Papers

Given a nonnegative matrix $X$, a factorization rank $r$ and {a positive integer $p$}, entrywise power matrix factorization (EPMF) looks for a low-rank matrix $X_r$ such that $X = |X_r|^{\circ p}$ (exact case) or $X \approx |X_r|^{\circ p}$ (approximate case), where $(\cdot)^{\circ p}$ denotes the componentwise exponent. EPMF includes the modulus model ($p=1$) and componentwise square factorization ($p=2$) as special cases, the latter being closely related to the square root rank. We analyze the computational complexity of the exact decision problem and the Frobenius-norm approximation problem, and establish a complete complexity landscape. In the exact case, we show that EPMF is equivalent to the combinatorial problem of flipping the signs of the entries of a given matrix $X$ to obtain a rank-$r$ matrix, which we refer to as the low-rank matrix signing (LRMS) problem. We first show that LRMS, and hence exact EPMF, is strongly NP-hard, improving a weak NP-hardness result for the square-root-rank (Math. Prog., 2015). We then show that LRMS can be solved in polynomial time when $r$ is fixed. Moreover, when the rank $r$ is part of the input, we show that for generic matrices the algorithm is fixed-parameter tractable (FPT) in the parameter $r$; in fact, the running time is fixed-parameter linear in the number of entries of the input matrix. In the approximate case using the Frobenius norm as an error measure, we show that EPMF is NP-hard, already when $r=2$, the smallest nontrivial case.

We establish a correspondence between two structures arising in the geometry of hyperplane arrangements: Euclidean $\vee$-systems and real polyhedral Kähler (PK) arrangements. We prove that every irreducible Euclidean $\vee$-system determines a real PK arrangement, and conversely that every real PK arrangement arises this way. As a consequence, we show that, up to equivalence, there are exactly three irreducible rank-three Euclidean $\vee$-systems whose vectors have equal length; their arrangements are the mirrors of the reflection groups of the regular tetrahedron, cube, and icosahedron. The correspondence also yields a description of the moduli space of Euclidean $\vee$-systems in a fixed projective class: it is homeomorphic to the relative interior of a polytope. We also give a direct proof that the hyperplane arrangement associated with a Euclidean $\vee$-system is simplicial. Among the currently known simplicial line arrangements, we identify precisely those that arise from $\vee$-systems. As a consequence, we prove that the Schreiber–Veselov catalog is complete for irreducible rank-three Euclidean $\vee$-systems with at most $27$ vectors.

For the Erdős–Frankl–Pach problem on uniform set systems of bounded VC-dimension, the Ahlswede–Khachatrian/Mubayi–Zhao construction has long served as the standard lower-bound benchmark. We develop a recursive lifting method that goes beyond this benchmark in every dimension \(d\ge3\), proving that for every \(d\ge3\) and \(n\ge d+3\), \[ M_d(n)\ge \binom{n-1}{d}+\binom{n-4}{d-2}+M_{d-3}(n-5). \] The proof is elementary and proceeds through explicit trace obstructions. We also record a further recursive improvement in the concluding remarks.

This paper aims to provide a direct combinatorial proof of the Gessel-Wachs formula for $q$-derangement numbers in the setting of decorated permutations, without using the $q$-binomial inversion formula. Decorated permutations, introduced by Postnikov in his study of the totally nonnegative Grassmannian, provide a natural framework for Chen's signed fixed-point model. Our proof is based on a major-index generating function for decorated permutations with a fixed number of signed fixed points, together with a sign-reversing and descent-set-preserving involution, thereby answering a question raised by Chen. This involution was discovered through human–AI collaboration. Moreover, our framework yields an immediate proof of a result of Désarménien and Wachs concerning the equidistribution between descent classes of derangements and descent classes of desarrangements. We also supply combinatorial proofs of two recurrence relations for the $q$-derangement numbers in the setting of desarrangements with the aid of the insertion lemma for ordinary permutations.

Let \(\mathcal{B}_n\) be the Boolean lattice of all subsets of \([n]\) and let \(\mathcal{P}_{n;\ell,u}\) be the subposet of \(\mathcal{B}_n\) induced by the consecutive levels \(\ell,\ell+1,\ldots,u\). We determine $\nu_{n;\ell,u}$, the maximum size of a family of pairwise disjoint maximal intervals in $\mathcal P_{n;\ell,u}$, whenever \(u\le ({n+\ell^2})/({\ell+1})\). This completely settles Engel's problem~[Combin. Probab. Comput., 1996]. The proof is constructive. We also record consequences for weakly cross-intersecting set-pair systems and discuss the three-level case.

