Faster algorithms for sparse ILP and hypergraph multi-packing/multi-cover problems

Abstract

In our paper, we consider the following general problems: check feasibility, count the number of feasible solutions, find an optimal solution, and count the number of optimal solutions in ${{\,\mathrm{\mathcal {P}}\,}}\cap {{\,\mathrm{\mathbb {Z}}\,}}^n$, assuming that ${{\,\mathrm{\mathcal {P}}\,}}$ is a polyhedron, defined by systems $A x \le b$ or $Ax = b,\, x \ge 0$ with a sparse matrix A. We develop algorithms for these problems that outperform state-of-the-art ILP and counting algorithms on sparse instances with bounded elements in terms of the computational complexity. Assuming that the matrix A has bounded elements, our complexity bounds have the form $s^{O(n)}$, where s is the minimum between numbers of non-zeroes in columns and rows of A, respectively. For $s = o\bigl (\log n \bigr )$, this bound outperforms the state-of-the-art ILP feasibility complexity bound $(\log n)^{O(n)}$, due to Reis & Rothvoss (in: 2023 IEEE 64th Annual symposium on foundations of computer science (FOCS), IEEE, pp. 974–988). For $s = \phi ^{o(\log n)}$, where $\phi$ denotes the input bit-encoding length, it outperforms the state-of-the-art ILP counting complexity bound $\phi ^{O(n \log n)}$, due to Barvinok et al. (in: Proceedings of 1993 IEEE 34th annual foundations of computer science, pp. 566–572, https://doi.org/10.1109/SFCS.1993.366830, 1993), Dyer, Kannan (Math Oper Res 22(3):545–549, https://doi.org/10.1287/moor.22.3.545, 1997), Barvinok, Pommersheim (Algebr Combin 38:91–147, 1999), Barvinok (in: European Mathematical Society, ETH-Zentrum, Zurich, 2008). We use known and new methods to develop new exponential algorithms for Edge/Vertex Multi-Packing/Multi-Cover Problems on graphs and hypergraphs. This framework consists of many different problems, such as the Stable Multi-set, Vertex Multi-cover, Dominating Multi-set, Set Multi-cover, Multi-set Multi-cover, and Hypergraph Multi-matching problems, which are natural generalizations of the standard Stable Set, Vertex Cover, Dominating Set, Set Cover, and Maximum Matching problems.

Appendices

Appendix A: Proof of Theorem 10

Proof

Fix a parameter r and let ${{\,\mathrm{\mathcal {I}}\,}}= \{-r, \dots , r\}$. For $a \in {{\,\mathrm{\mathcal {A}}\,}}$, denote ${{\,\mathrm{\mathcal {H}}\,}}_a = \{x \in {{\,\mathrm{\mathcal {I}}\,}}^n :a^\top x = 0\}$, and let

$$\begin{aligned} {{\,\mathrm{\mathcal {N}}\,}}= {{\,\mathrm{\mathcal {I}}\,}}^n \setminus \bigcup \limits _{a \in {{\,\mathrm{\mathcal {A}}\,}}} {{\,\mathrm{\mathcal {H}}\,}}_a. \end{aligned}$$

Consider a polynomial $f :{{\,\mathrm{\mathbb {R}}\,}}^n \rightarrow {{\,\mathrm{\mathbb {R}}\,}}$ given by the formula

$$\begin{aligned} f(x) = \prod \limits _{a \in {{\,\mathrm{\mathcal {A}}\,}}} a^\top x. \end{aligned}$$

