arXiv subjects

Siyue Liu

Publications and source records attributed to Siyue Liu.

15 recordsLinked to original sources

arXiv2026

Asymptotically optimal packings of arithmetic progressions with prime differences

For a positive integer $n$, put $A_d=\{id:1\le i\le\lfloor n/d\rfloor\}$ for $1\le d\le n$ and $B_d=\{id:1\le i\le n\}$ for $d\in\mathbb{N}$. For $D\subseteq\{1,\ldots,n\}$, let $m_D(n)$ be the minimum length of an integer interval containing pairwise disjoint shifted copies of $A_d$ for all $d\in D$. For a finite set $E\subseteq\mathbb{N}$, define $M_E(n)$ analogously using $B_e$, $e\in E$. Let $\mathcal{P}(x)=\{p\le x:p\text{ is prime}\}$. We prove $m_{\mathcal{P}(\sqrt n)}(n)=\left(\frac43+o(1)\right)\frac{n^{3/2}}{\ln n}$ and $M_{\mathcal{P}(n)}(n)=\left(\frac16+o(1)\right)\frac{n^3}{\ln n}$ as $n\to\infty$. These asymptotic formulas attain the known lower bounds and settle two conjectures of Alon, D\k{e}bski, Grytczuk and Przyby\l{}o concerning prime differences. The proof combines a cyclic phase-selection principle with lattice covering estimates and a decomposition into regular blocks of primes.

math.NT↗

arXiv2026

Feedback edge set in bipartite digraph

Let $\beta(G)$ denote the minimum size of a feedback edge set of a digraph $G$, and let $\gamma(G)$ denote the number of unordered pairs of nonadjacent vertices. Motivated by the Chudnovsky--Seymour--Sullivan conjecture for $3$-free digraphs, we study the corresponding feedback-edge problem for bipartite digraphs. In the bipartite setting, $\gamma(G)$ is taken to count only nonadjacent pairs with ends in distinct partite sets. We prove that every $4$-free bipartite digraph $G$ satisfies $\beta(G)\le \gamma(G)/2$. We also determine the exact Tur\'an number of $2k$-free strong bipartite digraphs with partite sets $X$ and $Y$: if $|X|,|Y|\ge k+1$, then the maximum number of edges is $$(|X|-(k-1))(|Y|-(k-1))+2k-2.$$ Finally, for the extremal case $k=2$, we analyze the structure of $4$-free strong bipartite Tur\'an digraphs and prove the sharper bound $\beta(G)\le \gamma(G)/3$ for all such digraphs. This constant is attained by a natural balanced three-block construction.

math.CO↗

arXiv2026

Weighted Equitability and Matroid-Constrained Discrepancy

We prove weighted matroid equitability. Let $M=(E,\mathcal{I})$ be a matroid whose ground set can be partitioned into $k$ bases, and assign a nonnegative weight to every element. Then $E$ has a partition into $k$ bases such that the weights of any two bases differ by at most the largest element weight. We present two proofs based on the localized exchange theorem of Akrami, Liu, Raj, and V\'egh. The first is existential, while the second is constructive and leads to a strongly polynomial-time algorithm. As applications, we obtain an additive guarantee for matroid-constrained makespan minimization for identical machines and a strongly polynomial-time algorithm for finding an EF1 allocation under a matroid constraint and identical additive valuations. We further generalize the Beck--Fiala framework in discrepancy theory to settings with matroid constraints. Given a nonnegative matrix of column sparsity $\Delta$, we show that a fractional basis can be rounded to a basis of no larger cost while preserving every row sum within additive error $2\Delta$ times the largest matrix entry. Motivated by the $2$-sparse prefix Beck--Fiala conjecture, we formulate a conjecture on prefix-constrained matroid bases and prove a discrepancy bound $O(\log n)$. Finally, we give a counterexample to the weighted carpooling conjecture, thereby also disproving a conjecture by Morell and Skutella on single-source unsplittable flows with two-sided discrepancy bounds.