We propose several approaches for bounding the minim\-um distances of the family of quasi-cyclic LDPC codes in the 5G NR standard. In particular, we show that the high-rate [9984, 8448] and the low-rate [25344, 8448] BG1 5G LDPC codes have minimum distances in the ranges {8..14} and {22..57}, respectively. Also we propose a new early termination approach based on circulant modular reduction, which significantly lowers syndrome calculation complexity for the LDPC decoder.

This paper develops a systematic method for determining particular solutions of the $k$th-order linear nonhomogeneous recurrence relation $$a_n + c_1 a_{n-1} + \cdots + c_k a_{n-k} = \sum_{j=1}^J p_j(n){r_j}^n$$ with $n \geq k$, $c_k \neq 0$, $r_j \neq 0$. Here each $p_j(n)$ is a polynomial. The main result is the following: for the characteristic polynomial $c(t)=t^k+c_1t^{k-1}+\cdots+c_k$, if $s_j$ denotes the multiplicity of $r_j$ as a root of $c(t)$ ($s_j=0$ when $r_j$ is not a root), then there exists a particular solution of the form $q_n=\sum_{j=1}^J b_j(n)n^{s_j}r_j^n$, where each $b_j(n)$ is a polynomial of the same degree as $p_j(n)$. This result parallels the method of undetermined coefficients for linear ODEs with constant coefficients and yields a systematic procedure for determining the form of particular solutions.

We consider the set of matchings of a graph and a local change operation, called a flip, between them. In the combinatorial setting, the base graphs are either complete graphs or complete bipartite graphs, and in the geometric setting, the graphs are embedded on point sets in the plane, with the requirement that edges must be drawn as straight lines and must not cross. For base graphs with an even number of vertices, we consider perfect matchings, i.e., all vertices are matched, and for base graphs with an odd number of vertices, we consider almost-perfect matchings, i.e., all but one vertex of the graph are matched. A 2-flip between two perfect matchings exchanges two edges, and a 1-flip between two almost-perfect matchings exchanges one edge. The corresponding flip graph has the set of perfect or almost-perfect matchings as vertices, with pairs of them connected by an edge if they differ in a 2-flip or 1-flip, respectively. In this work, we provide a comprehensive picture of Hamiltonicity properties of these flip graphs. We prove that the flip graphs in the combinatorial setting are Hamilton-connected, i.e., they admit a Hamilton path between any two vertices, or, if the flip graphs are bipartite, we prove that they are Hamilton-laceable, i.e., they admit a Hamilton path between any two vertices from different partition classes. In the geometric setting, we prove that any path in them misses exponentially many vertices, in particular, they have no Hamilton paths or cycles. For points in convex position and almost-perfect matchings under 1-flips, we complement this by constructing a cycle in the flip graph that visits almost all vertices.

Let $P_n=\mathrm{Cay}(S_n,\{r_2,\ldots,r_n\})$ be the pancake graph, with prefix reversals acting on the right. Conjugating Zaks' suffix-reversal permutation Gray code by the full reversal gives a distinguished Hamiltonian cycle $Z_n$ in $P_n$. We determine the stabilizer of this particular cycle under the left regular action of $S_n$. If $\rho=r_{n-1}r_n=[n,1,2,\ldots,n-1]$, then, for every $n\ge3$, $\mathrm{Stab}_{L(S_n)}(Z_n)=\langle L_\rho,L_{r_n}\rangle\cong D_n$, where $D_n$ denotes the dihedral group of order $2n$. The inclusion $\supseteq$ follows from the recursive block decomposition $W_n=(W_{n-1}r_n)^{n-1}W_{n-1}$ and from the palindromy $W_n^R=W_n$. The reverse inclusion follows from a general cyclic-order rigidity lemma: if a Hamiltonian cycle on a finite group is invariant under $L_a$, with $\mathrm{ord}(a)\ge3$, then every left translation preserving the same cycle conjugates $a$ to $a$ or $a^{-1}$. For $n\ge5$, Deng-Zhang's automorphism theorem gives the same stabilizer inside $\mathrm{Aut}(P_n)$; the exceptional ranks are handled separately. We also compute the compression factor of $Z_n$: it is $n$ for $n\ge4$ and $6$ for $n=3$.

