Arc connectivity and submodular flows in digraphs

Let $D=(V,A)$ be a digraph. For an integer $k\geq 1$, a $k$-arc-connected flip is an arc subset of $D$ such that after reversing the arcs in it the digraph becomes (strongly) $k$-arc-connected. The first main result of this paper introduces a sufficient condition for the existence of a $k$-arc-connected flip that is also a submodular flow for a crossing submodular function. More specifically, given some integer $\tau\geq 1$, suppose $d_A^+(U)+(\frac{\tau}{k}-1)d_A^-(U)\geq \tau$ for all $U\subsetneq V, U\neq \emptyset$, where $d_A^+(U)$ and $d_A^-(U)$ denote the number of arcs in $A$ leaving and entering $U$, respectively. Let $\mathcal{C}$ be a crossing family over ground set $V$, and let $f:\mathcal{C}\to \mathbb{Z}$ be a crossing submodular function such that $f(U)\geq \frac{k}{\tau}(d_A^+(U)-d_A^-(U))$ for all $U\in \mathcal{C}$. Then $D$ has a $k$-arc-connected flip $J$ such that $f(U)\geq d_J^+(U)-d_J^-(U)$ for all $U\in \mathcal{C}$. The result has several applications to Graph Orientations and Combinatorial Optimization. In particular, it strengthens Nash-Williams' so-called weak orientation theorem, and proves a weaker variant of Woodall's conjecture on digraphs whose underlying undirected graph is $\tau$-edge-connected. The second main result of this paper is even more general. It introduces a sufficient condition for the existence of capacitated integral solutions to the intersection of two submodular flow systems. This sufficient condition implies the classic result of Edmonds and Giles on the box-total dual integrality of a submodular flow system. It also has the consequence that in a weakly connected digraph, the intersection of two submodular flow systems is totally dual integral.Comment: 29 pages, 4 figure

Geometric rescaling algorithms for submodular function minimization

We present a new class of polynomial-time algorithms for submodular function minimization (SFM) as well as a unified framework to obtain strongly polynomial SFM algorithms. Our algorithms are based on simple iterative methods for the minimum-norm problem, such as the conditional gradient and Fujishige–Wolfe algorithms. We exhibit two techniques to turn simple iterative methods into polynomial-time algorithms. First, we adapt the geometric rescaling technique, which has recently gained attention in linear programming, to SFM and obtain a weakly polynomial bound O((n4 · EO + n5)log(nL)). Second, we exhibit a general combinatorial black box approach to turn εL-approximate SFM oracles into strongly polynomial exact SFM algorithms. This framework can be applied to a wide range of combinatorial and continuous algorithms, including pseudo-polynomial ones. In particular, we can obtain strongly polynomial algorithms by a repeated application of the conditional gradient or of the Fujishige–Wolfe algorithm. Combined with the geometric rescaling technique, the black box approach provides an O((n5 · EO + n6)log2n) algorithm. Finally, we show that one of the techniques we develop in the paper can also be combined with the cutting-plane method of Lee et al., yielding a simplified variant of their O(n3log2n · EO + n4logO(1)n) algorithm

Rescaling algorithms for linear conic feasibility

We propose simple polynomial-time algorithms for two linear conic feasibility problems. For a matrix A ∈ R m× n, the kernel problem requires a positive vector in the kernel of A, and the image problem requires a positive vector in the image of A T. Both algorithms iterate between simple first-order steps and rescaling steps. These rescalings improve natural geometric potentials. If Goffin's condition measure ρ A is negative, then the kernel problem is feasible, and the worst-case complexity of the kernel algorithm is O((m 3n + mn 2)log|ρ A| −1); if ρ A > 0, then the image problem is feasible, and the image algorithm runs in time O(m 2n 2 log ρ A −1). We also extend the image algorithm to the oracle setting. We address the degenerate case ρA = 0 by extending our algorithms to find maximum support nonnegative vectors in the kernel of A and in the image of A T. In this case, the running time bounds are expressed in the bit-size model of computation: for an input matrix A with integer entries and total encoding length L, the maximum support kernel algorithm runs in time O((m 3n + mn 2)L), whereas the maximum support image algorithm runs in time O(m 2n 2L). The standard linear programming feasibility problem can be easily reduced to either maximum support problems, yielding polynomial-time algorithms for linear programming