cs.DS↗

arXiv2026

Semi-Streaming Algorithms for Submodular Maximization under Random Arrival Order

We study random order semi-streaming algorithms for submodular maximization under a wide range of combinatorial constraint classes, including matroids, matroid $p$-parity, $p$-exchange systems and $p$-systems. For most of these classes of constraints, our results are the first improvement over what is known to be achievable for adversarial order. For matroids, matching and $p$-matchoids, previous random order results were known, and we improve over some of these as well. In the case of matroids, our improved results show a separation between adversarial and random order semi-streaming algorithms, and exponentially improve the number of passes necessary for getting $1 - 1/e - \varepsilon$ approximation for maximizing a monotone submodular function subject to a matroid constraint. We also prove a new hardness result showing a similar separation for $p$-systems. Our results are based on two new technical tools. One tool provides a general way to translate offline algorithms for many classes of constraints into random order semi-streaming algorithms. The other tool is a semi-streaming variant of a recently proposed offline algorithm for matroid constraints.

cs.DS↗

arXiv2026

Open-Domain Safety Policy Construction

Moderation layers are increasingly a core component of many products built on user- or model-generated content. However, drafting and maintaining domain-specific safety policies remains costly. We present Deep Policy Research (DPR), a minimal agentic system that drafts a full content moderation policy based on only human-written seed domain information. DPR uses a single web search tool and lightweight scaffolding to iteratively propose search queries, distill diverse web sources into policy rules, and organize rules into an indexed document. We evaluate DPR on (1) the OpenAI undesired content benchmark across five domains with two compact reader LLMs and (2) an in-house multimodal advertisement moderation benchmark. DPR consistently outperforms definition-only and in-context learning baselines, and in our end-to-end setting it is competitive with expert-written policy sections in several domains. Moreover, under the same seed specification and evaluation protocol, DPR outperforms a general-purpose deep research system, suggesting that a task-specific, structured research loop can be more effective than generic web research for policy drafting. We release our experiment code at https://github.com/xiaowu0162/deep-policy-research.

cs.CL↗

arXiv2026

Lattice Structure and Efficient Basis Construction for Strongly Connected Orientations

Let $\vec{G}=(V,E^+\cup E^-)$ be a bidirected graph whose underlying undirected graph $G=(V,E)$ is $2$-edge-connected. A strongly connected orientation (SCO) is defined as a subset of arcs that contains exactly one of $e^+,e^-$ for every $e\in E$ and induces a strongly connected subgraph of $\vec{G}$. Given a family $\mathcal{F}$ of proper subsets of $V$, we call an SCO tight if there is exactly one arc entering $U$ for every $U\in \mathcal{F}$. We give a polynomial-time algorithm to construct a set $\mathcal{B}$ consisting of tight SCO's which forms an integral basis for the linear hull of tight SCO's. That is, $\mathcal{B}$ is a linearly independent subset of tight SCO's, and every integer vector in the linear hull of tight SCO's can be written as an integral combination of $\mathcal{B}$. This extends the main result of Abdi, Conu\'ejols, Liu and Silina (IPCO 2025), who gave a non-constructive proof of the existence of such a basis in an equivalent setting. While the previous proof uses polyhedral theory, our proof is purely combinatorial and yields a polynomial-time algorithm. As an application of our algorithm, we show that parity-constrained tight strongly connected orientations can be solved in deterministic polynomial time. Along the way, we discover appealing connections to the theory of perfect matching lattices.

math.CO↗

arXiv2025

Weighted Chairman Assignment and Flow-Time Scheduling

Given positive integers $m, n$, a fractional assignment $x \in [0,1]^{m \times n}$ and weights $d \in \mathbb{R}^n_{>0}$, we show that there exists an assignment $y \in \{0,1\}^{m \times n}$ so that for every $i\in[m]$ and $t\in [n]$, \[ \Big|\sum_{j \in [t]} d_j (x_{ij} - y_{ij}) \Big| < \max_{j \in [n]} d_j. \] This generalizes a result of Tijdeman (1973) on the unweighted version, known as the chairman assignment problem. This also confirms a special case of the single-source unsplittable flow conjecture with arc-wise lower and upper bounds due to Morell and Skutella (IPCO 2020). As an application, we consider a scheduling problem where jobs have release times and machines have closing times, and a job can only be scheduled on a machine if it is released before the machine closes. We give a $3$-approximation algorithm for maximum flow-time minimization.