We formulate compactifications of continuous maps naturally. We consider real analytic functions mainly. We are interested in topological properties and combinatorial ones of explicit resulting maps. For understanding them, we use their Reeb spaces, being quotient spaces of the spaces of the domains of the functions and defined by the equivalence relation identifying two points in same components of their level sets. They are known to be $0$- or $1$-dimensional (metrizable) cell-complexes, in our situations or more general certain tame cases. Reeb spaces have been important in understanding topological properties and combinatorial ones of functions and spaces roughly, since the last century. These compactifications have been explicitly studied by the author previously and recently. We have obtained real algebraic functions whose Reeb spaces are not so complicated and which seem to be of most natural and simplest. We present new discussions and examples.

We prove a sparse embedding theorem for induced embeddings of bounded-degree graphs. The theorem applies to pairs $G\subseteq \Gamma$: the graph $G$ supplies the positive edges of the target graph, while the ambient graph $\Gamma$ supplies the induced constraints which must be avoided. Its main feature is that these two types of constraints are kept separate throughout the embedding process. As an application, we show that, for every fixed $\Delta,r\ge2$, there are constants $A,C>0$ such that every $n$-vertex graph $H$ with maximum degree at most $\Delta$ satisfies $r_{\text{ind}}(H;r)\le C n^{\Delta+2}(\log n)^A$. This improves the exponent in the polynomial bound of Conlon, Fox and Zhao for bounded-degree induced Ramsey numbers. The proof combines the aforementioned embedding theorem with a sparse random transference argument, in which the random host is used only to certify robust deterministic hypotheses for every colour class.

The Alon–Tarsi number $AT(G)$ of a graph $G$, defined via the graph polynomial, is a strengthening of the list chromatic number $\chi_{\ell}(G)$. We study the Alon–Tarsi number of squares of planar graphs. The square of a graph $G$ is the graph obtained by joining every pair of vertices whose distance in $G$ is at most $2$. Recently, Kim and Luo (2026) proved that $\chi_{\ell}(G^2)\le 6$ for every subcubic planar graph containing no $k$-cycles for $4\le k\le 8$. We strengthen this result by proving that $AT(G^2)\le 6$ for every such graph $G$.

Let $f_r(k)$ be the smallest $n$ such that every $r$-coloring of $\{1,2,\ldots,n\}$ has a monochromatic solution to the equation \[\frac{1}{x_1}+\frac{1}{x_2}+\cdots+\frac{1}{x_k}=\frac{1}{x_{k+1}}, \] where $x_1,x_2,\ldots,x_k$ are not necessarily distinct. In this paper, we prove that $f_r(2)\geq 4^r/2$ for all $r\geq1$, and $f_r(k)\geq(2^r-1)k^r$ for all $k\geq3$ and $r\geq1$. When $r=2$, we show that, if $k=3\cdot2^m$ for some positive integer $m$, then $f_2(k)=3k^2$; and if $k=p^m$ for some odd prime number $p$ and positive integer $m$, then $f_2(k)\geq3k^2+1$. We also provide new computational results for $f_2(k)$ and $f_3(k)$, as well as a generalization of our lower bounds for $f_2(k)$ to equations with general coefficients.

Addressing a question posed by Erdős and Hajnal, Chen and Ma proved that, for all $n \ge 600$, the complete bipartite graph $K_{n,n+1}$ is the unique graph on $2n+1$ vertices with at least $n^2+n$ edges that contains no two vertices of equal degree joined by a path of length three. In this paper, we extend this result and prove that for every fixed integer \(\ell\ge 2\) and sufficiently large \(n\), the unique \(2n\)-vertex graph with at least \((n^2+n)/2\) edges that contains no two vertices of equal degree joined by a path of length \(2\ell\) is the half graph \(H_n\). This resolves the problem posed by Chen and Ma, as well as a related question of Attwa, Azócar Carvajal, Boyadzhiyska, Pierron, and Taraz concerning paths of even length with equal-degree endpoints.