On finding exact solutions of linear programs in the oracle model

We consider linear programming in the oracle model: mincT x s.t. x ∊ P, where the polyhedron P = {x ∊ ℝn: Ax ≤ b} is given by a separation oracle that returns violated inequalities from the system Ax ≤ b. We present an algorithm that finds exact primal and dual solutions using O(n2 log(n/δ)) oracle calls and O(n4 log(n/δ) + n6 log log(1/δ)) arithmetic operations, where δ is a geometric condition number associated with the system (A, b). These bounds do not depend on the cost vector c. The algorithm works in a black box manner, requiring a subroutine for approximate primal and dual solutions; the above running times are achieved when using the cutting plane method of Jiang, Lee, Song, and Wong (STOC 2020) for this subroutine. Whereas approximate solvers may return primal solutions only, we develop a general framework for extracting dual certificates based on the work of Burrell and Todd (Math. Oper. Res. 1985). Our algorithm works in the real model of computation, and extends results by Grötschel, Lovász, and Schrijver (Prog. Comb. Opt. 1984), and by Frank and Tardos (Combinatorica 1987) on solving LPs in the bit-complexity model. We show that under a natural assumption, simultaneous Diophantine approximation in these results can be avoided

Mixed-integer vertex covers on bipartite graphs

Let $A$ be the edge-node incidence matrix of a bipartite graph \n$G = (U, V ; E)$, $I$ be a subset of the nodes of $G$, and $b$ be a vector such \nthat $2b$ is integral. We consider the following mixed-integer set: \n X(G, b, I) = {x : Ax ≥ b, x ≥ 0, x_i integer for all i ∈ I}. \nWe characterize conv$(X(G, b, I))$ in its original space. That is, we describe a matrix $(C, d)$ such that conv(X(G, b, I)) = {x : Cx ≥ d}. This \nis accomplished by computing the projection onto the space of the $x$-variables of an extended formulation, given in [1], for $conv(X(G, b, I))$. \nWe then give a polynomial-time algorithm for the separation problem for conv$(X(G, b, I))$, thus showing that the problem of optimizing a linear \nfunction over the set $X(G, b, I)$ is solvable in polynomial time. \

Characterization of tellurum inclusions in CdZnTe ingots grown by vertical bridgam technique

CdZnTe (CZT) crystals are employed for the preparation of room temperature operating X-ray detectors [1]. The functioning of the devices without refrigeration is made possible by growing high resistivity (>1010 Ohm.cm) ingots. This is usually reached by contemporarily doping with group III or group VII elements and using tellurium deviated charge. This second condition is responsible for the presence in crystals of a large number of tellurium inclusions. These can be incorporated at the growing interface or can form during cooling as a result of the retrograde behavior of the liquidus curve [2]. Unfortunately, inclusions severely limit the performances of CZT-based detectors, in particular in the case of imaging devices. In fact because of the role of diffusion of the electrons drifting from the cathode to the anode, tellurium inclusions act as traps for the charge carriers. Consequently the detector response close to the inclusions is deteriorated [3]. Hence, monitoring tellurium inclusion density is very important for assessing the material quality, selecting the best region in CZT wafer and for studying the formation mechanisms of inclusions during growth. Tellurium inclusions presence can be revealed by means of optical transmission microscopy in the near-infrared, in fact tellurium inclusions are opaque to the IR, while the CZT matrix is transparent. We developed an instrument for 3D mapping of inclusions mounted on a standard optical microscope with automatic vertical movement. Pictures are taken at different focal planes. Images are then elaborated by a dedicated software that ascribes each inclusion to the proper focal plane. As a result, all inclusions are counted and precisely localized in 3D. Using different objective lenses of the microscope it is possible to choose the optimal compromise between resolution and extent of the monitored area. However, at high magnification it is possible to map inclusions down to 1 micron diameter. The spatial position information of tellurium inclusion obtained by 3D IR mapping was used to select a single inclusion in the sample and then acquire photoluminescence (PL) map in the selected region. The inclusion was placed very close to the surface (few microns) by etching the sample. A correlation was set between the PL spectra emission and the presence of tellurium inclusion. [1] Knoll GF (2000) Radiation Detection and Measurements. Wiley [2] Rudolph P (1995), J. Cryst. Growth 147:297-304. [3] Carini GA (2006), Appl. Phys. Lett. 88:143515-143526. [4] Zambelli N (2011), J. Cryst. Growth 318:1167-1170

