Papers

Edge deletion problems ask for the fewest edges whose deletion gives a graph with a specified property. We study two such problems on block graphs. First making the graph claw-free, and second making it a line graph. Since every line graph is claw-free, the second problem is at least as difficult as the first. We show that on block graphs the two optimum values are always equal, and both are attained by the line graph of a forest. The main step is a completion argument, in which starting from any claw-free solution, we put back edges to get some nice structure inside each block still maintaining the claw-free property. This reduces both edge-deletion problems to the same problem of selecting vertices within the blocks of the input graph. We solve it by dynamic programming in $O(n+m)$ time, which also produces an optimal deletion set and a forest whose line graph is what remains after the completion argument. We also extend the method to $K_{1,r+1}$-free edge deletion for every fixed $r$.

We study Bounded-Density Vertex Deletion (BDVD), where one asks whether at most \(k\) vertices can be deleted so that every remaining subgraph has density at most a rational input threshold \(\tau_\rho\). We address the twin-cover and distance-to-cluster cases left open by Raupach, Breitkopf, Herrmann, and Nichterlein (MFCS 2026). We prove fixed-parameter tractability parameterized solely by the twin-cover number, for unrestricted input density. For distance to cluster, we establish fixed-parameter tractability parameterized by \(\operatorname{dc}(G)+\lfloor2\tau_\rho+1\rfloor\), and complement this with \(\mathsf{W[1]}\)-hardness parameterized by \(\operatorname{dc}(G)\) alone when the target density is part of the input, even for integral targets. Thus distance to cluster alone is fixed-parameter tractable throughout \(\tau_\rho<1\), while it is insufficient for unrestricted input density. Finally, we settle the combined parameter treewidth plus solution size, an open problem posed by Bazgan, Nichterlein, and Vazquez Alferez (JCSS 2025). We prove that BDVD is \(\mathsf{W[1]}\)-hard parameterized by \(\operatorname{td}(G)+k\), and hence also by \(\operatorname{tw}(G)+k\), already for targets strictly between \(1\) and \(2\).

Given $n$ distributions (hypotheses) $\mathcal H=\{H_1,\ldots,H_n\}$ and samples from an unknown target distribution $P$, proper hypothesis selection seeks a distribution whose total variation distance to $P$ is close to the best achievable within the known distributions. We give an algorithm that, using $O(\varepsilon^{-2}\log(n/\delta))$ target samples, running in $O(n\varepsilon^{-2}\log(n/\delta))$ time, with probability at least $1-\delta$, returns $H_{\hat i}$ satisfying $d_\mathrm{TV}(P,H_{\hat i})\leq 3\min_{i\in[n]}d_\mathrm{TV}(P,H_i)+\varepsilon.$ The algorithm achieves the optimal approximation factor and sample complexity. The running time improves the dependence on the failure probability from polynomial to logarithmic in the state-of-the-art bound $\tilde{O}(n\varepsilon^{-2}\delta^{-1})$ of Aamand, Aliakbarpour, Chen, and Silwal (COLT 2026), thereby answering their Question~2.

In this note, we provide a deterministic algorithm that computes the $(\min, +)$-Matrix Product $C = A * B$ of two $N \times N$ matrices $A$ and $B$ with polynomially bounded integer entries in $\tilde{O}(N^{2.9964})$ time, improving upon the exponent of $O(N^{2.9995})$ time obtained in the recent breakthrough by Anthropic (Alman and Vassilevska Williams, 2026). Our algorithm uses a phased approach first proposed by Bringmann, Grandoni, Saha, and Vassilevska Williams in 2016. The phased approach produces an approximation $\tilde{C}$ of $C$ in truly subcubic time such that the number of remaining triples $(i,k,j): A_{i,k}+B_{k,j} < \tilde{C}(i,j)$ is truly subcubic. This provided strong evidence that a truly subcubic $(\min, +)$-Matrix Product and therefore, a truly subcubic algorithm for All Pairs Shortest Paths (APSP) may exist. However, to find the remaining triples in subcubic time, a sparse certificate is needed whose construction was not known. We use the thin matrix product theorem of the recent breakthrough to provide such a certificate. This gives a simple modular subcubic algorithm for APSP.

