A New Contraction Technique with Applications to Congruency-Constrained Cuts

Abstract

Minimum cut problems are among the most classical problems in Combinatorial Optimization and are used in a wide set of applications. Some of the best-known efficiently solvable variants include global minimum cuts, minimum s-t cuts, and minimum odd cuts in undirected graphs. We study a problem class that can be seen to generalize the above variants, namely finding congruency-constrained minimum cuts, i.e., we consider cuts whose number of vertices is congruent to r modulo m, for some integers r and m. Apart from being a natural generalization of odd cuts, congruency-constrained minimum cuts exhibit an interesting link to a long-standing open problem in Integer Programming, namely whether integer programs described by an integer constraint matrix with bounded subdeterminants can be solved efficiently.

We develop a new contraction technique inspired by Karger’s celebrated contraction algorithm for minimum cuts, that, together with further insights, leads to a polynomial time randomized approximation scheme for congruency-constrained minimum cuts for any constant modulus m. Instead of contracting edges of the original graph, we use splitting-off techniques to create an auxiliary graph on a smaller vertex set, which is used for performing random edge contractions. This way, a well-structured distribution of candidate pairs of vertices to be contracted is obtained, where the involved pairs are generally not connected by an edge. As a byproduct, our technique reveals new structural insights into near-minimum odd cuts, and, more generally, near-minimum congruency-constrained cuts.

R. Zenklusen—Supported by Swiss National Science Foundation grant 200021_165866.

A Missing Proofs

1.1 A.1 Proof of Lemma 1

Let $\mathcal {R}(\beta ,q)=\{R_1,R_2,\ldots ,R_{2m_q-1}\}$ with distinct $u_i\in R_i$ for all $i\in [2m_q-1]$ as given in item (ii) of Definition 1. We distinguish two cases: Either, there are $m_q$ many vertices among the $u_i$ with $u_i\in C'$, or there are $m_q$ many with $u_i\not \in C'$.

In the first case, assume w.l.o.g. that $u_1,\ldots ,u_{m_q}\in C'$, and let for $k\in \{0,\ldots ,m_q-1\}$. We show that for some k, the set has the desired properties. First observe that all $C_k$ are cuts, as $C_0=C'$ is a cut, and $u_{1}\notin C_k \ni u_{m_q}$ for $k\in [m_q-1]$. Moreover, $k\leqslant m_q-1$ implies

$$\begin{aligned} w(\delta (C_k)) \leqslant w(\delta (C')) + {\textstyle \sum }_{i=1}^k w(\delta (R_i)) \leqslant w(\delta (C'))+(m_q-1)\beta . \end{aligned}$$

(5)

Using $m_q=\frac{m}{\gcd (m,q)}=\frac{m}{m'}$, we see that (5) is precisely point (i) of Lemma 1 for $C_k$. To conclude, we show that there exists k such that $C_k$ satisfies $\gamma (C_k)\equiv r\pmod {m}$, i.e., point (ii). Using that $\gamma (u)\equiv 0\pmod {m}$ for all $u\in R_i\setminus \{u_i\}$, and $u_i\in C'$ for all $i\in [m_q]$, we obtain $\gamma (C_k)\equiv \gamma (C')-\sum _{i=1}^k \gamma (u_i)\equiv \gamma (C')-kq\pmod {m}$. It thus suffices to find $k\in \{0,\ldots ,m_q-1\}$ with $\gamma (C')-kq\equiv r\pmod {m}$, or equivalently,

$$\begin{aligned} kq \equiv \gamma (C')-r \pmod {m}. \end{aligned}$$

(6)

By assumption, $\gamma (C')-r\equiv 0\!\pmod {m'}$, so , and because $m'=\gcd (m,q)$. Dividing (6) by $m'$, we obtain the equivalent equation $kq'\equiv r'\pmod {m_q}$, which has a solution $k\in \{0,\ldots ,m_q-1\}$ as $\gcd (q',m_q)=1$.

The second case, i.e., $u_1,\ldots ,u_{m_q}\notin C'$, is similar: $C_k$ always is a cut because $C_0=C'$ is a cut, and $u_1\in C_k\not \ni u_{m_q}$ for $k\ge 1$. Equation (5) remains true and implies point (i). For point (ii), we use $\gamma (C_k)\equiv \gamma (C')+\sum _{i=1}^k\gamma (u_i)$, and the above analysis results in $kq'\equiv -r'\pmod {m_q}$, admitting a solution $k\in \{0,\ldots ,m_q-1\}$.

