arXiv subjects

Giacomo Zambelli

Publications and source records attributed to Giacomo Zambelli.

11 recordsLinked to original sources

arXiv2026

On finding exact solutions of linear programs in the oracle model

We consider linear programming in the oracle model: $\max\{c^\top x \,:\, x\in P\}$, where the polyhedron $P=\{x\in\mathbb{R}^n\,:\, Ax\le b\}$ is given by a separation oracle. We present an algorithm that finds exact primal and dual solutions using $O(n^2\log(n/δ))$ oracle calls and $O(n^4\log(n/δ)+n^5\log\log(1/δ))$ arithmetic operations, where $δ$ is a geometric condition number associated with the system $(A,b)$. These bounds do not depend on the cost vector $c$ and do not require a priori knowledge of $δ$. For rational data, $\log(1/δ)$ is polynomially bounded in the encoding size of $(A,b)$, thus providing a polynomial-time algorithm. 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 strengthens results by Grötschel, Lovász, and Schrijver (Prog. Comb. Opt. 1984), and by Frank and Tardos (Combinatorica 1987) that rely on bit-complexity arguments. Our algorithm avoids rounding-based arguments such as simultaneous Diophantine approximation and uses geometric arguments instead.

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arXiv2023

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 $τ\geq 1$, suppose $d_A^+(U)+(\fracτ{k}-1)d_A^-(U)\geq τ$ 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 $f(U)\geq \frac{k}τ(d_A^+(U)-d_A^-(U))$ 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 $τ$-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.

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arXiv2020

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. Firstly, we adapt the geometric rescaling technique, which has recently gained attention in linear programming, to SFM and obtain a weakly polynomial bound $O(({n}^4\cdot \textrm{EO} + {n}^5)\log ({n} L))$. Secondly, we exhibit a general combinatorial black-box approach to turn $\varepsilon L$-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 $O(({n}^5\cdot \textrm{EO} +{n}^6)\log^2{n})$ 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, Sidford, and Wong \cite{LSW}, yielding a simplified variant of their $O(n^3 \log^2 n \cdot \textrm{EO} + n^4\log^{O(1)} n)$ algorithm.

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arXiv2019

Rescaling Algorithms for Linear Conic Feasibility

We propose simple polynomial-time algorithms for two linear conic feasibility problems. For a matrix $A\in \mathbb{R}^{m\times n}$, the kernel problem requires a positive vector in the kernel of $A$, and the image problem requires a positive vector in the image of $A^\top$. 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 $O\left((m^3n+mn^2)\log{|ρ_A|^{-1}}\right)$; if $ρ_A>0$, then the image problem is feasible and the image algorithm runs in time $O\left(m^2n^2\log{ρ_A^{-1}}\right)$. We also extend the image algorithm to the oracle setting. We address the degenerate case $ρ_A=0$ by extending our algorithms to find maximum support nonnegative vectors in the kernel of $A$ and in the image of $A^\top$. 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 $O\left((m^3n+mn^2)L\right)$, while the maximum support image algorithm runs in time $O\left(m^2n^2L\right)$. The standard linear programming feasibility problem can be easily reduced to either maximum support problems, yielding polynomial-time algorithms for Linear Programming.

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arXiv2018

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 {\em Gomory Mixed-Integer (GMI) cut} for a possibly {\em 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.

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arXiv2017

Convex Sets and Minimal Sublinear Functions

We show that, given a closed convex set $K$ containing the origin in its interior, the support function of the set $\{y\in K^*: \exists x\in K\mbox{ such that } \langle x,y \rangle =1\}$ is the pointwise smallest among all sublinear functions $σ$ such that $K=\{x: σ(x)\leq 1\}$.

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arXiv2017

A Counterexample to a Conjecture of Gomory and Johnson

In Mathematical Programming 2003, Gomory and Johnson conjecture that the facets of the infinite group problem are always generated by piecewise linear functions. In this paper we give an example showing that the Gomory-Johnson conjecture is false.

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arXiv2017

Unique lifting of integer variables in minimal inequalities

This paper contributes to the theory of cutting planes for mixed integer linear programs (MILPs). Minimal valid inequalities are well understood for a relaxation of an MILP in tableau form where all the nonbasic variables are continuous; they are derived using the gauge function of maximal lattice-free convex sets. In this paper we study lifting functions for the nonbasic integer variables starting from such minimal valid inequalities. We characterize precisely when the lifted coefficient is equal to the coefficient of the corresponding continuous variable in every minimal lifting. The answer is a nonconvex region that can be obtained as a finite union of convex polyhedra. We then establish a necessary and sufficient condition for the uniqueness of the lifting function.

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arXiv2017

Minimal inequalities for an infinite relaxation of integer programs

We show that maximal $S$-free convex sets are polyhedra when $S$ is the set of integral points in some rational polyhedron of $\mathbb{R}^n$. This result extends a theorem of Lovász characterizing maximal lattice-free convex sets. Our theorem has implications in integer programming. In particular, we show that maximal $S$-free convex sets are in one-to-one correspondence with minimal inequalities.

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arXiv2017

Maximal lattice-free convex sets in linear subspaces

We consider a model that arises in integer programming, and show that all irredundant inequalities are obtained from maximal lattice-free convex sets in an affine subspace. We also show that these sets are polyhedra. The latter result extends a theorem of Lovász characterizing maximal lattice-free convex sets in $\mathbb{R}^n$.

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arXiv2013

A polynomial projection-type algorithm for linear programming

We propose a simple O([n^5/\log n]L) algorithm for linear programming feasibility, that can be considered as a polynomial-time implementation of the relaxation method. Our work draws from Chubanov's "Divide-and-Conquer" algorithm [4], where the recursion is replaced by a simple and more efficient iterative method. A similar approach was used in a more recent paper of Chubanov [6].

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