Clustering is a basic tool for grouping data into meaningful classes, but in many applications the resulting clusters must also respect the fairness conditions among different groups. We study this direction for the group-fair Euclidean $k$-center problem in fixed dimension, where the input points are partitioned into disjoint color classes, and every cluster must satisfy given lower and upper bounds on the fraction of each color. We give a separator-based recursive approximation algorithm for the continuous-center setting. The main idea is to first reduce the continuous search space to a finite candidate center set and then use the geometric sphere separators to break the instance into smaller subproblems. Our algorithm gives a bicriteria approximation: the clustering radius is within a factor of $(1+\epsilon)$ of the optimum exactly fair solution, while the fairness bounds are relaxed only by a multiplicative factor. For fixed dimension, number of colors, and approximation parameters, the running time is polynomial in input size and subexponential in $k$. A recent IJCAI-ECAI2026 result gives a $(1+\epsilon)$-approximation for group-fair $k$-center in bounded doubling metrics with running time $(k/\epsilon)^{\mathcal{O}(k)}n^{\mathcal{O}(1)}$. In the fixed-dimensional Euclidean setting considered here, we improve the dependence on $k$ to subexponential, while allowing continuous centers. The fairness guarantees of the two results are different: the previous work allows an additive fairness violation, whereas our algorithm uses a controlled multiplicative relaxation.

We study multi-robot motion planning in a polygonal environment with polygonal robots. The setting of the problem consists of a polygonal domain $\mathcal{W}$ with $n$ vertices and $k$ robots $R_1,\dots,R_k$ translating in $\mathcal{W}$ each of which is represented by a polygon of constant complexity. Given a source configuration $\mathbf{s}$ and a target configuration $\mathbf{t}$ of the robots in $\mathcal{W}$, the goal of the problem is to find a collision-free motion plan from $\mathbf{s}$ to $\mathbf{t}$ that is shortest with respect to some cost function. We present the first fully polynomial-time approximation schemes (FPTAS) for motion planning with any fixed number $k$ of robots, under the min-sum objective, the min-max objective, and the min-makespan objective. Here, the min-sum variant aims to minimize the total traveling distance of all robots, the min-max variant aims to minimize the maximum traveling distance of a single robot, and the min-makespan variant aims to minimize the total time cost under the assumption that all robots move with a maximum speed $1$. The algorithm directly generalizes to the problem in any fixed dimension $d$, with a polytope environment and polytope robots. Our result settles an open question of Agarwal, Halperin, Sharir, Steiger [SODA'24] and Agarwal, Holmgren, Steiger [SODA'26], who gave an FPTAS for $k=2$ unit-square robots and a bicriteria-approximation algorithm for a fixed number $k$ of unit-square or unit-disk robots, respectively. Compared to the previous results, our algorithm is substantially simpler.

We give a randomized polynomial-time $0.48$-approximation for maximizing a nonnegative non-monotone $k$-submodular function under a matroid constraint, for every $k\ge2$. The algorithm combines a top-two label rule with a nested-chain potential and maximum-weight matroid bases. Its analysis reduces the approximation guarantee to a one-dimensional time allocation: a uniform local lower bound leads to a linear differential equation, concavity closes the remaining interval, and two elementary budgets complete the proof. A strict saving in the time budget absorbs sampling and discretization errors, yielding the exact $0.48$ guarantee in the value-oracle model.

Network coding allows intermediate nodes to combine messages, whereas routing only forwards them. The Li-Li conjecture asserts that network coding and fractional routing have the same capacity region in undirected multiple-unicast networks. We give a counterexample to this conjecture and apply it to external-memory integer sorting. Since Aggarwal and Vitter posed the question in 1988, whether integer sorting can beat the external-memory comparison bound has remained a central and extensively studied open problem. We give a deterministic sorting algorithm: for $N$ integers of $\Theta(\log N)$ bits, a memory holding $M$ integers, and block transfers of $B$ integers, where $2B\le M\le N$, the number of block transfers is $O\left(\frac{N}{B}\left[1+\frac{\log N}{(\log(M/B))^{1-\varepsilon}(\log M)^\varepsilon}\right]\right)$, where $\varepsilon>0$ is an absolute constant. In a range of parameter regimes, this bound is asymptotically smaller than the classical comparison-sorting bound. We also use Lean 4 to verify the sorting algorithm's correctness and complexity bounds, and the strict advantage of coding over routing.