cs.DS↗

arXiv2025

Matroids are Equitable

We show that if the ground set of a matroid can be partitioned into $k\ge 2$ bases, then for any given subset $S$ of the ground set, there is a partition into $k$ bases such that the sizes of the intersections of the bases with $S$ may differ by at most one. This settles the matroid equitability conjecture by Fekete and Szab\'o (Electron. J. Comb. 2011) in the affirmative. We also investigate equitable splittings of two disjoint sets $S_1$ and $S_2$, and show that there is a partition into $k$ bases such that the sizes of the intersections with $S_1$ may differ by at most one and the sizes of the intersections with $S_2$ may differ by at most two; this is the best one can hope for arbitrary matroids. We also derive applications of this result into matroid constrained fair division problems. We show that there exists a matroid-constrained fair division that is envy-free up to one item if the valuations are identical and tri-valued additive. We also show that for bi-valued additive valuations, there exists a matroid-constrained allocation that provides everyone their maximin share.

math.CO↗

arXiv2025

Approximately Packing Dijoins via Nowhere-Zero Flows

In a digraph, a dicut is a cut where all the arcs cross in one direction. A dijoin is a subset of arcs that intersects each dicut. Woodall conjectured in 1976 that in every digraph, the minimum size of a dicut equals to the maximum number of disjoint dijoins. However, prior to our work, it was not even known whether at least $3$ disjoint dijoins exist in an arbitrary digraph whose minimum dicut size is sufficiently large. By building connections with nowhere-zero (circular) $k$-flows, we prove that every digraph with minimum dicut size $τ$ contains $\left\lfloor\fracτ{k}\right\rfloor$ disjoint dijoins if the underlying undirected graph admits a nowhere-zero (circular) $k$-flow. The existence of nowhere-zero $6$-flows in $2$-edge-connected graphs (Seymour 1981) directly leads to the existence of $\left\lfloor\fracτ{6}\right\rfloor$ disjoint dijoins in a digraph with minimum dicut size $τ$, which can be found in polynomial time as well. The existence of nowhere-zero circular $\frac{2p+1}{p}$-flows in $6p$-edge-connected graphs (Lovász et al. 2013) directly leads to the existence of $\left\lfloor\frac{τp}{2p+1}\right\rfloor$ disjoint dijoins in a digraph with minimum dicut size $τ$ whose underlying undirected graph is $6p$-edge-connected. We also discuss reformulations of Woodall's conjecture into packing strongly connected orientations.

math.CO↗

arXiv2025

Minimum Cost Nowhere-zero Flows and Cut-balanced Orientations

Flows and colorings are disparate concepts in graph algorithms -- the former is tractable while the latter is intractable. Tutte introduced the concept of nowhere-zero flows to unify these two concepts. Jaeger showed that nowhere-zero flows are equivalent to cut-balanced orientations. Motivated by connections between nowhere-zero flows, cut-balanced orientations, Nash-Williams' well-balanced orientations, and postman problems, we study optimization versions of nowhere-zero flows and cut-balanced orientations. Given a bidirected graph with asymmetric costs on two orientations of each edge, we study the min cost nowhere-zero $k$-flow problem and min cost $k$-cut-balanced orientation problem. We show that both problems are NP-hard to approximate within any finite factor. Given the strong inapproximability result, we design bicriteria approximations for both problems: we obtain a $(6,6)$-approximation to the min cost nowhere-zero $k$-flow and a $(k,6)$-approximation to the min cost $k$-cut-balanced orientation. For the case of symmetric costs (where the costs of both orientations are the same for every edge), we show that the nowhere-zero $k$-flow problem remains NP-hard and admits a $3$-approximation.