Clearly, f is a homogeneous polynomial with $\deg (f) = |{{{\,\mathrm{\mathcal {A}}\,}}}| = m$. Let ${{\,\mathrm{\mathcal {R}}\,}}= \{ x \in {{\,\mathrm{\mathcal {I}}\,}}^{n} :f(x) = 0\}$ be the roots of f inside ${{\,\mathrm{\mathcal {I}}\,}}^{n}$. Note that ${{\,\mathrm{\mathcal {R}}\,}}= \bigcup _{a \in {{\,\mathrm{\mathcal {A}}\,}}} {{\,\mathrm{\mathcal {H}}\,}}_a$ and ${{\,\mathrm{\mathcal {N}}\,}}= {{\,\mathrm{\mathcal {I}}\,}}^n \setminus {{\,\mathrm{\mathcal {R}}\,}}$. Due to the known Schwartz–Zippel lemma, $|{{{\,\mathrm{\mathcal {R}}\,}}}| \le \deg (f) \cdot |{{{\,\mathrm{\mathcal {I}}\,}}}|^{n-1} = m \cdot (2r+1)^{n-1}$. Therefore, $|{{{\,\mathrm{\mathcal {N}}\,}}}| \ge (2 r + 1)^n - m \cdot (2 r + 1)^{n-1} = (2 r + 1)^{n-1} \cdot (2 r + 1 - m)$, and consequently

$$\begin{aligned} \frac{|{{{\,\mathrm{\mathcal {N}}\,}}}|}{|{{{\,\mathrm{\mathcal {I}}\,}}^n}|} \ge \frac{2r +1 - m}{ 2 r + 1} = 1 - \frac{m}{2 r + 1}. \end{aligned}$$

Assign $r:= m$. After that, the previous inequality becomes $\frac{|{{{\,\mathrm{\mathcal {N}}\,}}}|}{|{{{\,\mathrm{\mathcal {I}}\,}}^n}|} > 1/2$. Now, to find a vector z that can satisfy the claims

• 1. $a^\top z \not = 0$, for any $a \in {{\,\mathrm{\mathcal {A}}\,}}$;

• 2. $\Vert z\Vert _{\infty } \le m$;

we uniformly sample points z inside ${{\,\mathrm{\mathcal {I}}\,}}^n$. With a probability at least 1/2 it will satisfy the first claim. The second claim is satisfied automatically. Therefore, the expected number of sampling iterations is O(1). The arithmetic complexity of a single iteration is clearly bounded by $O(n \cdot m)$, which completes the proof. $\square$

Appendix B: Proof of the Lemma 2

1.1 A recurrent formula for the generating function of a group polyhedron

Let ${{\,\mathrm{\mathcal {G}}\,}}$ be an arbitrary finite Abelian group and $g_1,\dots ,g_n \in {{\,\mathrm{\mathcal {G}}\,}}$. Let additionally $r_i = |{\langle g_i \rangle }|$ be the order of $g_i$, for $i \in \{1, \dots , n\}$, and $r_{\max } = \max _{i} \{r_i\}$. For $g' \in {{\,\mathrm{\mathcal {G}}\,}}$ and $k \in \{1, \dots , n\}$, let ${{\,\mathrm{\mathcal {M}}\,}}(k,g')$ be the solutions set of the following system:

$$\begin{aligned} {\left\{ \begin{array}{ll} \sum \limits _{i = 1}^k x_i g_i = g'\\ x \in {{\,\mathrm{\mathbb {Z}}\,}}_{\ge 0}^k. \end{array}\right. } \end{aligned}$$

(B1)

Consider the formal power series ${{\,\mathrm{\mathfrak {f}}\,}}_k(g';{{\,\mathrm{\textbf{x}}\,}}) = \sum \limits _{z \in {{\,\mathrm{\mathcal {M}}\,}}(k,g') \cap {{\,\mathrm{\mathbb {Z}}\,}}^k} {{\,\mathrm{\textbf{x}}\,}}^z.$ For $k = 1$, we clearly have

$$\begin{aligned} {{\,\mathrm{\mathfrak {f}}\,}}_1(g';{{\,\mathrm{\textbf{x}}\,}}) = \frac{x_1^s}{1 - x_1^{r_1}},\quad \text {where } s = \min \{x_1 \in {{\,\mathrm{\mathbb {Z}}\,}}_{\ge 0} :x_1 g_1 = g'\}. \end{aligned}$$

(B2)

