Circuits in Extended Formulations

Circuits and extended formulations are classical concepts in linear programming theory. The circuits of a polyhedron are the elementary difference vectors between feasible points and include all edge directions. We study the connection between the circuits of a polyhedron $P$ and those of an extended formulation of $P$, i.e., a description of a polyhedron $Q$ that linearly projects onto $P$. It is well known that the edge directions of $P$ are images of edge directions of $Q$. We show that this `inheritance' under taking projections does not extend to the set of circuits. We provide counterexamples with a provably minimal number of facets, vertices, and extreme rays, including relevant polytopes from clustering, and show that the difference in the number of circuits that are inherited and those that are not can be exponentially large in the dimension. We further prove that counterexamples exist for any fixed linear projection map, unless the map is injective. Finally, we characterize those polyhedra $P$ whose circuits are inherited from all polyhedra $Q$ that linearly project onto $P$. Conversely, we prove that every polyhedron $Q$ satisfying mild assumptions can be projected in such a way that the image polyhedron $P$ has a circuit with no preimage among the circuits of $Q$. Our proofs build on standard constructions such as homogenization and disjunctive programming

The role of rationality in integer-programming relaxations

For a finite set X⊂Zd that can be represented as X=Q∩Zd for some polyhedron Q, we call Q a relaxation of X and define the relaxation complexity rc(X) of X as the least number of facets among all possible relaxations Q of X. The rational relaxation complexity rcQ(X) restricts the definition of rc(X) to rational polyhedra Q. In this article, we focus on X=Δd, the vertex set of the standard simplex, which consists of the null vector and the standard unit vectors in Rd. We show that rc(Δd)≤d for every d≥5. That is, since rcQ(Δd)=d+1, irrationality can reduce the minimal size of relaxations. This answers an open question posed by Kaibel and Weltge (Math Program 154(1):407–425, 2015). Moreover, we prove the asymptotic statement rc(Δd)∈O(dlog(d)√/), which shows that the ratio rc(Δd)rcQ(Δd)/ goes to 0, as d→∞

Advances on Strictly Δ\Delta-Modular IPs

There has been significant work recently on integer programs (IPs) $\min\{c^\top x \colon Ax\leq b,\,x\in \mathbb{Z}^n\}$ with a constraint marix $A$ with bounded subdeterminants. This is motivated by a well-known conjecture claiming that, for any constant $\Delta\in \mathbb{Z}_{>0}$, $\Delta$-modular IPs are efficiently solvable, which are IPs where the constraint matrix $A\in \mathbb{Z}^{m\times n}$ has full column rank and all $n\times n$ minors of $A$ are within $\{-\Delta, \dots, \Delta\}$. Previous progress on this question, in particular for $\Delta=2$, relies on algorithms that solve an important special case, namely strictly $\Delta$-modular IPs, which further restrict the $n\times n$ minors of $A$ to be within $\{-\Delta, 0, \Delta\}$. Even for $\Delta=2$, such problems include well-known combinatorial optimization problems like the minimum odd/even cut problem. The conjecture remains open even for strictly $\Delta$-modular IPs. Prior advances were restricted to prime $\Delta$, which allows for employing strong number-theoretic results. In this work, we make first progress beyond the prime case by presenting techniques not relying on such strong number-theoretic prime results. In particular, our approach implies that there is a randomized algorithm to check feasibility of strictly $\Delta$-modular IPs in strongly polynomial time if $\Delta\leq4$

Reconfiguration of basis pairs in regular matroids

In recent years, combinatorial reconfiguration problems have attracted great attention due to their connection to various topics such as optimization, counting, enumeration, or sampling. One of the most intriguing open questions concerns the exchange distance of two matroid basis sequences, a problem that appears in several areas of computer science and mathematics. In 1980, White proposed a conjecture for the characterization of two basis sequences being reachable from each other by symmetric exchanges, which received a significant interest also in algebra due to its connection to toric ideals and Gr\"obner bases. In this work, we verify White's conjecture for basis sequences of length two in regular matroids, a problem that was formulated as a separate question by Farber, Richter, and Shan and Andres, Hochst\"attler, and Merkel. Most of previous work on White's conjecture has not considered the question from an algorithmic perspective. We study the problem from an optimization point of view: our proof implies a polynomial algorithm for determining a sequence of symmetric exchanges that transforms a basis pair into another, thus providing the first polynomial upper bound on the exchange distance of basis pairs in regular matroids. As a byproduct, we verify a conjecture of Gabow from 1976 on the serial symmetric exchange property of matroids for the regular case.Comment: 28 pages, 6 figure

Regular Matroids Have Polynomial Extension Complexity

We prove that the extension complexity of the independence polytope of every regular matroid on n elements is O(n6). Past results of Wong and Martin on extended formulations of the spanning tree polytope of a graph imply a O(n2) bound for the special case of (co)graphic matroids. However, the case of a general regular matroid was open, despite recent attempts. We also consider the extension complexity of circuit dominants of regularmatroids, for which we give a O(n2) bound

Congruency-Constrained TU Problems Beyond the Bimodular Case

A long-standing open question in Integer Programming is whether integer programs with constraint matrices with bounded subdeterminants are efficiently solvable. An important special case thereof are congruency-constrained integer programs $\min\{c^\top x\colon\ Tx\leq b,\ \gamma^\top x\equiv r\pmod{m},\ x\in\mathbb{Z}^n\}$with a totally unimodular constraint matrix$T$. Such problems have been shown to be polynomial-time solvable for$m=2$, which led to an efficient algorithm for integer programs with bimodular constraint matrices, i.e., full-rank matrices whose$n\times n$subdeterminants are bounded by two in absolute value. Whereas these advances heavily relied on existing results on well-known combinatorial problems with parity constraints, new approaches are needed beyond the bimodular case, i.e., for$m>2$. We make first progress in this direction through several new techniques. In particular, we show how to efficiently decide feasibility of congruency-constrained integer programs with a totally unimodular constraint matrix for$m=3$. Furthermore, for general$m$, our techniques also allow for identifying flat directions of infeasible problems, and deducing bounds on the proximity between solutions of the problem and its relaxation