Abstract
There has been significant work recently on integer programs (IPs) \(\min \{c^\top x :Ax\le b,\,x\in \mathbb {Z}^n\}\) with a constraint marix A with bounded subdeterminants. This is motivated by a well-known conjecture claiming that, for any constant \(\varDelta \in \mathbb {Z}_{>0}\), \(\varDelta \)-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 \(\{-\varDelta , \dots , \varDelta \}\). Previous progress on this question, in particular for \(\varDelta =2\), relies on algorithms that solve an important special case, namely strictly \(\varDelta \)-modular IPs, which further restrict the \(n\times n\) minors of A to be within \(\{-\varDelta , 0, \varDelta \}\). Even for \(\varDelta =2\), such problems include well-known combinatorial optimization problems like the minimum odd/even cut problem. The conjecture remains open even for strictly \(\varDelta \)-modular IPs. Prior advances were restricted to prime \(\varDelta \), 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 \(\varDelta \)-modular IPs in strongly polynomial time if \(\varDelta \le 4\).
Funded through the Swiss National Science Foundation grants 200021_184622 and P500PT_206742, the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 817750), and the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – EXZ-2047/1 – 390685813.
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Notes
- 1.
A weaker variant of the conjecture claims efficient solvability of IPs with totally \(\varDelta \)-modular constraint matrices, where all subdeterminants are bounded by \(\varDelta \) in absolute value. The conjecture involving \(\varDelta \)-modular matrices implies the weaker variant. Indeed, an IP \(\min \{c^\top x :Ax \le b, x\in \mathbb {Z}^n\}\) with a totally \(\varDelta \)-modular constraint matrix can be reformulated as \(\min \{c^\top (x^+-x^-) :A(x^+-x^-)\le b, x^+, x^- \in \mathbb {Z}_{\ge 0}^n\}\). It is not hard to see that the constraint matrix of the new LP remains totally \(\varDelta \)-modular; moreover, it has full column rank because of the non-negativity constraints.
- 2.
To capture an MCCTU problem as a strictly \(\varDelta \)-modular IP, replace each congruency constraint \(\gamma _i^\top x \equiv r_i\pmod {m_i}\) by an equality constraint \(\gamma _i^\top x + m_i y_i = r\) with \(y_i\in \mathbb {Z}\). The corresponding constraint matrix is strictly \(\varDelta \)-modular for \(\varDelta =\prod _{i=1}^q m_i\).
- 3.
In fact, the proof in [24] claims a reduction to a submodular minimization problem, but shows the stronger one presented here.
- 4.
We recall that a lattice \(\mathcal {L}\subseteq 2^N\) is a set family such that for any \(A,B\in \mathcal {L}\), we have \(A\cap B, A\cup B\in \mathcal {L}\). We assume such a lattice to be given by a compact encoding in a directed acyclic graph H on the vertex set N such that \(X\subseteq N\) is an element of the lattice if and only if \(\delta _H^-(X)=\emptyset \) (cf. [20, Section 10.3]). Here, as usual, in a digraph \(G=(V,A)\) and for \(X\subseteq V\), we denote by \(\delta ^+(X)\) and \(\delta ^-(X)\) the arcs in A leaving and entering X, respectively. Moreover, we write \(\delta ^\pm (v){:}{=}\delta ^\pm (\{v\})\) for \(v\in V\).
- 5.
For simplicity, we use a notion of a 3-sum that allows one or both of \(ef^\top \) and \(gh^\top \) to be zero matrices. Typically, those cases would be called 2- and 1-sums, respectively.
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Nägele, M., Nöbel, C., Santiago, R., Zenklusen, R. (2023). Advances on Strictly \(\varDelta \)-Modular IPs. In: Del Pia, A., Kaibel, V. (eds) Integer Programming and Combinatorial Optimization. IPCO 2023. Lecture Notes in Computer Science, vol 13904. Springer, Cham. https://doi.org/10.1007/978-3-031-32726-1_28
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