Finally, given $\mathcal {R}(\beta ,q)$ and $C'$, checking which of the two cases applies can be done in polynomial time, as well as solving the respective congruence equation for k. Thus, a cut C with the desired properties can be obtained efficiently.   $\square$

1.2 A.2 Sketch of proof of Theorem 7

As indicated earlier, Theorem 7 is a consequence of splitting-off techniques from Graph Theory, a fundamental tool dating back to the ’70s [14,15,16]. Typically, a graph is modified by repeatedly splitting off two edges from a vertex v, i.e., replacing two non-parallel edges $\{v,x\}$ and $\{v,y\}$ by a new edge $\{x,y\}$, or deleting two parallel edges incident to v. Denoting for a graph $G=(V,E)$ and $A,B\subseteq V$, Lovász proved the following.

Theorem 8

(Lovász [14]). Let $G=(V,E)$ be Eulerian, let $Q\subseteq V$, and let $v\in V\setminus Q$. For every edge $\{v,x\}\in E$, there exists another edge $\{v,y\}\in E$ such that the graph $G'$ arising from G by splitting off $\{v,x\}$ and $\{v,y\}$ from v satisfies

$$\begin{aligned} \mu _G(\{q\},Q\setminus \{q\}) = \mu _{G'}(\{q\},Q\setminus \{q\})\qquad \forall q\in Q. \end{aligned}$$

Iterative applications of Theorem 8 for fixed $Q\subseteq V$ and $v\in V\setminus Q$ result in a new graph on the vertex set $V\setminus v$ only, without changing the value of minimum cuts separating a single vertex q from $Q\setminus \{q\}$, for all $q\in Q$. We aim for a generalization of this statement to a weighted setting, where the graph $G=(V,E)$ has edge weights $w:E\rightarrow \mathbb {R}_{\geqslant 0}$, a splitting operation consists of decreasing the weight on two edges $\{v,x\}$ and $\{v,y\}$ by some $\varepsilon >0$ while increasing the weight on the edge $\{x,y\}$ by $\varepsilon$, and we want the weighted cut values $\mu _{G,w}(\{q\},Q\setminus \{q\})$ to be invariant. We claim that this is achieved by Algorithm 2. We highlight that efficient weighted versions of other splitting-off results (than Theorem 8) have already been considered by Frank [7], and our method is heavily inspired by Frank’s approach.

In the inner for loop in Algorithm 2, if $q\in \{x,y\}$, then $c_2^q = \mu _{G,w}(\{q,v\},(Q\setminus \{q\})\cup \{x,y\})$ is the value of an infeasible cut problem (because both arguments of $\mu _{G,w}$ contain q), which we interpret as $\infty$.

In each iteration of the outer for loop in Algorithm 2, we split off $\varepsilon \geqslant 0$ from $\{v,x\}$ and $\{v,y\}$, with $\varepsilon$ chosen maximal so that all weights remain non-negative and the connectivities of interest are preserved. This choice of $\varepsilon$ implies that once the outer for loop terminated, there is no pair of edges incident to v from which a positive weight can be split off. Uniformly scaling all weights of this remaining graph to even integral weights (which we interpret as edge multiplicities) and employing Theorem 8, we can prove that there can only be a single edge with positive weight incident to v in the remaining graph, which we can thus safely delete without affecting connectivities within $V\setminus \{v\}$.

The following lemma summarizes the guarantees that we thereby obtain for Algorithm 2. A formal proof is deferred to a long version of this paper.

Lemma 2

Let $G=(V,E)$ be a graph with edge weights $w:E\rightarrow \mathbb {R}_{\geqslant 0}$, let $Q\subsetneq V$ and $v\in V\setminus Q$. On this input, Algorithm 2 returns, in running time dominated by $\mathcal {O}(|V|^3)$ many minimum cut computations in (G, w), a graph $H=(V\setminus \{v\}, F)$ with edge weights $w_{H}:F\rightarrow \mathbb {R}$ such that

• (i) $\mu _{H,w_{H}}(\{q\}, Q\setminus \{q\}) = \mu _{G,w}(\{q\}, Q\setminus \{q\})$ for all $q\in Q$, and

• (ii) $w_{H}(\delta _{H}(C\setminus \{v\})) \leqslant w(\delta _G(C))$ for all $C\subseteq V$.

Applying Lemma 2 iteratively for all $v\in V\setminus Q$ immediately yields Theorem 7.