If such s does not exist, we put ${{\,\mathrm{\mathfrak {f}}\,}}_1(g';{{\,\mathrm{\textbf{x}}\,}}):= 0$. Clearly, the series ${{\,\mathrm{\mathfrak {f}}\,}}_1(g';{{\,\mathrm{\textbf{x}}\,}})$ absolutely converges to the corresponding right-hand side function for any $x_1 \in {{\,\mathrm{\mathbb {C}}\,}}$ with $|{x_1^{r_1}}| < 1$. For any value of $x_k \in {{\,\mathrm{\mathbb {Z}}\,}}_{\ge 0}$, the system (B1) can be rewritten as

$$\begin{aligned} {\left\{ \begin{array}{ll} \sum \limits _{i = 1}^{k-1} x_i g_i = g' - x_k g_k\\ x \in {{\,\mathrm{\mathbb {Z}}\,}}_{\ge 0}^{k-1}. \end{array}\right. } \end{aligned}$$

Hence, for $k \ge 1$, we have

$$\begin{aligned} {{\,\mathrm{\mathfrak {f}}\,}}_k(g';{{\,\mathrm{\textbf{x}}\,}})= & {} \frac{ {{\,\mathrm{\mathfrak {f}}\,}}_{k-1}(g';{{\,\mathrm{\textbf{x}}\,}}) + x_{k} \cdot {{\,\mathrm{\mathfrak {f}}\,}}_{k-1}(g' - g_k;{{\,\mathrm{\textbf{x}}\,}}) + \dots + x_{k}^{r_k - 1} \cdot {{\,\mathrm{\mathfrak {f}}\,}}_{k-1}(g' - g_k \cdot (r_k - 1);{{\,\mathrm{\textbf{x}}\,}})}{1 - x_k^{r_k}} \nonumber \\= & {} \frac{1}{1 - x_{k}^{r_k}} \cdot \sum _{i = 0}^{r_k - 1} x_k^i \cdot {{\,\mathrm{\mathfrak {f}}\,}}_{k-1}(g' - i \cdot g_k;{{\,\mathrm{\textbf{x}}\,}}). \end{aligned}$$

(B3)

$$\begin{aligned} \text {Consequently,} \quad {{\,\mathrm{\mathfrak {f}}\,}}_k(g';{{\,\mathrm{\textbf{x}}\,}}) = \frac{\sum \nolimits _{i_1 = 0}^{r_1-1}\dots \sum \nolimits _{i_k = 0}^{r_k-1} \epsilon _{i_1,\dots ,i_k} x_1^{i_1} \dots x_k^{i_k}}{(1 - x_1^{r_1})(1 - x_2^{r_2})\dots (1 - x_k^{r_k})}, \end{aligned}$$

(B4)

where the numerator is a polynomial with coefficients $\epsilon _{i_1,\dots ,i_k} \in \{0,1\}$ and degree at most $(r_1 - 1) \dots (r_k - 1)$. Since a sum of absolutely convergent series is absolutely convergent, it follows from the induction principle that the series ${{\,\mathrm{\mathfrak {f}}\,}}_k(g';{{\,\mathrm{\textbf{x}}\,}})$ absolutely converges to the right-hand side of the formula (B4) when $|{x_i^{r_i}}| < 1$ for each $i \in \{1, \dots , k\}$.

1.2 The group ${{\,\mathrm{\mathcal {G}}\,}}$, induced by the SNF, of A

Recall that $A \in {{\,\mathrm{\mathbb {Z}}\,}}^{n \times n}$, $0 < \Delta = |{\det (A)}|$, and $h_1, \dots , h_n$ are the columns of $A^*:= \Delta \cdot A^{-1}$. The vector $c \in {{\,\mathrm{\mathbb {Z}}\,}}^n$ is chosen, such that $\langle c, h_i \rangle > 0$, for each $i \in \{1, \dots , n\}$, and $\psi = \max _i |{ \langle c, h_i \rangle }|$. Additionally, let $S = P A Q$ be the SNF of A, where $P,Q \in {{\,\mathrm{\mathbb {Z}}\,}}^{n \times n}$ are unimodular, and $\sigma = S_{n n}$.

