The difference between an ordinary and symbolic power of a homogeneous ideal in a polynomial ring is an important and challenging problem underlying many projects in commutative algebra and algebraic geometry. When restricting to monomial ideals, insight has been gained by exploiting techniques from combinatorics and graph theory. Recall that, by J. Herzog et al. [Adv. Math. 210, No. 1, 304–322 (2007; Zbl 1112.13006)], if \(I\) is a monomial ideal in \(R = K[x_1, \ldots, x_n]\) with \(K\) a field, then the \(k\)th symbolic power of \(I\) is \[I^{(k)} = \bigcap_{P \in \mathrm{Min}(I)} I^k_P.\] Also, we let \(\overline{I^k}\) denote the integral closure of the ordinary power \(I^k\).
In this paper, the authors use combinatorial optimization to connect ordinary powers of \(I\), symbolic powers of \(I\), and integral closures of \(I\) with certain matching numbers, covering numbers and their fractional versions. Four optimal values of programming problems are considered as follows. Let \(x^{\mathbf{a_1}}, \ldots, x^{\mathbf{a_m}}\) be the minimal monomial generators of \(I\) and \(M\) be the matrix whose \(i\)th column is \(\mathbf{a_i}\) be the “exponent matrix” of \(I\). We have \begin{align*} \nu_{\mathbf a}(I) & = \nu_{\mathbf a}(M) = \max \{ \mathbf{1}^m \cdot \mathbf{ y} \mid \mathbf{y} \in \mathbb N^m, M \cdot\mathbf{y} \leq \mathbf{ a} \} \\ \nu_{\mathbf a}^*(I) & = \nu_{\mathbf a}^*(M) = \max \{\mathbf{1}^m \cdot \mathbf{y} \mid \mathbf{y} \in \mathbb R_{\geq 0}^m, M \cdot \mathbf{y} \leq\mathbf{a} \} \\ \tau_{\mathbf a}(I) & = \tau_{\mathbf a}(M) = \min \{\mathbf{a} \cdot\mathbf{z} \mid\mathbf{ z} \in \mathbb N^n, M^T \cdot\mathbf{z} \geq\mathbf{1}^m \} \\ \tau_{\mathbf a}^*(I) & = \tau_{\mathbf a}^*(M) = \min \{\mathbf{a} \cdot \mathbf{z} \mid\mathbf{z} \in \mathbb R_{\geq 0}^n, M^T \cdot\mathbf{z} \geq\mathbf{1}^m \}. \end{align*}
The authors show that these optimal values can be used to gain information about \(I^k, \overline{I^k}\), and \(I^{(k)}\). Much of the work relies on the following proposition:
Proposition 1. Let \(I \subseteq R\) be a monomial ideal. Then

1.

\(x^{\mathbf a} \in I^k\) if and only if \(\nu_{\mathbf a}(I) \geq k\)

2.

\(x^{\mathbf a} \in \overline{I^k}\) if and only if \(\nu^*_{\mathbf a}(I) \geq k\)

3.

If \(I\) is square-free, then \(x^{\mathbf a} \in I^{(k)}\) if and only if \(\tau_{\mathbf a}(I) \geq k\).

In the special case that \(I\) is a square-free monomial ideal, the above optimal values are equal to well-known invariants from graph theory. In this case, \(I\) is naturally the edge ideal of a hypergraph \(\mathcal H\). The authors relate the optimal values via the following proposition:
Proposition. If \(I\) is a square-free monomial ideal, and hence an edge ideal of a hypergraph \(\mathcal H\), then \begin{align*} \nu_{\mathbf a}(I) & = \text{matching number of \(\mathcal H^{\mathbf a}\)} \\ \nu^*_{\mathbf a}(I) & = \text{fractional matching number of \(\mathcal H^{\mathbf a}\)} \\ \tau_{\mathbf a}(I) & = \text{covering number of \(\mathcal H^{\mathbf a}\)} \\ \tau^*_{\mathbf a}(I) & = \text{fractional covering of \(\mathcal H^{\mathbf a}\)} \end{align*} where \(\mathcal H^{\mathbf a}\) is a hypergraph which is a parallelization of \(\mathcal H\).
It is known that \[\nu_{\mathbf a}(I) \leq \nu_{\mathbf a}^*(I) = \tau_{\mathbf a}^*(I) \leq \tau_{\mathbf a}(I).\] The “gaps” between these invariants are well-understood. The authors apply known estimates of these gaps to Proposition 1 to obtain new containments between ordinary, symbolic, and integral powers as follows:
Theorem. Let \(I\) be a square-free monomial ideal and \(r\) be the maximum degree of the minimal generators of \(I\). For any \(k \geq 1\) we have

1.

\(\overline{I^{(r-1)(k-1)}+\lceil \frac{k}{r} \rceil} \subseteq I^k\);

2.

\(I^{(\lceil(1+\frac{1}{2} + \cdots + \frac{1}{r})k\rceil)} \subseteq \overline{I^k}\);

3.

\(I^{(rk-r+1)} \subseteq I^k\).

Approaching the relationship between combinatorial optimization and containments between various powers of ideals through a different lens, the authors use known membership criteria between \(I^k, I^{(k)}\) and \(\overline{I^k}\) for a monomial ideal \(I\) to obtain information about the gaps between the optimal values \(\nu_{\mathbf a}(I), \nu_{\mathbf a}^*(I), \tau_{\mathbf a}(I), \tau_{\mathbf a}^*(I)\).
Theorem. If \(M\) is an \(n \times m\) matrix of non-negative integers, then for all \(\mathbf{a} \in \mathbb N^n\) we have \[\nu_{\mathbf a}^*(M) < \nu_{\mathbf a}(M) + \min\{m,n\}.\]
Theorem. Let \(M\) be the incidence matrix of a simple hypergraph \(\mathcal H\). Let \(h\) be the maximal size of a minimal cover of \(\mathcal H\). Then for all \(\mathbf{a} \in \mathbb N^n\), \[\tau_{\mathbf a}(M) \leq h \nu_{\mathbf a}(M).\]
The paper contains further connections between algebra and combinatorics by considering properties of hypergraphs such as Mengerian and Fulkersionian. In particular, the authors demonstrate that their approaches give immediate proofs of previously known equalities [I. Gitler et al., Beitr. Algebra Geom. 48, No. 1, 141–150 (2007; Zbl 1114.13007); J. Herzog et al., Trans. Am. Math. Soc. 360, No. 12, 6231–6249 (2008; Zbl 1155.13003); Ngo Viet Trung, Vietnam J. Math. 34, No. 4, 489–494 (2006; Zbl 1112.13011)] for powers of edge ideals of hypergraphs in these cases.
Finally, the paper closes with a translation of the Conforti-Cornuéjols Conjecture that a hypergraph with packing property is Mengerian via the following conjecture from [N. V. Trung, “Square-free monomial ideals and hypergraphs”, in: Notes for the Workshop on Integral Closure, Multiplier Ideals and Cores. AIM (2006), https://www.aimath.org/WWN/integralclosure/Trung.pdf].
Conjecture. Let \(I\) be a square-free monomial ideal. Let mon-grade\((I)\) denote the maximal length of a regular sequence of monomials in \(I\), If \[\operatorname{mon-grade}(J) = \operatorname{ht}(J)\] for all monomial ideals \(J\) obtained from \(I\) by setting some variables equal to 0, 1, then \(I\) is a normal ideal.