Computing the top-$k$ simple shortest paths between two vertices of a graph, i.e., solving the P-KSSP problem, is a fundamental primitive in graph analytics. In modern scenarios, however, queries are rarely one-shot: parameters change as applications adapt their information needs, and recomputing a solution from scratch at every request quickly becomes a bottleneck. We study P-KSSP in a dynamic setting, focusing on changes of the parameter $k$: given a top-$k$ solution for a pair of vertices and a new value $k'$, the goal is to compute a top-$k'$ solution reusing the previously computed paths. Our approach rests on the pseudo-tree, a compact representation of a collection of simple $s$–$t$ paths: we show that building it from an arbitrary, unstructured solution amounts to trie construction, and give two constructions running in time linear in the size of the collection. On top of it, we design UPD, a dynamic algorithm that updates a solution upon a change of $k$; we prove its correctness, analyze its complexity, and show that it is a general, solver-agnostic scheme, which starts from the paths alone, is parametric in the routine that generates candidate paths and combinable with known preprocessing techniques, as well as with an online pruning rule that we formalize. While a worst-case update is asymptotically comparable to recomputing from scratch, an extensive experimental evaluation shows that a variant of UPD is the fastest method in the large majority of tested configurations, often by a wide margin (with speedups over recomputation that grow with $k$ and $k'$ and exceed $60\times$), yielding a practical recipe for the configuration to deploy.

We resolve the problem of unweighted online stochastic matching with a known arrival law by establishing its optimal competitive ratio. For Poisson arrivals, we achieve $1-e/e^e\approx0.8206$, matching the upper bound of Chierichetti, Giacchini, Panconesi, and Vattani [ICALP 2025]. For exactly $n$ i.i.d.~arrivals, we obtain the same guarantee up to an additive $O(n^{-1/2})$ loss. Our key technical contribution is an exact correspondence between Poisson matching and exponential scheduling, preserving both online performance and the expected offline optimum. To prove the latter, we show that the distribution of the maximum of a random supermodular set function is determined by its joint distributions along chains of nested sets, a result of independent interest. This reduces the analysis to chains, where classical Poisson–exponential duality applies. The offline equivalence extends to arbitrary nonnegative edge weights. We then design a posterior allocation policy for exponential scheduling and use the correspondence to transfer its optimal competitive guarantee to online matching. Additionally, we show that the optimal competitive ratio under an unknown arrival law is strictly lower than $1-e/e^e$, establishing a strict separation between the known and unknown arrival models.

First defined by Bouchitté and Todinca, Potential Maximal Cliques (PMC) are a well studied notion in graph theory due to its numerous applications. The PMCs of an undirected graph $G$ are defined as the maximal cliques of the minimal triangulations of $G$. Their number can be exponential in the size of the graph. PMCs were crucial in the design of a polynomial-time algorithm which solves Maximum Weight Independent Set on $P_6$-free graphs (Grzesik, Klimošová, Pilipczuk, Pilipczuk, 2020). They also play a central role in the proposal of a polynomial-time algorithm to solve Largest Induced Subgraph of treewidth $\le t$ (Fomin, Todinca, Villanger, 2013). Last but not least, certainly one of the main application of PMCs is the existence of an exact dynamic-programming algorithm to determine the treewidth of a graph (Fomin, Kratsch, Todinca, Villanger, 2008) which uses its list of PMCs as an input. In fact, the enumeration of PMCs stands as the bottleneck for the computation of treewidth with this method: the PMC-enumeration step takes output-quadratic time (Bouchitté, Todinca, 2002) while the dynamic-programming one takes a time linear in the number of PMCs. This two-step procedure was proven to be among the most performing approaches in practice to determine treewidth. Answering a longstanding open question, we propose an algorithm to enumerate PMCs in output-linear time, i.e. a running time linear in the number of PMCs. This offers nice perspectives not only from the theoretical viewpoint - better running time for the PMC-based methods determining treewidth, for Minimum Fill-In and Largest Induced Subgraph of treewidth $\le t$ - but also from the practical side, with the possibility to propose top-performance implementation.

We study proportional cake cutting under an equal-length constraint, where each of $n$ agents must receive a bundle of length exactly $1/n$ that is worth at least $1/n$ to that agent. Our main result determines the exact worst-case number of cuts needed to guarantee such an allocation, namely $2n-2$ for every $n$. This improves on the previously known quadratic upper bound. The proof combines the constrained necklace-splitting theorem with a recursive assignment argument based on Hall's theorem. The same bound extends to the stronger requirement of envy-freeness when $n$ is a prime power, and an $O(n^{1.525})$ upper bound holds for arbitrary $n$. We then ask whether such allocations can be computed. We first show that no deterministic protocol in the Robertson–Webb model terminates after finitely many queries on every instance and always returns an exact equal-length proportional allocation, even for two agents with identical valuations. For every $\varepsilon>0$, however, $O(n^2/\varepsilon)$ queries suffice to produce an allocation in which every agent receives a bundle of length exactly $1/n$ and value at least $1/n-\varepsilon$, again using at most $2n-2$ cuts. Finally, we augment the Robertson–Webb model with two-knife queries. We show that recursive balanced halving attains the optimal bound of $2n-2$ cuts when $n$ is a power of two. For arbitrary $n$, we give a procedure using $O(n\log n)$ cuts.