Let us consider the sets ${{\,\mathrm{\mathcal {M}}\,}}(k,g')$, induced by the group system (B1) with ${{\,\mathrm{\mathcal {G}}\,}}= {{\,\mathrm{\mathbb {Z}}\,}}^{n}/S{{\,\mathrm{\mathbb {Z}}\,}}^n$ and $g_i = P_{* i} \bmod S{{\,\mathrm{\mathbb {Z}}\,}}^n$. Note that $r_i \le \sigma$, for each $i \in \{1, \dots , n\}$. Additionally, let us consider a new formal series, defined by

$$\begin{aligned} {\hat{{{\,\mathrm{\mathfrak {f}}\,}}}}_k(g';{{\,\mathrm{\textbf{x}}\,}}) = \sum \limits _{z \in {{\,\mathrm{\mathcal {M}}\,}}(k,g') \cap {{\,\mathrm{\mathbb {Z}}\,}}^k} {{\,\mathrm{\textbf{x}}\,}}^{-\sum \limits _{i=1}^k h_i z_i}, \end{aligned}$$

which can be derived from the series ${{\,\mathrm{\mathfrak {f}}\,}}_k(g';{{\,\mathrm{\textbf{x}}\,}})$ by the monomial substitution $x_i \rightarrow {{\,\mathrm{\textbf{x}}\,}}^{-h_i}$. For ${\hat{{{\,\mathrm{\mathfrak {f}}\,}}}}_k(g';{{\,\mathrm{\textbf{x}}\,}})$, the formulae (B2), (B3) and (B4) become:

$$\begin{aligned}{} & {} {\hat{{{\,\mathrm{\mathfrak {f}}\,}}}}_1(g'; {{\,\mathrm{\textbf{x}}\,}}) = \frac{{{\,\mathrm{\textbf{x}}\,}}^{- s h_1}}{1 - {{\,\mathrm{\textbf{x}}\,}}^{-r_1 h_1}},\quad \text {where } s = \min \{y_1 \in {{\,\mathrm{\mathbb {Z}}\,}}_{\ge 0} :y_1 g_1 = g' \},\end{aligned}$$

(B5)

$$\begin{aligned}{} & {} {\hat{{{\,\mathrm{\mathfrak {f}}\,}}}}_k(g';{{\,\mathrm{\textbf{x}}\,}}) = \frac{1}{1 - {{\,\mathrm{\textbf{x}}\,}}^{-r_k h_k}} \cdot \sum \limits \limits _{i = 0}^{r_k-1} {{\,\mathrm{\textbf{x}}\,}}^{- i h_k} \cdot \hat{{\,\mathrm{\mathfrak {f}}\,}}_{k-1}(g' - i \cdot g_k; {{\,\mathrm{\textbf{x}}\,}}) \text { and} \end{aligned}$$

(B6)

$$\begin{aligned}{} & {} {\hat{{{\,\mathrm{\mathfrak {f}}\,}}}}_k(g';{{\,\mathrm{\textbf{x}}\,}}) = \frac{\sum \nolimits _{i_1 = 0}^{r_1-1}\dots \sum \nolimits _{i_k = 0}^{r_k-1} \epsilon _{i_1,\dots ,i_k} {{\,\mathrm{\textbf{x}}\,}}^{-(i_1 h_1 + \dots + i_k h_k)}}{(1 - {{\,\mathrm{\textbf{x}}\,}}^{-r_1 h_1})(1 - {{\,\mathrm{\textbf{x}}\,}}^{-r_2 h_2}) \dots (1 - {{\,\mathrm{\textbf{x}}\,}}^{-r_k h_k})}. \end{aligned}$$

(B7)

