standard model, agents may condition their actions on the entire execution history, and both the known- and unknown-distance variants admit expected rendezvous time $\Theta(d)$. We study the role of memory by introducing oblivious self-distance strategies, in which an agent's decision depends only on her position relative to her own starting location. For an initial separation of $2d$, let $R_d$ denote the optimal oblivious expected rendezvous time in the known-distance setting. We develop two finite-state frameworks based on absorbing Markov chains. Truncated chains give computable upper bounds through finite-support strategies, while weak-peek chains give lower bounds through a revealed-information relaxation. Together, they provide a mechanism for certifying optimality. Using that mechanism, we determine $R_1$ exactly and prove that it is attained by a finite-support strategy. For $d=2,\ldots,6$, numerical optimization gives the same truncation structure and objective values, yielding rigorous upper bounds below $7.83d^2$. We do not prove that the computed weak-peek minimizers are global, but the stability of the computations leads us to conjecture that they are, in which case the corresponding truncated strategies are optimal. We also prove that $R_d=\Theta(d^2)$. In the unknown-distance setting, we construct a universal strategy, independent of $d$, with expected rendezvous time $O(d^{2+\eta})$ for every fixed $\eta>0$. Thus, under the memory restriction, the known-distance rendezvous time becomes quadratic, while near-quadratic performance remains possible even without knowing $d$. The asymptotic analysis uses birth-death Markov chains and their electrical-network interpretation.

In a 2021 paper, Jankiewicz, Norin, and Wise asked whether there exists a finite $4$-connected graph of girth at least four and nonnegative Charney–Davis curvature such that no $4$-connected ordinary subgraph admits a legal system. We construct such a graph by starting from the hexagonal prism and attaching three $K_{3,4}$-based caps along pairwise disjoint induced $4$-cycles. The key structural input is a restriction theorem showing that a legal system on an induced-$4$-cycle amalgam restricts to each side, so the obstruction carried by the negatively curved prism survives the attachments. The resulting $33$-vertex graph is $4$-regular and $4$-connected, has girth four and Charney–Davis curvature one, and, by $4$-regularity, is its own unique $4$-connected ordinary subgraph.

Clustering systems provide a natural way to encode structural information contained in phylogenetic networks. In this note, we study the clustering systems of normal and tree-child networks through an overlap-based property of set systems, called not-overlap-covered (NOC). We show that the NOC property is equivalent to inclusion-visibility, a memberwise formulation of the strict-compatibility condition previously used for tree-child clustering systems. We characterize normal networks as precisely the semi-regular networks whose clustering systems satisfy NOC. Consequently, a clustering system is realized by a normal network if and only if it satisfies NOC, or equivalently, if every one of its clusters is inclusion-visible. In this case, the Hasse diagram provides a canonical normal realization. These are exactly the clustering systems realized by tree-child networks. The NOC formulation yields a sharp quadratic upper bound on the number of distinct clusters of tree-child and normal networks and a direct polynomial-time recognition algorithm. Finally, we explore several consequences of the NOC perspective beyond the phylogenetic setting. These include connections to the enumeration of normal networks, an order-theoretic interpretation of inclusion-visibility, structural properties of NOC set systems, and a tractable special case of Minimum Set Cover, which is NP-hard in general.

Finding large lattice subgraphs in sparse Erdős–Rényi random graphs is a classical problem at the interface of random graph theory and algorithms. General bounded-degree embedding and universality theorems give powerful results for broad graph families, but when specialized to square grids they operate at densities substantially larger than the grid-emergence scale. In this paper we exploit the specific geometry of the square grid. We introduce the Quarantined Local Search, a three-phase local algorithm that separates the construction of an initial boundary from the later corner-closure process and controls adaptive negative exposure through bounded pair-test histories. We prove that, for every fixed $\delta\in (0,1)$, there exists $C_\delta>0$ such that the algorithm embeds a $k\times k$ square grid with $k^2\le (1-\delta)n$ in $G(n,p)$ with high probability whenever $p\ge C_\delta\sqrt{\ln k/n}$. Thus, for $k^2=\Theta(n)$, a density of order $\sqrt{\log n/n}$ is sufficient, a factor of order $\sqrt{\log n}$ above the corresponding $n^{-1/2}$ emergence scale.

