Faster Algorithms for Integer Programs with Block Structure

We consider integer programming problems $\max \{ c^T x : \mathcal{A} x = b, l \leq x \leq u, x \in \mathbb{Z}^{nt}\}$ where $\mathcal{A}$ has a (recursive) block-structure generalizing "$n$-fold integer programs" which recently received considerable attention in the literature. An $n$-fold IP is an integer program where $\mathcal{A}$ consists of $n$ repetitions of submatrices $A \in \mathbb{Z}^{r \times t}$ on the top horizontal part and $n$ repetitions of a matrix $B \in \mathbb{Z}^{s \times t}$ on the diagonal below the top part. Instead of allowing only two types of block matrices, one for the horizontal line and one for the diagonal, we generalize the $n$-fold setting to allow for arbitrary matrices in every block. We show that such an integer program can be solved in time $n^2 t^2 {\phi} \cdot (rs{\Delta})^{\mathcal{O}(rs^2+ sr^2)}$ (ignoring logarithmic factors). Here ${\Delta}$ is an upper bound on the largest absolute value of an entry of $\mathcal{A}$ and ${\phi}$ is the largest binary encoding length of a coefficient of $c$. This improves upon the previously best algorithm of Hemmecke, Onn and Romanchuk that runs in time $n^3t^3 {\phi} \cdot {\Delta}^{\mathcal{O}(t^2s)}$. In particular, our algorithm is not exponential in the number $t$ of columns of $A$ and $B$. Our algorithm is based on a new upper bound on the $l_1$-norm of an element of the "Graver basis" of an integer matrix and on a proximity bound between the LP and IP optimal solutions tailored for IPs with block structure. These new bounds rely on the "Steinitz Lemma". Furthermore, we extend our techniques to the recently introduced "tree-fold IPs", where we again present a more efficient algorithm in a generalized setting

New Bounds on Augmenting Steps of Block-Structured Integer Programs

Iterative augmentation has recently emerged as an overarching method for solving Integer Programs (IP) in variable dimension, in stark contrast with the volume and flatness techniques of IP in fixed dimension. Here we consider 4-block n-fold integer programs, which are the most general class considered so far. A 4-block n-fold IP has a constraint matrix which consists of n copies of small matrices A, B, and D, and one copy of C, in a specific block structure. Iterative augmentation methods rely on the so-called Graver basis of the constraint matrix, which constitutes a set of fundamental augmenting steps. All existing algorithms rely on bounding the ??- or ?_?-norm of elements of the Graver basis. Hemmecke et al. [Math. Prog. 2014] showed that 4-block n-fold IP has Graver elements of ?_?-norm at most ?_FPT(n^{2^{s_D}}), leading to an algorithm with a similar runtime; here, s_D is the number of rows of matrix D and ?_FPT hides a multiplicative factor that is only dependent on the small matrices A,B,C,D, However, it remained open whether their bounds are tight, in particular, whether they could be improved to ?_FPT(1), perhaps at least in some restricted cases. We prove that the ?_?-norm of the Graver elements of 4-block n-fold IP is upper bounded by ?_FPT(n^{s_D}), improving significantly over the previous bound ?_FPT(n^{2^{s_D}}). We also provide a matching lower bound of ?(n^{s_D}) which even holds for arbitrary non-zero lattice elements, ruling out augmenting algorithm relying on even more restricted notions of augmentation than the Graver basis. We then consider a special case of 4-block n-fold in which C is a zero matrix, called 3-block n-fold IP. We show that while the ?_?-norm of its Graver elements is ?(n^{s_D}), there exists a different decomposition into lattice elements whose ?_?-norm is bounded by ?_FPT(1), which allows us to provide improved upper bounds on the ?_?-norm of Graver elements for 3-block n-fold IP. The key difference between the respective decompositions is that a Graver basis guarantees a sign-compatible decomposition; this property is critical in applications because it guarantees each step of the decomposition to be feasible. Consequently, our improved upper bounds let us establish faster algorithms for 3-block n-fold IP and 4-block IP, and our lower bounds strongly hint at parameterized hardness of 4-block and even 3-block n-fold IP. Furthermore, we show that 3-block n-fold IP is without loss of generality in the sense that 4-block n-fold IP can be solved in FPT oracle time by taking an algorithm for 3-block n-fold IP as an oracle

Nonlinear Integer Programming

Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject to integrality requirements for the variables. This chapter is dedicated to this topic. The primary goal is a study of a simple version of general nonlinear integer problems, where all constraints are still linear. Our focus is on the computational complexity of the problem, which varies significantly with the type of nonlinear objective function in combination with the underlying combinatorial structure. Numerous boundary cases of complexity emerge, which sometimes surprisingly lead even to polynomial time algorithms. We also cover recent successful approaches for more general classes of problems. Though no positive theoretical efficiency results are available, nor are they likely to ever be available, these seem to be the currently most successful and interesting approaches for solving practical problems. It is our belief that the study of algorithms motivated by theoretical considerations and those motivated by our desire to solve practical instances should and do inform one another. So it is with this viewpoint that we present the subject, and it is in this direction that we hope to spark further research.Comment: 57 pages. To appear in: M. J\"unger, T. Liebling, D. Naddef, G. Nemhauser, W. Pulleyblank, G. Reinelt, G. Rinaldi, and L. Wolsey (eds.), 50 Years of Integer Programming 1958--2008: The Early Years and State-of-the-Art Surveys, Springer-Verlag, 2009, ISBN 354068274

A Quantum Inspired Bi-level Optimization Algorithm for the First Responder Network Design Problem

In the aftermath of a sudden catastrophe, First Responders (FR) strive to promptly reach and rescue immobile victims. Simultaneously, other mobile individuals take roads to evacuate the affected region, or access shelters. The escalated traffic congestion significantly hinders critical FR operations if they share some of the same roads. A proposal from the Turkish Ministry of Transportation and Infrastructure being discussed for implementation is to allocate a subset of road segments for use by FRs only, mark them clearly, and pre-communicate them to the citizens. For the FR paths under consideration: (i) there should exist an FR path from designated entry points to each demand point in the network, and (ii) evacuees try to leave the network (through some exit points following the selfish routing principle) in the shortest time possible when they know that certain segments are not available to them. We develop a mixed integer non-linear programming formulation for this First Responder Network Design Problem (FRNDP). We solve FRNDP using a novel hybrid quantum-classical heuristic building on the Graver Augmented Multi-Seed Algorithm (GAMA). Using the flow-balance constraints for the FR and evacuee paths, we use a Quadratic Unconstrained Binary Optimization (QUBO) model to obtain a partial Graver Bases to move between the feasible solutions of FRNDP. To efficiently explore the solution space for high-quality solutions, we develop a novel bi-level nested GAMA within GAMA: GAGA. We test GAGA on random graph instances of various sizes and instances related to an expected Istanbul earthquake. Comparing GAGA against a state-of-the-art exact algorithm for traditional formulations, we find that GAGA offers a promising alternative approach. We hope our work encourages further study of quantum (inspired) algorithms to tackle complex optimization models from other application domains.Comment: 28 pages, 5 figure

Equivariant lattice bases

We study lattices in free abelian groups of infinite rank that are invariant under the action of the infinite symmetric group, with emphasis on finiteness of their equivariant bases. Our framework provides a new method for proving finiteness results in algebraic statistics. As an illustration, we show that every invariant lattice in $\mathbb{Z}^{(\mathbb{N}\times[c])}$, where $c\in\mathbb{N}$, has a finite equivariant Graver basis. This result generalizes and strengthens several finiteness results about Markov bases in the literature.Comment: 31 page