We give a dynamic graph algorithm which maintains a 3-coloring in bipartite graphs under edge insertions such that the coloring only changes within a radius of $O(\log n)$ around an inserted edge. In the context of locality theory in distributed graph algorithms, this gives an upper bound on the locality under the dynamic-LOCAL model. Our bound on the locality is tight, as the problem admits a matching $\Omega(\log n)$ lower bound in the stronger online-LOCAL model, by a result of Chang et al. [PODC 2024]. Thus, our result closes the gap between dynamic and online locality of this problem.

We study non-adaptive bit-probe schemes for static set membership. Each query reads a fixed number of bits and answers by strict majority. We revisit the random bipartite-graph construction of Garg and Radhakrishnan [STACS, 2017]. The main observation is that a slightly stronger expansion property already bounds the number of exceptional elements. This makes a separate probabilistic estimate for exceptional elements unnecessary. A detailed union-bound calculation then gives, for every fixed $t\ge5$, a logarithm-free existence bound of $O_t(m^{1/c}n^{1-1/c})$, where $c=\lfloor(t-1)/2\rfloor$. For odd $t$, this removes the logarithmic factor from the general upper bound of Garg and Radhakrishnan [STACS, 2017].

The Emergency Vertex Cover problem (Em-VC) assigns budgets to stations at graph vertices to cover every edge while minimizing the total budget. The cost of covering an edge from a station is the shortest-path distance from the station to the nearer endpoint plus the edge length. We study how restricting the number of stations affects the cost of ordinary and fault-tolerant coverage. In ordinary Em-VC, we prove that the optimal solution using at most two stations costs at most $5/4$ times the unrestricted optimum, improving the known $4/3$ guarantee for a single station. A recurrence extends this bound to larger station limits, giving approximation factors that converge to $\sqrt{5}-1$. Budget-threshold enumeration computes the restricted optimum in polynomial time for every fixed station limit. Under $k$-fold coverage, every edge must be covered by at least $k$ distinct stations, ensuring coverage after any $k-1$ station failures. We prove that selecting the $k$ vertices with the smallest full-coverage costs yields a $2$-approximation relative to the unrestricted optimum for $k$-fold coverage. When $k=2$, we combine upper bounds derived from the structure of the covering balls to improve the approximation guarantee to $5/3$.

We study learning-augmented streaming algorithms for estimating the value of Boolean Maximum Constraint Satisfaction Problems (Max-CSPs). Specifically, we consider streaming algorithms equipped with an $\varepsilon$-accurate oracle: for each variable, the oracle outputs a label in $\{-1,1\}$ that agrees with a fixed optimal assignment with probability $1/2+\varepsilon$, and disagrees otherwise. Prior work by Dong, Peng, and Vakilian [ITCS 2025] showed that, for the special case of Max-Cut, such an oracle enables a $(1/2+\Omega(\varepsilon^2))$-approximation using $\operatorname{poly}(1/\varepsilon)$ words of space in insertion-only streams, and $\operatorname{poly}(1/\varepsilon,\log n)$ words of space in dynamic streams. We substantially generalize and strengthen their result, provided that a highly accurate estimate of $\varepsilon$ is available. For any Boolean Max-$2$CSP and any $\eta>0$ independent of $\varepsilon$, we obtain, with high constant probability, a single-pass $(1-\eta)$-approximation using $\operatorname{poly}(1/\varepsilon,1/\eta)$ words of space in insertion-only streams, and $\operatorname{poly}(1/\varepsilon,1/\eta,\log n)$ words of space in dynamic streams, where $n$ is the number of variables. We further extend our techniques to general Boolean Max-$k$CSPs for $k\ge 3$, with slightly worse space complexity.

