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Grossman et al. show that the subdeterminants of the incidence matrix of a graph can be characterized using the graph's odd cycle packing number. In particular, a graph's incidence matrix is totally unimodular if and only if the graph is bipartite. We extend the characterization of total unimodularity to disjoint hypergraphs, i.e., hypergraphs whose hyperedges of size at least four are pairwise disjoint. Disjoint hypergraphs interpolate between graphs and hypergraphs, which correspond to arbitrary {0,1}-matrices. We prove that total unimodularity for disjoint hypergraphs is equivalent to forbidding both odd cycles and a structure we call an ''odd tree house''. Our result extends to disjoint directed hypergraphs, i.e., those whose incidence matrices allow for {-1,0, 1}-entries. As a corollary, we resolve a conjecture on almost totally unimodular matrices, formulated by Padberg (1988) and Cornu\'ejols & Zuluaga (2000), in the special case where columns with at least four non-zero elements have pairwise disjoint supports., Comment: 49 pages, 14 figures

When modeling the demand in revenue management systems, a natural approach is to focus on a canonical interval of time, such as a week, so that we forecast the demand over each week in the selling horizon. Ideally, we would like to use random variables with general distributions to model the demand over each week. The current demand can give a signal for the future demand, so we also would like to capture the dependence between the demands over different weeks. Prevalent demand models in the literature, which are based on a discrete-time approximation to a Poisson process, are not compatible with these needs. In this paper, we focus on revenue management models that are compatible with a natural approach for forecasting the demand. Building such models through dynamic programming is not difficult. We divide the selling horizon into multiple stages, each stage being a canonical interval of time on the calendar. We have random number of customer arrivals in each stage, whose distribution is arbitrary and depends on the number of arrivals in the previous stage. The question we seek to answer is the form of the corresponding fluid approximation. We give the correct fluid approximation in the sense that it yields asymptotically optimal policies. The form of our fluid approximation is surprising as its constraints use expected capacity consumption of a resource up to a certain time period, conditional on the demand in the stage just before the time period in question. As the resource capacities and number of stages increase with the same rate, our performance guarantee converges to one. To our knowledge, this result gives the first asymptotically optimal policy under dependent demands with arbitrary distributions. Our computational experiments indicate that using the correct fluid approximation can make a dramatic impact in practice.

Conforti et al. give a compact extended formulation for a class of bimodular-constrained integer programs, namely those that model the stable set polytope of a graph with no disjoint odd cycles. We extend their techniques to design compact extended formulations for the integer hull of translated polyhedral cones whose constraint matrix is strictly $\Delta$-modular and has rows that represent a cographic matroid. Our work generalizes the important special case from Conforti et al. concerning $4$-connected graphs with odd cycle transversal number at least $4$. We also discuss the necessity of our assumptions.

Advancements in deep learning-based 3D object detection necessitate the availability of large-scale datasets. However, this requirement introduces the challenge of manual annotation, which is often both burdensome and time-consuming. To tackle this issue, the literature has seen the emergence of several weakly supervised frameworks for 3D object detection which can automatically generate pseudo labels for unlabeled data. Nevertheless, these generated pseudo labels contain noise and are not as accurate as those labeled by humans. In this paper, we present the first approach that addresses the inherent ambiguities present in pseudo labels by introducing an Evidential Deep Learning (EDL) based uncertainty estimation framework. Specifically, we propose MEDL-U, an EDL framework based on MTrans, which not only generates pseudo labels but also quantifies the associated uncertainties. However, applying EDL to 3D object detection presents three primary challenges: (1) relatively lower pseudolabel quality in comparison to other autolabelers; (2) excessively high evidential uncertainty estimates; and (3) lack of clear interpretability and effective utilization of uncertainties for downstream tasks. We tackle these issues through the introduction of an uncertainty-aware IoU-based loss, an evidence-aware multi-task loss function, and the implementation of a post-processing stage for uncertainty refinement. Our experimental results demonstrate that probabilistic detectors trained using the outputs of MEDL-U surpass deterministic detectors trained using outputs from previous 3D annotators on the KITTI val set for all difficulty levels. Moreover, MEDL-U achieves state-of-the-art results on the KITTI official test set compared to existing 3D automatic annotators., Comment: Accepted at ICRA 2024. Code: https://github.com/paathelb/MEDL-U

