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.

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) algorithm.

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.

In this paper we study systems of the form $b\leq Mx\leq d$, $l\leq x\leq u$, where M is obtained from a totally unimodular matrix with two nonzero elements per row by multiplying by 2 some of its columns, and where $b,d,l,u$ are integral vectors. We give an explicit description of a totally dual integral system that describes the integer hull of the polyhedron P defined by the above inequalities. Since the inequalities of such a totally dual integral system are Chvátal inequalities for P, our result implies that the matrix M has cut-rank 1. We also derive a strongly polynomial time algorithm to find an integral optimal solution for the dual of the problem of minimizing a linear function with integer coefficients over the aforementioned totally dual integral system.

In a graph G, an odd hole is an induced odd cycle of length at least 5. A clique of G is a set of pairwise adjacent vertices. In this paper we consider the class ${\cal C}_k$ of graphs whose cliques have a size bounded by a constant k. Given a graph G in ${\cal C}_k$, we show how to recognize in polynomial time whether G contains an odd hole.