More Like this

1. Spherical convex hull of random points on a wedge

   https://doi.org/10.1007/s00208-023-02704-9  

   Besau, Florian ; Gusakova, Anna ; Reitzner, Matthias ; Schütt, Carsten ; Thäle, Christoph ; Werner, Elisabeth M. ( August 2023 , Mathematische Annalen)

   Consider two half-spaces$$H_1^+$$$H_1^{+}$and$$H_2^+$$$H_2^{+}$in$${\mathbb {R}}^{d+1}$$$R^{d+1}$whose bounding hyperplanes$$H_1$$$H_1$and$$H_2$$$H_2$are orthogonal and pass through the origin. The intersection$${\mathbb {S}}_{2,+}^d:={\mathbb {S}}^d\cap H_1^+\cap H_2^+$$$S_{2,+}^d:=S^d\cap H_1^{+}\cap H_2^{+}$is a spherical convex subset of thed-dimensional unit sphere$${\mathbb {S}}^d$$$S^d$, which contains a great subsphere of dimension$$d-2$$$d-2$and is called a spherical wedge. Choosenindependent random points uniformly at random on$${\mathbb {S}}_{2,+}^d$$$S_{2,+}^d$and consider the expected facet number of the spherical convex hull of these points. It is shown that, up to terms of lower order, this expectation grows like a constant multiple of$$\log n$$$logn$. A similar behaviour is obtained for the expected facet number of a homogeneous Poisson point process on$${\mathbb {S}}_{2,+}^d$$$S_{2,+}^d$. The result is compared to the corresponding behaviour of classical Euclidean random polytopes and of spherical random polytopes on a half-sphere.

    

   more » « less
2. From approximate to exact integer programming

   https://doi.org/10.1007/s10107-024-02084-1  

   Dadush, Daniel ; Eisenbrand, Friedrich ; Rothvoss, Thomas ( April 2024 , Mathematical Programming)

   Approximate integer programming is the following: For a given convex body$$K \subseteq {\mathbb {R}}^n$$$K\subseteq R^n$, either determine whether$$K \cap {\mathbb {Z}}^n$$$K\cap Z^n$is empty, or find an integer point in the convex body$$2\cdot (K - c) +c$$$2·(K-c)+c$which isK, scaled by 2 from its center of gravityc. Approximate integer programming can be solved in time$$2^{O(n)}$$$2^{O\left(n\right)}$while the fastest known methods for exact integer programming run in time$$2^{O(n)} \cdot n^n$$$2^{O\left(n\right)}·n^n$. So far, there are no efficient methods for integer programming known that are based on approximate integer programming. Our main contribution are two such methods, each yielding novel complexity results. First, we show that an integer point$$x^* \in (K \cap {\mathbb {Z}}^n)$$$x^{\ast }\in (K\cap Z^n)$can be found in time$$2^{O(n)}$$$2^{O\left(n\right)}$, provided that theremaindersof each component$$x_i^* \mod \ell $$$x_i^{\ast }mod\ell$for some arbitrarily fixed$$\ell \ge 5(n+1)$$$\ell \ge 5(n+1)$of$$x^*$$$x^{\ast }$are given. The algorithm is based on acutting-plane technique, iteratively halving the volume of the feasible set. The cutting planes are determined via approximate integer programming. Enumeration of the possible remainders gives a$$2^{O(n)}n^n$$$2^{O\left(n\right)}{n}^n$algorithm for general integer programming. This matches the current best bound of an algorithm by Dadush (Integer programming, lattice algorithms, and deterministic, vol. Estimation. Georgia Institute of Technology, Atlanta, 2012) that is considerably more involved. Our algorithm also relies on a newasymmetric approximate Carathéodory theoremthat might be of interest on its own. Our second method concerns integer programming problems in equation-standard form$$Ax = b, 0 \le x \le u, \, x \in {\mathbb {Z}}^n$$$Ax=b,0\le x\le u,x\in Z^n$. Such a problem can be reduced to the solution of$$\prod _i O(\log u_i +1)$$${\prod }_{i}O(log{u}_i+1)$approximate integer programming problems. This implies, for example thatknapsackorsubset-sumproblems withpolynomial variable range$$0 \le x_i \le p(n)$$$0\le x_i\le p\left(n\right)$can be solved in time$$(\log n)^{O(n)}$$${(logn)}^{O\left(n\right)}$. For these problems, the best running time so far was$$n^n \cdot 2^{O(n)}$$$n^n·2^{O\left(n\right)}$.

    

   more » « less
3. Fertilitopes

   https://doi.org/10.1007/s00454-023-00488-y  

   Defant, Colin ( June 2023 , Discrete & Computational Geometry)

   We introduce tools from discrete convexity theory and polyhedral geometry into the theory of West’s stack-sorting map s. Associated to each permutation$$\pi $$$\pi$is a particular set$$\mathcal V(\pi )$$$V\left(\pi \right)$of integer compositions that appears in a formula for the fertility of $$\pi $$$\pi$, which is defined to be$$|s^{-1}(\pi )|$$$|s^{-1}\left(\pi \right)|$. These compositions also feature prominently in more general formulas involving families of colored binary plane trees calledtroupesand in a formula that converts from free to classical cumulants in noncommutative probability theory. We show that$$\mathcal V(\pi )$$$V\left(\pi \right)$is a transversal discrete polymatroid when it is nonempty. We define thefertilitopeof$$\pi $$$\pi$to be the convex hull of$$\mathcal V(\pi )$$$V\left(\pi \right)$, and we prove a surprisingly simple characterization of fertilitopes as nestohedra arising from full binary plane trees. Using known facts about nestohedra, we provide a procedure for describing the structure of the fertilitope of$$\pi $$$\pi$directly from$$\pi $$$\pi$using Bousquet-Mélou’s notion of the canonical tree of $$\pi $$$\pi$. As a byproduct, we obtain a new combinatorial cumulant conversion formula in terms of generalizations of canonical trees that we callquasicanonical trees. We also apply our results on fertilitopes to study combinatorial properties of the stack-sorting map. In particular, we show that the set of fertility numbers has density 1, and we determine all infertility numbers of size at most 126. Finally, we reformulate the conjecture that$$\sum _{\sigma \in s^{-1}(\pi )}x^{\textrm{des}(\sigma )+1}$$${\sum }_{\sigma \in s^{-1}\left(\pi \right)}{x}^{\text{des}\left(\sigma \right)+1}$is always real-rooted in terms of nestohedra, and we propose natural ways in which this new version of the conjecture could be extended.