An integer matrix $\mathbf{A}$ is $\Delta$-modular if the determinant of each $\text{rank}(\mathbf{A}) \times \text{rank}(\mathbf{A})$ submatrix of $\mathbf{A}$ has absolute value at most $\Delta$. The study of $\Delta$-modular matrices appears in the theory of integer programming, where an open conjecture is whether integer programs defined by $\Delta$-modular constraint matrices can be solved in polynomial time if $\Delta$ is considered constant. The conjecture is only known to hold true when $\Delta \in \{1,2\}$. In light of this conjecture, a natural question is to understand structural properties of $\Delta$-modular matrices. We consider the column number question -- how many nonzero, pairwise non-parallel columns can a rank-$r$ $\Delta$-modular matrix have? We prove that for each positive integer $\Delta$ and sufficiently large integer $r$, every rank-$r$ $\Delta$-modular matrix has at most $\binom{r+1}{2} + 80\Delta^7 \cdot r$ nonzero, pairwise non-parallel columns, which is tight up to the term $80\Delta^7$. This is the first upper bound of the form $\binom{r+1}{2} + f(\Delta)\cdot r$ with $f$ a polynomial function. Underlying our results is a partial list of matrices that cannot exist in a $\Delta$-modular matrix. We believe this partial list may be of independent interest in future studies of $\Delta$-modular matrices.

We develop a technique that can be applied to provide improved upper bounds for two important questions in linear integer optimization. - Proximity bounds: Given an optimal vertex solution for the linear relaxation, how far away is the nearest optimal integer solution (if one exists)? - Flatness bounds: If a polyhedron contains no integer point, what is the smallest number of integer parallel hyperplanes defined by an integral, non-zero, normal vector that intersect the polyhedron? This paper presents a link between these two questions by refining a proof technique that has been recently introduced by the authors. A key technical lemma underlying our technique concerns the areas of certain convex polygons in the plane: if a polygon $K\subseteq\mathbb{R}^2$ satisfies $\tau K \subseteq K^{\circ}$, where $\tau$ denotes $90^{\circ}$ counterclockwise rotation and $K^{\circ}$ denotes the polar of $K$, then the area of $K^{\circ}$ is at least 3., Comment: This manuscripts greatly builds upon a related IPCO2022 paper. For all of the overlapping material, the current manuscript provides improved results. Additionally, the current manuscript derives new connections to flatness and generalizations to {0,k,2k} modular problems

The intersection cut framework was introduced by Balas in 1971 as a method for generating cutting planes in integer optimization. In this framework, one uses a full-dimensional convex $S$-free set, where $S$ is the feasible region of the integer program, to derive a cut separating $S$ from a non-integral vertex of a linear relaxation of $S$. Among all $S$-free sets, it is the inclusion-wise maximal ones that yield the strongest cuts. Recently, this framework has been extended beyond the integer case in order to obtain cutting planes in non-linear settings. In this work, we consider the specific setting when $S$ is defined by a homogeneous quadratic inequality. In this 'quadratic-free' setting, every function $\Gamma: D^m \to D^n$, where $D^k$ is the unit disk in $\mathbb{R}^k$, generates a representation of a quadratic-free set. While not every $\Gamma$ generates a maximal quadratic free set, it is the case that every full-dimensional maximal quadratic free set is generated by some $\Gamma$. Our main result shows that the corresponding quadratic-free set is full-dimensional and maximal if and only if $\Gamma$ is non-expansive and satisfies a technical condition. This result yields a broader class of maximal $S$-free sets than previously known. Our result stems from a new characterization of maximal $S$-free sets (for general $S$ beyond the quadratic setting) based on sequences that 'expose' inequalities defining the $S$-free set.

