Graph Partitioning With Limited Moves

TL;DR

The paper studies the r-move $k$-partitioning problem, a budgeted variant of the Multiway cut that starts from an initial partition and seeks a lower-cut solution by moving at most $r$ non-terminal nodes. It develops multiple algorithmic strategies: a polynomial-time $3(r+1)$-approximation via an extended CKR LP, an $FPTAS$ for constant $r$, and a constant-factor bicriteria approach that trades off the number of moves against cut quality, plus specialized results for $k=2$. The work establishes $W[1]$-hardness in parameter $r$ and proves an integrality gap of at least $r+1$ for the underlying LP, alongside reductions from the Densest $r$-Subgraph to demonstrate hardness both with and without terminals. Numerically, the proposed methods show strong performance on synthetic and real-like graphs, suggesting practical utility for network design, local search, and learning-augmented optimization contexts where limited reconfiguration is feasible.

Abstract

In many real world networks, there already exists a (not necessarily optimal) $k$-partitioning of the network. Oftentimes, one aims to find a $k$-partitioning with a smaller cut value for such networks by moving only a few nodes across partitions. The number of nodes that can be moved across partitions is often a constraint forced by budgetary limitations. Motivated by such real-world applications, we introduce and study the $r$-move $k$-partitioning~problem, a natural variant of the Multiway cut problem. Given a graph, a set of $k$ terminals and an initial partitioning of the graph, the $r$-move $k$-partitioning~problem aims to find a $k$-partitioning with the minimum-weighted cut among all the $k$-partitionings that can be obtained by moving at most $r$ non-terminal nodes to partitions different from their initial ones. Our main result is a polynomial time $3(r+1)$ approximation algorithm for this problem. We further show that this problem is $W[1]$-hard, and give an FPTAS for when $r$ is a small constant.

Figures (5)

• Figure 1: The example graph for the proof of Lemma \ref{['lem:integrality_gap']}.
• Figure 2: Reduction graph $G'$ created from a densest $r$-subgraph instance $G$. The solid-line edges do not depend on $G$. For each edge $e=(u,v)$ in $G$, nodes $u$ and $v$ in $A$ are connected to a node $e$ in $B$.
• Figure 3: Reduction graph $G'$ created from a densest $r$-subgraph instance $G$.
• Figure 4: Performance of rounding (Algorithm \ref{['alg:r_approx']}) and greedy algorithms with respect to the solution of LP \ref{['tab:maxmove-lp']} for the explained stochastic block graphs.
• Figure 5: Performance of the FPTAS with two different values of $\alpha$ and greedy algorithm with respect to the solution of LP \ref{['tab:maxmove-lp']} for the email-Eu-core network graph.

Theorems & Definitions (29)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Definition 2.1: The $r$-move $k$-partitioning problem
• Lemma 2.1
• Lemma 2.2
• Lemma 4.1
• Lemma 4.2
• Lemma 4.3