In this note we explore the following simple question: How does the classic gambler's ruin phenomenon change if we allow the gambler financial leverage? We show that, with a moderate amount of leverage, it is possible for an (idealized) gambler to sustain seemingly improbable profits over a remarkably long time horizon without ever going bankrupt, even if the gambler makes only fair bets. With a leverage ratio of $L = 10$ (less than what Lehman Brothers Holdings had from 1995–2008), the probability of multiplying one's money by $1000\times$ without going bankrupt is roughly $50\%$. More generally, given $L$-fold leverage, and given the goal of multiplying one's money by a factor of $r > 1$, the optimal gambling strategy has probability roughly $r^{-1/L}$ of success. The results in this note are designed as part of the course development for the CMU randomized algorithms course.

Tokenization is an essential preprocessing step in almost all large language models (LLMs). One of the most popular frameworks is BPE tokenization, which consists of special token segmentation, pretokenization, and byte-pair encoding (BPE). However, existing algorithms are sequential, with each stage only processing complete output from the previous stage. In order to reduce latency and memory footprints, recent works (Jiang and Gong, ICML 2026) (Li, Yang and Mamouras, ASPLOS 2026) (Mamouras, Li and Yang, PLDI 2026) have initiated the study of online algorithms for subroutines such as pretokenization and BPE with a goal of streaming input to output as the input arrives online. However, the search for efficient online tokenization remains open. In this work, we formalize and resolve online tokenization. We introduce the notion of delay for online tokenization which characterizes the extent to which the algorithm output lags behind the (possibly computationally intractable) optimum. We give a near-linear time algorithm with optimal delay (up to constant factors). We complement this algorithm with a conditional lower bound: unless $\mathbf{P} = \mathbf{NP}$, there is no polynomial time algorithm that beats the delay of our algorithm by more than a constant factor. We implement a practical variant for tokenizers from OpenAI's tiktoken and HuggingFace's tokenizers. We obtain a $7.9\times$-$34.8\times$ latency advantage (measured as time to first token) over tiktoken (as well as the recent gigatoken tokenizer) and our implementation maintains approximately constant memory usage independent of input size. Our implementation also achieves a $2.4\times$-$7.0\times$ improvement in throughput over tiktoken and $15.8\times$-$38.1\times$ over tokenizers.

A $u\hbox{-}v$ separator in $G$ is a set $S\subseteq V(G)\setminus\{u,v\}$ such that there is no $u\hbox{-}v$ path in $G\setminus S$. A minimal separator in $G$ is a set $S$ such that for some $u,v\in V(G)\setminus S$, $S$ is a $u\hbox{-}v$ separator in $G$ but no proper subset of $S$ is. A set $S$ is bisimplicial if it is the union of at most two cliques. Recognizing graphs in which every minimal separator is the union of at most $k$ cliques is polynomial-time solvable for $k=1$ and NP-hard for every fixed $k\geq 3$. We settle the remaining case $k=2$ by giving a polynomial-time algorithm to decide whether every minimal separator in a graph is bisimplicial. A set $S$ is triad-free if its independence number $\alpha(S)$ is at most two. Given a graph $G$ and two vertices $u,v$, the problem of finding a $u\hbox{-}v$ separator $S$ with $\alpha(S)\leq k$ had the same complexity gap: it was known to be polynomial-time solvable for $k=1$ and NP-hard for every fixed $k\geq 3$. We resolve the remaining case $k=2$ by giving a polynomial-time algorithm that finds a triad-free $u\hbox{-}v$ separator or determines that none exists.

In submodular facility location (SFL), introduced by Svitkina and Tardos (SODA 2006), clients $\cal C$ in a metric space are assigned to facilities $\cal F$. A facility serving a set $S \subseteq \cal C$ of clients incurs cost $g(S)$, where $g$ is a monotone submodular function common to all facilities. The best known approximation factor for SFL is $O(\log\log|\cal C|)$, due to Abbasi, Adamczyk, Bosch-Calvo, Byrka, Grandoni, Sornat, and Tinguely (ICALP 2024), improving on the $O(\log|\cal C|)$-approximation of Svitkina and Tardos. We show that this is best possible up to a constant factor in the value-oracle model by showing that there exists no algorithm with only polynomially many queries to $g$ with approximation ratio $o(\log\log|\cal C|)$. This bound holds for algorithms with unlimited computation time, and when $g$ is restricted to matroid rank functions. We also show that the natural linear programming relaxation has an integrality gap of $\Theta(\log\log|\cal C|)$. Our lower bound follows by constructing an extension of Vondrák's symmetry-gap approach (FOCS 2009) for matroid rank functions.

