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 $\Delta\leq4$

How hard is it to verify flat affine counter systems with the finite monoid property ?

We study several decision problems for counter systems with guards defined by convex polyhedra and updates defined by affine transformations. In general, the reachability problem is undecidable for such systems. Decidability can be achieved by imposing two restrictions: (i) the control structure of the counter system is flat, meaning that nested loops are forbidden, and (ii) the set of matrix powers is finite, for any affine update matrix in the system. We provide tight complexity bounds for several decision problems of such systems, by proving that reachability and model checking for Past Linear Temporal Logic are complete for the second level of the polynomial hierarchy $\Sigma^P_2$, while model checking for First Order Logic is PSPACE-complete

Complexity Bounds for Block-IPs

We consider integer programs (IPs) with a certain block structure, called two-stage stochastic. A two-stage stochastic IP is an integer program of the form $\min\{c^Tx \mid Ax=b,\, \ell\leq x\leq u,\, x\in \mathbb{Z}^{s + nt}\}$ where the constraint matrix $A\in \mathbb{Z}^{rn \times s+tn}$ consists of blocks $A^{(i)} \in \mathbb{Z}^{r\times s}$ on a vertical line and blocks $B^{(i)}\in \mathbb{Z}^{r\times t}$ on the diagonal line aside. We improve the bound for the Graver complexity of two-stage stochastic IPs. Our bound of $3^{O(s^s(2r||A||_\infty+1)^{rs})}$ reduces the dependency from $rs^2$ to $rs$ and is asymptotically tight under the exponential time hypothesis in the case that $r=1$. The improved Graver complexity bound stems from improved bounds on the intersection for a class of structurally rich integer cones. Our bound of $3^{O(d\Delta)^d}$ for dimension $d$ and absolute entries bounded by $\Delta$ is independent of the number of intersected integer cones. We investigate special properties of this class, which is complemented by the fact that these properties do not hold for general integer cones. Moreover, we give structural characterizations of this class that admit their use for two-stage stochastic IPs

Extended formulations for a class of polyhedra with bimodular cographic constraint matrices

We are motivated by integer linear programs (ILPs) defined by constraint matrices with bounded determinants. Such matrices generalize the notion of totally-unimodular matrices. When the determinants are bounded by $2$, the matrix is called bimodular. Artmann et al. give a polynomial-time algorithm for solving any ILP defined by a bimodular constraint matrix. Complementing this result, Conforti et al. give a compact extended formulation for a particular class of bimodular-constrained ILPs, namely those that model the stable set polytope of a graph with odd cycle packing number $1$. We demonstrate that their compact extended formulation can be modified to hold for polyhedra such that (1) the constraint matrix is bimodular, (2) the row-matroid generated by the constraint matrix is cographic and (3) the right-hand side is a linear combination of the columns of the constraint matrix. This generalizes the important special case from Conforti et al. concerning 4-connected graphs with odd cycle transversal number at least four. Moreover, our results yield compact extended formulations for a new class of polyhedra

Characterization of matrices with bounded graver bases and depth parameters and applications to integer programming

An intensive line of research on fixed parameter tractability of integer programming is focused on exploiting the relation between the sparsity of a constraint matrix A and the norm of the elements of its Graver basis. In particular, integer programming is fixed parameter tractable when parameterized by the primal tree-depth and the entry complexity of A, and when parameterized by the dual tree-depth and the entry complexity of A; both these parameterization imply that A is sparse, in particular, the number of its non-zero entries is linear in the number of columns or rows, respectively. We study preconditioners transforming a given matrix to an equivalent sparse matrix if it exists and provide structural results characterizing the existence of a sparse equivalent matrix in terms of the structural properties of the associated column matroid. In particular, our results imply that the \u1d4c1₁-norm of the Graver basis is bounded by a function of the maximum \u1d4c1₁-norm of a circuit of A. We use our results to design a parameterized algorithm that constructs a matrix equivalent to an input matrix A that has small primal/dual tree-depth and entry complexity if such an equivalent matrix exists. Our results yield parameterized algorithms for integer programming when parameterized by the \u1d4c1₁-norm of the Graver basis of the constraint matrix, when parameterized by the \u1d4c1₁-norm of the circuits of the constraint matrix, when parameterized by the smallest primal tree-depth and entry complexity of a matrix equivalent to the constraint matrix, and when parameterized by the smallest dual tree-depth and entry complexity of a matrix equivalent to the constraint matrix

Polynomial Identity Testing and the Ideal Proof System: PIT is in NP if and only if IPS can be p-simulated by a Cook-Reckhow proof system

The Ideal Proof System (IPS) of Grochow & Pitassi (FOCS 2014, J. ACM, 2018) is an algebraic proof system that uses algebraic circuits to refute the solvability of unsatisfiable systems of polynomial equations. One potential drawback of IPS is that verifying an IPS proof is only known to be doable using Polynomial Identity Testing (PIT), which is solvable by a randomized algorithm, but whose derandomization, even into NSUBEXP, is equivalent to strong lower bounds. However, the circuits that are used in IPS proofs are not arbitrary, and it is conceivable that one could get around general PIT by leveraging some structure in these circuits. This proposal may be even more tempting when IPS is used as a proof system for Boolean Unsatisfiability, where the equations themselves have additional structure. Our main result is that, on the contrary, one cannot get around PIT as above: we show that IPS, even as a proof system for Boolean Unsatisfiability, can be p-simulated by a deterministically verifiable (Cook-Reckhow) proof system if and only if PIT is in NP. We use our main result to propose a potentially new approach to derandomizing PIT into NP

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