For an integer $L$, write $C_{\ge L}$ for the family of cycles of length at least $L$. For $L=2a$ let $H(n,L)=K_a+\overline K_{n-a}$, and for $L=2a+1$ let $H(n,L)$ be obtained from $K_a+\overline K_{n-a}$ by adding one edge inside the independent part. We prove sharp results for two even target graphs, namely even cycles $C_{2s}$ and even paths $P_{2r+1}$. For even cycles, with $s\ge3$ and $L\ge2s$, we have \[ \mathrm{ex}(n,C_{2s},C_{\ge L+1})=C_{2s}(H(n,L)) \] for all sufficiently large $n$. Together with the known $C_4$ case of Zhu, Győri, He, Lv, Salia and Xiao~[Bull. Lond. Math. Soc. 55 (2023)], this verifies the even-cycle case of their conjecture on $\mathrm{ex}(n,C_k,C_{\ge L+1})$. For even paths, with $r\ge2$ and $L\ge2r$, we have \[ \mathrm{ex}(n,P_{2r+1},C_{\ge L+1})=N(P_{2r+1},H(n,L)) \] for all sufficiently large $n$. We also derive the corresponding exact results when the forbidden graph is a path $P_{p+1}$, sharpening the relevant even-cycle and even-path asymptotic results of Győri, Salia, Tompkins and Zamora~[Discrete Math. Theor. Comput. Sci. 21 no. 1 (2019)].

Starting from the stability theorem of Erdős and Simonovits, stability problems for graphs forbidding a fixed subgraph have been studied in terms of edge numbers, spectral radii and subgraph counts. Let $\mathcal{N}(F,G)$ denote the number of unlabeled copies of $F$ in $G$. It is known that, for every fixed path $P_t$ and even cycle $C_{2a}$, the maximum number of copies in an $n$-vertex $C_{2\ell+1}$-free graph is attained by the bipartite Turán graph $T_{n,2}$. In this paper we obtain strong structural stability for $C_{2\ell+1}$-free graphs in terms of copies of paths and even cycles. For fixed $\ell\ge2$ and $3\le r\le2\ell-1$, we show that if an $n$-vertex $C_{2\ell+1}$-free graph contains at least as many copies of $P_t$ or $C_{2a}$ as the corresponding suspended extremal construction, then it has the corresponding suspension structure. This gives exact high-chromatic extremal theorems for paths and even cycles. We also prove a counting theorem for nearly complete bipartite graphs. It shows that, for every fixed matching-admissible connected bipartite graph $F$, both imbalance between the two parts and missing cross-edges decrease the number of copies of $F$ by a term with a specified main coefficient. This theorem is independent of the forbidden odd cycle and converts subgraph-count assumptions into the edge bounds needed for the structural theorem.

A dissociation set in a graph is a subset of vertices that induces a subgraph of maximum degree at most one, which is a natural generalization of the notion of an independent set. The dissociation number of a graph is defined as the maximum cardinality of a dissociation set. This paper studies the minimum spectral radius of connected graphs with a given order $n$ and a given dissociation number $\psi$. For $\psi=n-k$ with $k\ge 4$ fixed and $n$ sufficiently large, we establish both upper and lower bounds for this minimum spectral radius and prove the extremal graphs must belong to a specific graph class.

A classical theorem of Andrásfai, Erdős, and Sós states that every $n$-vertex graph $G$ with odd girth at least $2k+1$ and minimum degree $\delta(G)>\frac{2n}{2k+1}$ is bipartite (i.e., homomorphic to $K_2$). Messuti and Schacht proved that the same odd girth condition with $\delta(G)>\frac{3n}{4k}$ forces a homomorphism to $C_{2k+1}$. In this paper, we strengthen the above results by showing that every $n$-vertex graph $G$ with odd girth at least $2k+1$ and minimum degree $\delta(G)>\frac{4n}{6k-1}$ is homomorphic to the Möbius ladder on $4k$ vertices. This answers a question of Messuti and Schacht and generalizes a result of Brandt and Ribe-Baumann.

We establish a strict logical separation between two distinct phenomena in orthogonality hypergraphs: chromatic completeness, the possibility of assigning a single globally consistent nondegenerate spectrum to all contexts, and geometric coordinatizability, the existence of a faithful orthogonal representation by rays. A strong chromatic number larger than the Hilbert-space dimension obstructs only the former. It does not, by itself, obstruct the existence of a faithful orthogonal representation. We make this separation explicit by comparing two three-dimensional examples with the same strong chromatic number. A completed 25-ray version of the Yu-Oh configuration has strong chromatic number four and nevertheless possesses an explicit faithful orthogonal representation in R^3. Conversely, Greechie's G_{32} hypergraph also has strong chromatic number four, and has a separating and unital set of two-valued states, but we give an elementary algebraic proof that it admits no faithful orthogonal representation in C^3. The obstruction in G_{32} is therefore not chromatic but projective-geometric: the incidence relations force two distinct atoms to collapse onto the same ray.