Luminescence Properties of CZT Crystals in the Presence of Tellurium Inclusions

CZT is a widespread material for the realization of room temperature radiation detectors. The presence of defects and in particular secondary phases, like Te inclusions, represents nowadays a limit in the realization of high resolution devices. For the development of CZT detectors, in particular for high-flux applications, is very important to understand the role of deep levels, the influence of Te inclusions on the device performance and their correlations between Te inclusions and deep levels. Using a IR microscope recently developed at IMEM, it is possible to identify the 3D position of each inclusion in the bulk and reconstruct a 3D plot describing the spacial position of every inclusion. This permits to select a sigle inclusion in the sample, to place the inclusion very close to the surface (few microns) by etching the sample and hence to study the selected inclusion. In this way it is possible to perform photoluminescence and cathodoluminescence mapping in the inclusion region and investigate the behavior of the crystal-inclusion interface. The correlation between the deep level emission acquired at the micro-scale and the presence of tellurium inclusion is discussed

Geometric rescaling algorithms for submodular function minimization

We present a new class of polynomial-time algorithms for submodular function minimization (SFM), as well as a unified framework to obtain strongly polynomial SFM algorithms. Our new algorithms are based on simple iterative methods for the minimum-norm problem, such as the conditional gradient and the Fujishige-Wolfe algorithms. We exhibit two techniques to turn simple iterative methods into polynomial-time algorithms. Firstly, we use the geometric rescaling technique, which has recently gained attention in linear programming. We adapt this technique to SFM and obtain a weakly polynomial bound O((n4 · EO + n5) log(nL)). Secondly, we exhibit a general combinatorial black-box approach to turn any strongly polynomial εL-approximate SFM oracle into an strongly polynomial exact SFM algorithm. This framework can be applied to a wide range of combinatorial and continuous algorithms, including pseudopolynomial ones. In particular, we can obtain strongly polynomial algorithms by a repeated application of the conditional gradient or of the Fujishige-Wolfe algorithm. Combined with the geometric rescaling technique, the black-box approach provides a O((n5 · EO + n6) log2 n) algorithm. Finally, we show that one of the techniques we develop in the paper, “sliding”, can also be combined with the cutting-plane method of Lee, Sidford, and Wong [27], yielding a simplified variant of their O(n3 log2 n · EO + n4 logO(1) n) algorith

On matrices with the Edmonds-Johnson property arising from bidirected graphs

In this paper we study totally half-modular matrices obtained from {0,±1}-matrices with at most two nonzero entries per column by multiplying by 2 some of the columns. We give an excluded-minor characterization of the matrices in this class having strong Chv`atal rank 1. Our result is a special case of a conjecture by Gerards and Schrijver [11]. It also extends a well known theorem of Edmonds and Johnson [10]

Optimal cutting planes from the group relaxations

We study quantitative criteria for evaluating the strength of valid inequalities for Gomory and Johnson's finite and infinite group models and we describe the valid inequalities that are optimal for these criteria. We justify and focus on the criterion of maximizing the volume of the nonnegative orthant cut off by a valid inequality. For the finite group model of prime order, we show that the unique maximizer is an automorphism of the Gomory Mixed-Integer (GMI) cut for a possibly different finite group problem of the same order. We extend the notion of volume of a simplex to the infinite dimensional case. This is used to show that in the infinite group model, the GMI cut maximizes the volume of the nonnegative orthant cut off by an inequality