Clearly, here the absolute convergence takes place for the values of ${{\,\mathrm{\textbf{x}}\,}}$ with $|{{{\,\mathrm{\textbf{x}}\,}}^{- r_i h_i}}| < 1$, for each $i \in \{1, \dots , k\}$. Let us consider now the formal series

$$\begin{aligned} {{\,\mathrm{\mathfrak {g}}\,}}_k(g'; \tau ) = \sum \limits _{y \in {{\,\mathrm{\mathcal {M}}\,}}_k(g')} e^{- \tau \cdot \langle c, \sum _{i=1}^k h_i y_i \rangle }, \end{aligned}$$

which can be derived from ${\hat{{{\,\mathrm{\mathfrak {f}}\,}}}}_k(g';{{\,\mathrm{\textbf{x}}\,}})$ by the substitution $x_i \rightarrow e^{\tau \cdot c_i}$. For ${{\,\mathrm{\mathfrak {g}}\,}}_k(g'; \tau )$, the formulae (B5), (B6), and (B7) become:

$$\begin{aligned}{} & {} {{\,\mathrm{\mathfrak {g}}\,}}_1(g'; \tau ) = \frac{e^{- \langle c, s h_1 \rangle \cdot \tau }}{1 - e^{- \langle c, r_1 h_1 \rangle \cdot \tau }}, \end{aligned}$$

(B8)

$$\begin{aligned}{} & {} {{\,\mathrm{\mathfrak {g}}\,}}_k(g';\tau ) = \frac{1}{1 - e^{-\langle c, r_k h_k \rangle \cdot \tau }} \cdot \sum \limits _{i = 0}^{r_k-1} e^{- \langle c, i h_k \rangle \cdot \tau } \cdot {{\,\mathrm{\mathfrak {g}}\,}}_{k-1}(g' - i \cdot g_k; \tau ), \end{aligned}$$

(B9)

$$\begin{aligned}{} & {} {{\,\mathrm{\mathfrak {g}}\,}}_k(g';\tau ) = \frac{\sum \nolimits _{i_1 = 0}^{r_1-1}\dots \sum \nolimits _{i_k = 0}^{r_k-1} \epsilon _{i_1,\dots ,i_k} e^{-\langle c, i_1 h_1 + \dots + i_k h_k \rangle \cdot \tau } }{\bigl (1 - e^{-\langle c, r_1 h_1 \rangle \cdot \tau }\bigr )\bigl (1 - e^{-\langle c, r_2 h_2 \rangle \cdot \tau }\bigr ) \dots \bigl (1 - e^{- \langle c, r_k h_k \rangle \cdot \tau }\bigr )}. \end{aligned}$$

(B10)

Since the series ${\hat{{{\,\mathrm{\mathfrak {f}}\,}}}}_k(g';{{\,\mathrm{\textbf{x}}\,}})$ absolutely converges, when $|{{{\,\mathrm{\textbf{x}}\,}}^{- r_i h_i}}| < 1$, for each $i \in \{1, \dots , k\}$, the new one converges, for any $\tau > 0$. Since $\langle c,h_i \rangle \in {{\,\mathrm{\mathbb {Z}}\,}}_{\not =0}$, for each i, the number of terms $e^{-\langle c, \cdot \rangle \cdot \tau }$ is bounded by $2 \cdot k \cdot \sigma \cdot \psi + 1$. So, after combining similar terms, the numerator’s length becomes $O(k \cdot \sigma \cdot \psi )$. In other words, there exist coefficients $\epsilon _i \in {{\,\mathrm{\mathbb {Z}}\,}}_{\ge 0}$, such that

$$\begin{aligned} {{\,\mathrm{\mathfrak {g}}\,}}_k(g';\tau ) = \frac{\sum \nolimits _{i = - k \cdot \sigma \cdot \psi }^{k \cdot \sigma \cdot \psi } \epsilon _i \cdot e^{- i \cdot \tau }}{\bigl (1 - e^{-\langle c, r_1 \cdot h_1 \rangle \tau }\bigr )\bigl (1 - e^{-\langle c, r_2 h_2 \rangle \cdot \tau }\bigr ) \dots \bigl (1 - e^{- \langle c, r_k h_k \rangle \cdot \tau }\bigr )}. \end{aligned}$$

(B11)

The formulae (B8), (B9), and (B11) coincide with the desired formulae (17), (18), and (19). So, the proof of Lemma 2 is finished.