cs.DS↗

arXiv2025

On the Smallest Support Size of Integer Solutions to Linear Equations

In this note, we study the size of the support of integer solutions to linear equations $Ax=b, ~x\in\Z^n$ where $A\in\Z^{m\times n}, b\in\Z^n$. We give an upper bound on the smallest support size as a function of $A$, taken as a worst case over all $b$ such that the above system has a solution. This bound is asymptotically tight, and in fact matches the bound given in Aliev, Averkov, De Loera and Oertel Mathematical Programming 2022, while the proof presented here is simpler, relying only on linear algebra.

math.OC↗

arXiv2025

A Min-Max Relation on Dicuts and Dijoins in Weighted Chordal Digraphs

In a digraph, a dicut is a cut where all the arcs cross in one direction. A dijoin is a subset of arcs that intersects every dicut. Edmonds and Giles conjectured that in a weighted digraph, the minimum weight of a dicut is equal to the maximum size of a packing of dijoins. This has been disproved. However, the unweighted version conjectured by Woodall remains open. We prove that the Edmonds-Giles conjecture is true if the underlying undirected graph is chordal. We also give a strongly polynomial-time algorithm to construct such a packing.

math.CO↗

arXiv2024

Strongly connected orientations and integer lattices

Let $D=(V,A)$ be a digraph whose underlying undirected graph is $2$-edge-connected, and let $P$ be the polytope whose vertices are the incidence vectors of arc sets whose reversal makes $D$ strongly connected. We study the lattice theoretic properties of the integer points contained in a proper face $F$ of $P$ not contained in $\{x:x_a=i\}$ for any $a\in A,i\in \{0,1\}$. We prove under a mild necessary condition that $F\cap \{0,1\}^A$ contains an integral basis $B$, i.e., $B$ is linearly independent, and any integral vector in the linear hull of $F$ is an integral linear combination of $B$. This result is surprising as the integer points in $F$ do not necessarily form a Hilbert basis. In proving the result, we develop a theory similar to Matching Theory for degree-constrained dijoins in bipartite digraphs. Our result has consequences for head-disjoint strong orientations in hypergraphs, and also to a famous conjecture by Woodall that the minimum size of a dicut of $D$, say $\tau$, is equal to the maximum number of disjoint dijoins. We prove a relaxation of this conjecture, by finding for any prime number $p\geq 2$, a $p$-adic packing of dijoins of value $\tau$ and of support size at most $2|A|$. We also prove that the all-ones vector belongs to the lattice generated by $F\cap \{0,1\}^A$, where $F$ is the face of $P$ satisfying $x(\delta^+(U))=1$ for every dicut $\delta^+(U)$ with minimum size.

math.CO↗

arXiv2024

Convexification Techniques for Fractional Programs

This paper develops a correspondence relating convex hulls of fractional functions with those of polynomial functions over the same domain. Using this result, we develop a number of new reformulations and relaxations for fractional programming problems. First, we relate 0-1 problems involving a ratio of affine functions with the boolean quadric polytope, and use inequalities for the latter to develop tighter formulations for the former. Second, we derive a new formulation to optimize a ratio of quadratic functions over a polytope using copositive programming. Third, we show that univariate fractional functions can be convexified using moment hulls. Fourth, we develop a new hierarchy of relaxations that converges finitely to the simultaneous convex hull of a collection of ratios of affine functions of 0-1 variables. Finally, we demonstrate theoretically and computationally that our techniques close a significant gap relative to state-of-the-art relaxations, require much less computational effort, and can solve larger problem instances.

math.OC↗

arXiv2024

On the Congruency-Constrained Matroid Base

Consider a matroid where all elements are labeled with an element in $\mathbb{Z}$. We are interested in finding a base where the sum of the labels is congruent to $g \pmod m$. We show that this problem can be solved in $\tilde{O}(2^{4m} n r^{5/6})$ time for a matroid with $n$ elements and rank $r$, when $m$ is either the product of two primes or a prime power. The algorithm can be generalized to all moduli and, in fact, to all abelian groups if a classic additive combinatorics conjecture by Schrijver and Seymour holds true. We also discuss the optimization version of the problem.

math.CO↗