A branch-and-bound (BB) tree certifies a dual bound on the value of an integer program. In this work, we introduce the tree compression problem (TCP): Given a BB tree T that certifies a dual bound, can we obtain a smaller tree with the same (or stronger) bound by either (1) applying a different disjunction at some node in T or (2) removing leaves from T? We believe such post-hoc analysis of BB trees may assist in identifying helpful general disjunctions in BB algorithms. We initiate our study by considering computational complexity and limitations of TCP. We then conduct experiments to evaluate the compressibility of realistic branch-and-bound trees generated by commonly-used branching strategies, using both an exact and a heuristic compression algorithm., Comment: A short version of this article was accepted for publication at IPCO 2023. This extended version contains more detailed discussions and proofs, and new computational contributions and experiments

The Steinitz constant in dimension $d$ is the smallest value $c(d)$ such that for any norm on $\mathbb{R}^{ d}$ and for any finite zero-sum sequence in the unit ball, the sequence can be permuted such that the norm of each partial sum is bounded by $c(d)$. Grinberg and Sevastyanov prove that $c(d) \le d$ and that the bound of $d$ is best possible for arbitrary norms; we refer to their result as the Steinitz Lemma. We present a variation of the Steinitz Lemma that permutes multiple sequences at one time. Our result, which we term a colorful Steinitz Lemma, demonstrates upper bounds that are independent of the number of sequences. Many results in the theory of integer programming are proved by permuting vectors of bounded norm; this includes proximity results, Graver basis algorithms, and dynamic programs. Due to a recent paper of Eisenbrand and Weismantel, there has been a surge of research on how the Steinitz Lemma can be used to improve integer programming results. As an application we prove a proximity result for block-structured integer programs., Comment: Shortened proofs, fixed typos, and streamlined the argument in Section 3

Consider a linear program of the form $\max\;c^{\top}x:Ax\leq b$, where $A$ is an $m\times n$ integral matrix. In 1986 Cook, Gerards, Schrijver, and Tardos proved that, given an optimal solution $x^{*}$, if an optimal integral solution $z^{*}$ exists, then it may be chosen such that $\left\Vert x^{*}-z^{*}\right\Vert _{\infty}

Motivated by the poor performance of cross-validation in settings where data are scarce, we propose a novel estimator of the out-of-sample performance of a policy in data-driven optimization.Our approach exploits the optimization problem's sensitivity analysis to estimate the gradient of the optimal objective value with respect to the amount of noise in the data and uses the estimated gradient to debias the policy's in-sample performance. Unlike cross-validation techniques, our approach avoids sacrificing data for a test set, utilizes all data when training and, hence, is well-suited to settings where data are scarce. We prove bounds on the bias and variance of our estimator for optimization problems with uncertain linear objectives but known, potentially non-convex, feasible regions. For more specialized optimization problems where the feasible region is "weakly-coupled" in a certain sense, we prove stronger results. Specifically, we provide explicit high-probability bounds on the error of our estimator that hold uniformly over a policy class and depends on the problem's dimension and policy class's complexity. Our bounds show that under mild conditions, the error of our estimator vanishes as the dimension of the optimization problem grows, even if the amount of available data remains small and constant. Said differently, we prove our estimator performs well in the small-data, large-scale regime. Finally, we numerically compare our proposed method to state-of-the-art approaches through a case-study on dispatching emergency medical response services using real data. Our method provides more accurate estimates of out-of-sample performance and learns better-performing policies.

We study integer-valued matrices with bounded determinants. Such matrices appear in the theory of integer programs (IP) with bounded determinants. For example, Artmann et al. showed that an IP can be solved in strongly polynomial time if the constraint matrix is bimodular, that is, the determinants are bounded in absolute value by two. Determinants are also used to bound the $\ell_1$-distance between IP solutions and solutions of its linear relaxation. One of the first works to quantify the complexity of IPs with bounded determinants was that of Heller, who identified the maximum number of differing columns in a totally unimodular matrix. Each extension of Heller's bound to general determinants has been super-polynomial in the determinants or the number of equations. We provide the first column bound that is polynomial in both values. For integer programs with box constraints, our result gives the first $\ell_1$-distance bound that is polynomial in the determinants and the number of equations. Our result can also be used to derive a bound on the height of Graver basis elements that is polynomial in the determinants and the number of equations. Furthermore, we show a tight bound on the number of differing columns in a bimodular matrix; this is the first tight bound since Heller. Our analysis reveals combinatorial properties of bimodular IPs that may be of independent interest., Comment: Fixed typo and accepted for journal publication