We study the randomised and deterministic demand-query complexity of approximate bipartite matching with few query rounds. The demand-query model, introduced in [Blumrosen and Nisan SICOMP 2009; Nisan SODA 2021], is inspired by combinatorial auctions and welfare maximisation, where an oracle returns the first neighbouring vertex (item) of a vertex u (agent) according to a given permutation pi (preference ordering of the agent over the items). For randomised algorithms, our main result gives a trade-off between the number of rounds k, the per-round query budget q=n^{1+\delta}, and the achievable approximation ratio. Let 1 <= k <= log n, and -1 <= delta <= 1/k. Every k-round randomised algorithm using O(q) demand queries per round has a worst-case expected-size approximation factor of Omega(n^{1/2k-delta/2}/k). Conversely, we give a randomised algorithm achieving approximation factor O(n^{(1-k*delta)/(k+1)}/k) in expectation. For deterministic algorithms, for k*q >= n, we show that the optimal deterministic approximation factor is ~Theta(n^2/(q*k^2)), thereby improving the previous lower bound of [Nisan SODA 2021] by a polynomial factor over a wide range of parameters. In particular, for q=Theta(n), we improve the deterministic round-complexity lower bound for constant-factor approximation from ~Omega(n^{1/3}) to ~Omega(sqrt n). Our algorithms produce matching and vertex-cover certificates; the deterministic trade-off also extends to minimum vertex cover and maximum fractional matching. We extend the randomised upper bounds to the weaker Demand-OR model [Nisan 2021]. We prove a non-adaptive simulation of uniform random-neighbour queries using known techniques, avoiding the O(log n)-round penalty of the trivial binary-search simulation. Consequently, our upper bounds yield the same k-round approximation guarantees using O(q * log^2 n) Demand-OR queries per round.

In this paper, we study online paging with delay or deadlines. In online paging with delay, an online algorithm manages a cache of size $k$ and needs to serve incoming requests by adding the requested pages to the cache. In addition, it has the option to delay serving incoming requests at a cost that depends on the delay. We propose a new online algorithm, which combines a threshold rule with first-in, first-out (FIFO) replacement, and show that it is $(2k+2)$-competitive for cache size $k$. This result corrects a previously published claim that the competitive ratio is exactly $2k+1$ regardless of the number of holes (pages outside the cache). However, in the case of one hole, we show that the FIFO rule with a different threshold is indeed $(2k+1)$-competitive, giving a tight result. In the case of deadlines, the same analysis gives a deadline-triggered algorithm that is $(k+1)$-competitive in general and $k$-competitive if there is only one hole. Moreover, we show that the second hole makes paging with deadlines strictly harder than classical paging, where the competitive ratio is independent of the number of holes. For paging with deadlines, we prove a deterministic lower bound of $k+\tfrac12$, already for two holes. We show that, as a consequence, no threshold-triggered algorithm can achieve a competitive ratio less than $2k+\frac32$ in general.

Let $V_i(n)$ denote the minimum worst-case number of comparisons needed to find the $i$-th smallest of $n$ distinct elements. We construct algorithms showing $V_7(16)\le28$ and $V_8(16)\le29$, improving the previously reported upper bounds of 30 and 31. Together with known lower bounds, these results establish $V_7(16)=28$ and leave $V_8(16)\in\{28,29\}$. For 17 elements, we establish $V_5(17)=27$ and improve the upper bounds for ranks 6 through 9 to 29, 30, 31, and 31, respectively. Every new upper bound has an explicit decision-graph certificate verified by a static checker developed independently of the search. We also derive a lower bound for selection from a partially ordered state by combining the Fussenegger-Gabow weight with its order-reversed counterpart. All new upper bounds and this lower bound are formally verified in Lean 4. For $V_8(16)$, we show that a lower bound based solely on the total number of decision-tree leaves cannot establish $V_8(16)\ge29$. The algorithms were found using fixed-budget search, information-guided comparison ordering, and proof-number search. Proof-number search was decisive in finding the 28-comparison algorithm; in our experiments, refuting a comparison budget remained substantially more expensive than finding an algorithm within it.

We present an optimal tester for support size that determines whether an unknown probability distribution is supported on $n$ elements or $\varepsilon$-far away (in total variation distance) with sample complexity $\Theta(\frac{n}{\varepsilon\log(n)})$, improving the prior best $O(\frac{n}{\varepsilon\log(n)} \log(\min\{\frac{1}{\varepsilon}, n\}))$ due to Ferreira Pinto Jr. and Harms (STOC 2025). As a generalization, we provide a pretty good lower bound on support size.