Let $G$ be a simple graph with cover ideal $J(G)$ in a polynomial ring $S$ in $|V(G)|$ variables. For a matching $M$ of $G$, we denote by $\ell(M)$ the length of the longest $M$-alternating path in $G$. We define $\alpha_t(G)$ to be the maximum size of an ordered matching $M$ of $G$ such that $\ell(M) \le 2t-1$. We then prove that $$\operatorname{depth}(S/J(G)^{(t)}) \le |V(G)| - 1 - \alpha_t(G)$$ for all $t \ge 1$, where $J(G)^{(t)}$ denotes the $t$-th symbolic power of $J(G)$, and that equality holds when $G$ is a forest.

We study the triangular array $T(n,k;\mu)$ defined by the Graham–Knuth–Patashnik recurrences $$ T(n,k) \;=\; (\alpha n + \beta k + \gamma) \, T(n-1,k) + (\alpha' n + \beta' k + \gamma') \, T(n-1,k-1) $$ with initial condition $T(0,k)=\delta_{k,0}$ and parameters $\mu=(\alpha,\beta,\gamma,\alpha',\beta',\gamma')$, which are considered to be indeterminates. We first prove that, for any fixed $n\ge 0$, the sequence $(T(n,k;\mu))_{k\ge 0}$ is strongly log-concave with the coefficientwise partial order in the variables $\alpha,\beta,\gamma,\alpha',\beta',\gamma'$. Moreover, we show that the sequence of the corresponding row-generating polynomials $(P_n(x;\mu))_{n\ge 0}$ is strongly log-convex with the coefficientwise partial order in the variables $x$ and $\alpha,\beta,\gamma,\alpha',\beta',\gamma'$. Finally, we show that this sequence is coefficientwise Hankel-totally positive of order 2 with the same partial order.

We study continuous-time quantum walks on cycles. We prove two complementary results. Firstly, the cycle $C_9$ does not admit uniform mixing at any time. Using the similar idea and Dickson polynomials, we prove that $C_{15}$ does not admit uniform mixing at any time neither. Secondly, for every prime $p$, we show that the cycle $C_{p^2}$ admits $\epsilon$-uniform mixing.

For graphs $F$ and $H$, a spanning subgraph $G$ of $F$ is weakly $H$-saturated in $F$ if the edges in $E(F)\setminus E(G)$ can be added one at a time, each addition creating a new copy of $H$. Recently, Tancer and Tyomkyn proved that, given an $n$-vertex graph $F$, deciding whether $\mathrm{wsat}(F,K_3)=n-1$ is NP-hard. In this paper, we study the decision version of the weak saturation problem and show that, for every fixed integer $r\ge 3$, given a graph $F$ and an integer $k$, deciding whether $\mathrm{wsat}(F,H)\le k$ is NP-complete when $H\in\{K_r,K_{r,r}\}$. Our approach uses novel graph-theoretic and topological ideas and techniques, yielding new constructions that build on the construction of Tancer and Tyomkyn. In particular, our proofs bring the flag-no-square property, a fundamental property in topology that is of independent interest, into the study of weak saturation problem.

If a finite collection (with repetitions allowed) of unit fractions has sum larger than $K$, how close can a sum of some of them come to $1$ from below? Improving a bound of Erdős and Graham, we prove that the gap is at most $\exp(-c\sqrt{K\log K})$ for an absolute constant $c>0$ and all sufficiently large $K$. The proof combines equal fractions and applies a Fourier estimate to groups of denominators with comparable contributions. Elementary divisor counts then give the bound.

We introduce split-free cable expressions and their sequential restriction. Live cables are vertex blocks that future operations cannot split. The main result identifies sequential split-free cable width, up to an additive two, with active neighborhood-width, the minimum number of distinct nonzero future-neighborhood classes across a linear layout. The lower bound extracts such a layout from every cable play and includes a parity argument for prefixes inside a birth group. The upper bound compiles any active-neighborhood layout into a singleton-birth play. This gives an exponential upper bound and a linear lower bound in terms of linear rank-width. We prove that the complement of the seven-vertex cycle has linear rank-width two and sequential cable width exactly eight. This disproves the proposed affine upper estimate already at rank two. We also prove an unbounded separation between branching expression width and sequential width on trees.