On June 24th, Governor Cooper announced that North Carolina will not be moving into Phase 3 of its reopening process at least until July 17th. Given the recent increases in daily positive cases and hospitalizations, this decision was not surprising. However, given the political and economic pressures which are forcing the state to reopen, it is not clear what actions will help North Carolina to avoid the worst. We use a compartmentalized model to study the effects of social distancing measures and testing capacity combined with contact tracing on the evolution of the pandemic in North Carolina until the end of the year. We find that going back to restrictions that were in place during Phase 1 will slow down the spread but if the state wants to continue to reopen or at least remain in Phase 2 or Phase 3 it needs to significantly expand its testing and contact tracing capacity. Even under our best-case scenario of high contact tracing effectiveness, the number of contact tracers the state currently employs is inadequate., Comment: arXiv admin note: substantial text overlap with arXiv:2005.14700

The "New York State on Pause" executive order came into effect on March 22 with the goal of ensuring adequate social distancing to alleviate the spread of COVID-19. Pause will remain effective in New York City in some form until early June. We use a compartmentalized model to study the effects of testing capacity and social distancing measures on the evolution of the pandemic in the "post-Pause" period in the City. We find that testing capacity must increase dramatically if it is to counterbalance even relatively small relaxations in social distancing measures in the immediate post-Pause period. In particular, if the City performs 20,000 tests per day and relaxes the social distancing measures to the pre-Pause norms, then the total number of deaths by the end of September can reach 250,000. By keeping the social distancing measures to somewhere halfway between the pre- and in-Pause norms and performing 100,000 tests per day, the total number of deaths by the end of September can be kept at around 27,000. Going back to the pre-Pause social distancing norms quickly must be accompanied by an exorbitant testing capacity, if one is to suppress excessive deaths. If the City is to go back to the "pre-Pause" social distancing norms in the immediate post-Pause period and keep the total number of deaths by the end of September at around 35,000, then it should be performing 500,000 tests per day. Our findings have important implications on the magnitude of the testing capacity the City needs as it relaxes the social distancing measures to reopen its economy.

Lattice-free gradient polyhedra can be used to certify optimality for mixed-integer convex minimization models. We consider how to construct these polyhedra for unconstrained models with two integer variables under the assumption that all level sets are bounded. A classic result of Bell, Doignon, and Scarf states that a lattice-free gradient polyhedron with at most four facets exists in this setting. We present an algorithm for creating a sequence of gradient polyhedra, each of which has at most four facets, that finitely converges to a lattice-free gradient polyhedron. Each update requires constantly many gradient evaluations. Our updates imitate the gradient descent algorithm, and consequently, it yields a gradient descent type of algorithm for problems with two integer variables.

We refer to the distance between optimal solutions of integer programs and their linear relaxations as proximity. In 2018, Eisenbrand and Weismantel proved that proximity is independent of the dimension for programs in standard form. We improve their bounds using existing and novel results on the sparsity of integer solutions. We first bound proximity in terms of the largest absolute value of any full-dimensional minor in the constraint matrix, and this bound is tight up to a polynomial factor in the number of constraints. We also give an improved bound in terms of the largest absolute entry in the constraint matrix, after efficiently transforming the program into an equivalent one. Our results are stated in terms of general sparsity bounds, so any new results on sparse solutions immediately improves our work. Generalizations to mixed integer programs are also discussed.

We consider the asymptotic distribution of the IP sparsity function, which measures the minimal support of optimal IP solutions, and the IP to LP distance function, which measures the distance between optimal IP and LP solutions. We create a framework for studying the asymptotic distribution of general functions related to integer optimization. There has been a significant amount of research focused around the extreme values that these functions can attain, however less is known about their typical values. Each of these functions is defined for a fixed constraint matrix and objective vector while the right hand sides are treated as input. We show that the typical values of these functions are smaller than the known worst case bounds by providing a spectrum of probability-like results that govern their overall asymptotic distributions., Comment: Accepted for journal publication