Johnson graphs $J(n,k)$ are a classical family of highly symmetric networks known to be Hamiltonian-connected in the fault-free setting. In this paper, we investigate their Hamiltonian connectivity under three failure models, namely general edge faults, matching faults, and vertex faults. For general edge faults, we prove that $J(n,k)$ remains Hamiltonian-connected after the deletion of any set of at most $k(n-k)-3$ edges for $n\geq4$. Since $J(n,k)$ is $k(n-k)$-regular, this attains the natural degree-based upper bound for Hamiltonian connectivity. We then consider matching faults, which exclude the concentration of multiple faulty links at a single vertex and permit substantially larger fault sets. We show that $J(n,k)$ remains Hamiltonian-connected after the deletion of an arbitrary matching for $n\geq5$, including a perfect matching whenever one exists. For vertex failures, we prove that $J(n,k)$ is $(n-2)$-vertex-fault-tolerant Hamiltonian-connected for $n\geq5$. All three results are constructive and lead to recursive fault-tolerant Hamiltonian routing algorithms. Simulation results on Johnson graphs with up to $12{,}870$ vertices further show that the routing algorithms successfully construct fault-free Hamiltonian paths for all tested source-destination pairs, with measured execution times exhibiting near-linear growth with network size. These results establish a unified fault-tolerant Hamiltonian-connectivity framework for Johnson graphs under different failure patterns.

We introduce and study the rotating-memory Fibonacci numbers, a periodic variable-order analogue of the Fibonacci sequence in which the number of preceding terms used in the recurrence changes cyclically with the index. Despite this varying memory, the resulting sequences exhibit a remarkably rigid structure. We derive closed forms, rational generating functions, arithmetic properties, and exact growth behavior, and show that the sequence decomposes naturally into geometric subsequences. We also develop combinatorial interpretations in terms of periodically constrained tilings, and restricted compositions, including bijective explanations for the multiplicative structure of the sequence. In addition, the first two nonclassical periods admit natural geometry-driven realizations: the period-2 sequence arises from monomer–dimer tilings of a triangular chain, while the period-3 sequence is related to tilings of a double hexagon strip by single and double hexagons. These connections provide geometric interpretations of the rotating recurrence in which the periodic behavior is induced by the underlying structures themselves, and suggest a broader problem of constructing analogous models for higher periods.

We determine a sharp threshold for the adaptable 2-colorability of a random graph equipped with a uniformly random, not necessarily proper, red/blue coloring of the edges. To accomplish this, we characterize a family of subgraphs along with edge colorings whose inclusion or exclusion determines adaptable $2$-colorability. We further show that above the threshold, a long path with alternating edge colors is formed. We use this path to prove the existence of such a subgraph in the supercritical regime. We then provide and prove symmetric bounds on the critical window for $2$-adaptable colorability. Particularly, we prove bounds matching that of the critical windows for the giant component in the Erd$\H{o}$s-R$\'e$nyi random graph model as well as the satisfiability of a random $2$-SAT instance. Finally, we show that below the critical window, the solution space of adaptable $2$-colorings remains connected, that is one can travel from one adaptable $2$-coloring to another by a sequence of $2$-colorings which differ on $O(\log{n})$ many vertices.

We consider the two-machine open-shop scheduling problem in which one machine is unavailable during a fixed interval. We study the resumable setting: an operation interrupted by the unavailability interval may resume, without penalty, when the machine becomes available. The objective is to minimize the makespan. Although the problem is NP-hard and several approximation algorithms are known, whether it admits a fully polynomial-time approximation scheme (FPTAS) has remained open for two decades. We resolve this question affirmatively by giving the first FPTAS, thereby strengthening the previously known polynomial-time approximation scheme (PTAS). As an intermediate result, we develop a new pseudo-polynomial dynamic program with seven state dimensions, improving on the ten-dimensional formulation in the literature.

We show that there is a fixed planar graph $H$, namely the $5 \times 5$ grid, such that Max Independent Set remains NP-hard in $H$-induced-minor-free graphs. This refutes the Dallard–Milanič–Štorgel conjecture and a weakening of it by Gartland and Lokshtanov, and by Korhonen.

We design an adaptive two-sided error testing algorithm for the binary rank of a $0,1$ matrix $M$ with query complexity $O(d^3\log(d+1)/\epsilon^2)$, where $d$ is the tested binary rank bound and $\epsilon$ is the distance parameter. This answers an open question posed by Parnas, Ron and Shraibman~\cite{parnas2021property}, who asked if the binary rank can be tested with query complexity polynomial in $d$ and $1/\epsilon$. Furthermore, our testing algorithm can be used to find an approximate binary decomposition of $M$ with an additional $d(n+m)$ queries. That is, under the promise that the binary rank of $M$ is at most $d$, we show how to find, with probability at least $5/6$, two $0,1$ matrices $A',B'$ such that $M' = A' \cdot B'$ is a $0,1$ matrix which differs from $M$ on at most an $O(\epsilon)$ fraction of its entries. Our results also imply a testing algorithm with polynomial query complexity for the equivalent problem of testing if the edges of a bipartite graph can be partitioned into at most $d$ bicliques.