For Vertex Cover, a minimal blocking set of a graph $G$ is an inclusion-wise minimal vertex set contained in no minimum vertex cover of $G$. Bougeret et al. [SIDMA 2022] showed that, for all minor-closed graph classes $\mathcal{G}$, Vertex Cover has a polynomial kernel parameterized by the distance to $\mathcal{G}$ if and only if $\mathcal{G}$ has bounded minimal blocking sets. In this work, we explore how far this connection between minimal blocking sets and kernelization extends to $\mathcal{F}$-Minor Deletion parameterized by the deletion distance to $\mathcal{G}$. A minimal $\mathcal{F}$-blocking set of $G$ is a minimal vertex set not contained in any optimal $\mathcal{F}$-Minor Deletion solution of $G$. For connected families $\mathcal{F}$, we show that if some graph class $\mathcal{G}$ closed under disjoint union has unbounded minimal $\mathcal{F}$-blocking sets, then $\mathcal{F}$-Minor Deletion parameterized by the deletion distance to $\mathcal{G}$ has no polynomial kernel. This generalizes a result of Hols et al. [SIDMA 2022] for Vertex Cover. We then show that this criterion does not capture all kernelization lower bounds for $\mathcal{F}$-Minor Deletion, even for minor-closed graph classes. Motivated by the work of Dekker and Jansen [Discrete Appl. Math. 2024], we study the more general notion of extended $\mathcal{F}$-blocking sets and the kernelization lower bounds they yield. For every fixed integer $\eta\geq 1$, Pw-$\eta$ Deletion is the problem of deleting at most $k$ vertices to reduce the pathwidth to at most $\eta$. We show that it has no polynomial kernel parameterized by the distance to graphs with pathwidth at most $\eta+1$. Moreover, for connected families $\mathcal{F}$, we show that $\mathcal{F}$-Minor Deletion parameterized by the distance to graphs with pathwidth at most $\max_{F\in\mathcal{F}} pw(F)+1$ has no polynomial kernel.

The Held–Karp bound is a linear-programming lower bound for the metric traveling salesperson problem (TSP). We approximate this bound in the multi-pass insertion-only streaming model for metrics induced by connected positive-weight graphs. For an $n$-vertex, $m$-edge graph and $0<\eps\le1/2$, the algorithm returns a $(1+\eps)$ upper approximation with high probability, using $\wtO(\eps^{-4})$ passes, $\wtO(n\eps^{-2})$ words, and $\wtO(m\eps^{-4})$ arithmetic operations per pass. We replace the changing edge-weight vector by a short history of multiplicative-weights (MWU) updates. After initialization, updates repeat until a column is removed. We store each repeated update once, with its repetition count. The record length depends on the number of columns, not updates; the argument applies to any fixed nonnegative matrix. For Held–Karp, tree-path queries recover edge weights from these records. Separate search and update thresholds allow fresh sparsifiers at successive search stages. Controlled rounding gives a bounded-precision implementation. The estimate yields a $(3/2+\eps)$ graph-metric TSP value estimate and a $(7/5+\eps)$ estimate when the input graph is unweighted. Conversely, in weighted graph streams, every one-pass $(1+\eta)$ upper approximation to Held–Karp, for fixed $0\le\eta<1$, requires quadratic space. The algorithms estimate values; they do not construct tours within these bounds.