We construct random hives from deformed GUE minor processes. Starting from two independent diagonally deformed GUE matrices \[ X=\sqrt{n}(wG+uD),\qquad Y=\sqrt{n}(w'G'+u'D'), \] where \(D,D'\) are diagonal and have GUE spectra, we use their minor processes to form a double hive and then apply the octahedron recurrence. Under the matching condition \[ \frac{u}{w^2}=\frac{u'}{(w')^2}, \] we prove that the resulting hive law is close, in relative entropy, to a GUE hive law. More precisely, if \[ a^2=w^2+u^2,\qquad b^2=(w')^2+(u')^2, \] then the produced hive density $q_n$ satisfies \[ D_{\mathrm{KL}}\!\left( q_n\, \middle\|\, \operatorname{Density}\bigl(H_n(a\sqrt n,b\sqrt n,c_{**}\sqrt n)\bigr) \right) = O(n\log n). \] The third scale $c_{**}$ is determined by a limiting tetrahedral optimization problem; equivalently, writing \(\delta=u+u'\), \[ \delta^2 = \frac{ 2c_{**}^4(c_{**}^2-a^2-b^2) }{ (c_{**}^2-a^2+b^2)(c_{**}^2+a^2-b^2) }. \] Thus the construction realizes GUE hive laws, up to subleading relative entropy, throughout the right-angled and obtuse regime. The introduction records two explicit surface-tension approximations and numerical comparisons which motivated the construction.

We prove the remaining conjectural distribution-equivalence of mesh patterns of length 2 from 2019. This completes the classification of distribution-equivalence and Wilf-equivalence classes for mesh patterns of length 2. Consequently the numbers of such classes are 105 and 46, respectively. We approach this by constructing a marked occurrence bijection, in the spirit of the cluster method for consecutive patterns.

Seeking to bridge the structural divide between a permutation's cycle notation and its one-line notation, Berman and Tenner introduced a novel notion of permutation pattern known as the arrow pattern. Recently, Archer and Laudone initiated a systematic study of arrow pattern avoidance, leaving behind three intriguing conjectures. In this paper, we resolve all three conjectures. First, we enumerate all six subclasses of permutations that simultaneously avoid a classical pattern of length 3 and a fixed arrow pattern of length 3, thereby confirming the first two conjectures. Second, we settle the third conjecture (which involves a different arrow pattern) by providing two independent proofs. These proofs rely on a restriction of Biane's bijection to non-nesting involutions and Krattenthaler's bijection from 321-avoiding permutations to Dyck paths, respectively.

Modern supply chains span diverse operational environments, ranging from e-commerce distribution networks to customized production-to-order manufacturing lines. Across these settings, operational efficiency depends on coordinating two highly interdependent stages: order preparation and downstream delivery. Although these stages are traditionally managed in isolation, real-world fulfillment systems must satisfy stringent delivery expectations under dynamic stochastic order arrivals. To bridge this gap, we introduce the Dynamic Order Fulfillment Problem (DOFP), a new problem class unifying logistical challenges previously studied separately. We model DOFP as a Markov decision process whose state and decision spaces are partitioned into preparation and delivery sub-spaces, linked by synchronization constraints. While recent approaches attempt to optimize both fulfillment stages simultaneously over myopic rolling horizons, our framework isolates and optimizes the downstream delivery policy, treating preparation strictly as a state-level constraint filter. To solve this, we develop the Decomposition-Driven Framework with Value Function Approximation (DDF-VFA), which utilizes a novel policy-level decomposition. This design partitions the search into a delivery-stage master problem and a preparation-stage compatibility subproblem, iteratively refined via feedback loops. DDF-VFA executes this strategy by combining a large-neighborhood search over partial delivery decisions with a neural-network value function approximation for the cost-to-go. Numerical illustrations on two example variants using real-world datasets show that DDF-VFA consistently outperforms benchmarks that optimize the two stages independently or jointly without decomposition. Finally, the framework naturally scales to accommodate additional real-world complexities such as batched or multi-stage preparation.

Bergeron's conjecture states that, if $1\leq a<b<c<d$ are integers with $ad=bc$, then one has the coefficient-wise inequality ${\binom{b+c}b}_q \ge {\binom{a+d}a}_q$ among two Gaussian polynomials. It originated in algebraic combinatorics and is wide open. The corresponding inequality for binomial coefficients (i.e., the case $q=1$) must be known to experts, but we could not find it in the literature. We give two proofs, each generalizing the statement in a separate direction. Binomial coefficients (and Gaussian polynomials) are fundamental combinatorial objects, and so one naturally hopes to see a combinatorial proof of this inequality. However, this seems hard to come by. We nevertheless give a combinatorial proof of a special case.