    

   more » « less
4. On the stationarity for nonlinear optimization problems with polyhedral constraints

   https://doi.org/10.1007/s10107-023-01979-9  

   di Serafino, Daniela ; Hager, William W. ; Toraldo, Gerardo ; Viola, Marco ( May 2023 , Mathematical Programming)

   For polyhedral constrained optimization problems and a feasible point$$\textbf{x}$$$x$, it is shown that the projection of the negative gradient on the tangent cone, denoted$$\nabla _\varOmega f(\textbf{x})$$${\nabla }_{\Omega }f\left(x\right)$, has an orthogonal decomposition of the form$$\varvec{\beta }(\textbf{x}) + \varvec{\varphi }(\textbf{x})$$$\beta \left(x\right)+\phi \left(x\right)$. At a stationary point,$$\nabla _\varOmega f(\textbf{x}) = \textbf{0}$$${\nabla }_{\Omega }f\left(x\right)=0$so$$\Vert \nabla _\varOmega f(\textbf{x})\Vert $$$\Vert {\nabla }_{\Omega }f\left(x\right)\Vert$reflects the distance to a stationary point. Away from a stationary point,$$\Vert \varvec{\beta }(\textbf{x})\Vert $$$\Vert \beta \left(x\right)\Vert$and$$\Vert \varvec{\varphi }(\textbf{x})\Vert $$$\Vert \phi \left(x\right)\Vert$measure different aspects of optimality since$$\varvec{\beta }(\textbf{x})$$$\beta \left(x\right)$only vanishes when the KKT multipliers at$$\textbf{x}$$$x$have the correct sign, while$$\varvec{\varphi }(\textbf{x})$$$\phi \left(x\right)$only vanishes when$$\textbf{x}$$$x$is a stationary point in the active manifold. As an application of the theory, an active set algorithm is developed for convex quadratic programs which adapts the flow of the algorithm based on a comparison between$$\Vert \varvec{\beta }(\textbf{x})\Vert $$$\Vert \beta \left(x\right)\Vert$and$$\Vert \varvec{\varphi }(\textbf{x})\Vert $$$\Vert \phi \left(x\right)\Vert$.

    

   more » « less
5. Algorithms for covering multiple submodular constraints and applications

   https://doi.org/10.1007/s10878-022-00874-x  

   Chekuri, Chandra ; Inamdar, Tanmay ; Quanrud, Kent ; Varadarajan, Kasturi ; Zhang, Zhao ( June 2022 , Journal of Combinatorial Optimization)

   We consider the problem of covering multiple submodular constraints. Given a finite ground setN, a weight function$$w: N \rightarrow \mathbb {R}_+$$$w:N\to R_{+}$,rmonotone submodular functions$$f_1,f_2,\ldots ,f_r$$$f_1,f_2,\dots ,f_r$overNand requirements$$k_1,k_2,\ldots ,k_r$$$k_1,k_2,\dots ,k_r$the goal is to find a minimum weight subset$$S \subseteq N$$$S\subseteq N$such that$$f_i(S) \ge k_i$$$f_i\left(S\right)\ge k_i$for$$1 \le i \le r$$$1\le i\le r$. We refer to this problem asMulti-Submod-Coverand it was recently considered by Har-Peled and Jones (Few cuts meet many point sets. CoRR.arxiv:abs1808.03260Har-Peled and Jones 2018) who were motivated by an application in geometry. Even with$$r=1$$$r=1$Multi-Submod-Covergeneralizes the well-known Submodular Set Cover problem (Submod-SC), and it can also be easily reduced toSubmod-SC. A simple greedy algorithm gives an$$O(\log (kr))$$$O(log(kr\left)\right)$approximation where$$k = \sum _i k_i$$$k={\sum }_i{k}_i$and this ratio cannot be improved in the general case. In this paper, motivated by several concrete applications, we consider two ways to improve upon the approximation given by the greedy algorithm. First, we give a bicriteria approximation algorithm forMulti-Submod-Coverthat covers each constraint to within a factor of$$(1-1/e-\varepsilon )$$$(1-1/e-\epsilon )$while incurring an approximation of$$O(\frac{1}{\epsilon }\log r)$$$O(\frac{1}{ϵ}logr)$in the cost. Second, we consider the special case when each$$f_i$$$f_i$is a obtained from a truncated coverage function and obtain an algorithm that generalizes previous work on partial set cover (Partial-SC), covering integer programs (CIPs) and multiple vertex cover constraints Bera et al. (Theoret Comput Sci 555:2–8 Bera et al. 2014). Both these algorithms are based on mathematical programming relaxations that avoid the limitations of the greedy algorithm. We demonstrate the implications of our algorithms and related ideas to several applications ranging from geometric covering problems to clustering with outliers. Our work highlights the utility of the high-level model and the lens of submodularity in addressing this class of covering problems.

    

   more » « less