We give improved sublinear-time and semi-streaming approximation algorithms for maximum path cover, $(1,2)$-TSP, and Max-TSP. For each fixed $\eps>0$, our distance-query algorithm estimates the optimum cost of an $n$-vertex $(1,2)$-TSP instance from above within factor $2-\cstar+\eps=1.413184\ldots+\eps$, with high probability, in $n^{2-\Omega_\eps(1)}$ time, where $\cstar=2(\sec1-\tan1)$. This improves the $3/2+\eps$ factor of direct matching estimation in this subquadratic regime~\cite{BhattacharyaKissSaranurak2023} and of the faster near-linear estimator~\cite{BehnezhadEtAl2024}. A single random vertex order defines an implicit bipartite graph whose matchings project to path covers. An alternating-permutation analysis shows that its expected maximum matching size is at least $\cstar$ times the optimum path-cover size. On long paths, even the best of polynomially many uniform orders retains only a $\cstar+o(1)$ fraction with high probability. In semi-streaming, we construct deterministic $(5/7-\eps)$-approximate unweighted path covers and $(9/7+\eps+5/(7n))$-approximate $(1,2)$-TSP tours, improving the respective $2/3-\eps$ and $4/3+\eps+1/n$ guarantees in~\cite{AlipourFarokhnejadMoemke2025}. For nonnegative weights, we obtain $2/3-\eps$ for path cover and Max-TSP, improving the earlier asymptotic $7/12-\eps$ Max-TSP bound. These algorithms use $\poly(1/\eps)$ passes independent of $n$ and $O(n\log(n/\eps)\poly(1/\eps))$ words. We compute two approximate matchings separated by edge deletion. A cycle-compensation lemma accounts for the weight removed when breaking their cycles. For unweighted graphs, a third approximate matching on a simple auxiliary graph with at most $2n$ vertices chooses component connections; a Tutte–Berge count proves the stronger bound.

We construct an approximate sampler for finite doubly free-boundary Macdonald rail-yard processes with Macdonald, boundary, and edge parameters in $(0,1)$. The construction allows arbitrary finite sequences of additions and removals of horizontal or vertical strips and all four boundary types. It reverses a finite commutation–reflection history from a partition sequence whose entries are equal, with the common partition weighted by both boundaries. We derive local reconstruction kernels from the skew Cauchy, Littlewood, and branching identities. Bulk and boundary updates combine integer increments with conditional shape sampling at a prescribed size; same-sign swaps use finite conditional resampling. For $K$ reflection periods and an initial partition-size cutoff $M$, the algorithm terminates almost surely and samples the resulting approximating law exactly. We prove total variation convergence to the target process as $K,M\to\infty$ and give explicit bounds for both truncation errors, decaying geometrically in the cutoffs for fixed input parameters. Under fixed parameter bounds, we also bound the expected operation count of direct enumeration in a unit-cost model. We further construct an exact rejection sampler for the initial partition law without a size cutoff or evaluation of its normalizing constant. Using this sampler removes the size-truncation error, leaving only the error from the finite reflection history.

We show that the guaranties of the Gram-Schmidt Walk Algorithm of Bansal, Dadush, Garg, and Lovett can be achieved by an Ising model. Given vectors $v_1, \ldots, v_n$ of norm less than 1, we give a closed form for a centrally symmetric distribution on a vector $\sigma \in \{\pm 1 \}^n$ so that the signed sum $\sum_j \sigma(j) v_j$ is $1$-subgaussian. We show that the Glauber dynamics for this model mix rapidly provided that the norms of the input vectors are bounded below 1 by an inverse-polynomial. The proofs employ the theory of Lorentzian polynomials, and we use these techniques to show fast mixing bounds for other spin systems.

The Boolean Satisfiability (SAT) problem is a remarkably expressive computational problem, serving both as a cornerstone of complexity theory and as a unifying language for a wide range of search and reasoning tasks. While $\mathrm{NP}$-complete in general, SAT is well known to be linear-time solvable on instances with bounded treewidth. This motivates the search for "hidden" tree-like structures in formulas—for instance by detecting so-called weak $t$-treewidth backdoors. Such backdoors are sets of $k$ variables that can be assigned in such a way that the remaining instance has treewidth at most $t$ and is satisfiable whenever the original instance is. Ever since Gaspers and Szeider [ICALP 2012] showed that finding a weak $t$-treewidth backdoor for a SAT formula is $\mathrm{W}[2]$-hard with respect to $k$, tractability results have been restricted to structurally sparse formulas—most notably, a linear-time algorithm for $d$-SAT (bounded clause-length SAT) by Fomin, Lokshtanov, Misra, Ramanujan, and Saurabh [SODA 2015] and a non-uniform polynomial-time algorithm when the incidence graph of the input formula belongs to a nowhere-dense graph class (or more generally is $K_{d,d}$-free) by Lokshtanov, Panolan, and Ramanujan [ICALP 2022]. In this paper, we push the tractability boundaries of finding weak $t$-treewidth backdoors beyond sparsity and into the regime of "well-behaved" dense formulas. Aiming towards relaxing structural input restrictions from nowhere-dense to edge-stable graph classes, as featured in the theory of beyond sparsity for FO model checking, or to bounded VC-dimension set systems, we identify signed semi-ladder-freeness as the correct dense structural restriction that establishes the boundary between tractability and intractability along this line.