Automated Discovery of Branching Rules with Optimal Complexity for the Maximum Independent Set Problem

TL;DR

This work automates the discovery of optimal branching rules for the Maximum Independent Set problem within a branch-and-reduce framework. By formulating local subgraph constraints as reduced $\alpha$-tensors and translating rule selection into a weighted minimum set covering problem, it yields provably optimal branching factors $\gamma$ and significantly fewer branches than expert-designed rules. The methodology relies on integer programming (with LP relaxations as fast approximations) to generate on-the-fly rules, and demonstrates state-of-the-art performance on 3-regular graphs with average complexities around $O(1.0441^n)$. The approach is general, extensible to other combinatorial problems, and supported by an open-source Julia implementation.

Abstract

The branching algorithm is a fundamental technique for designing fast exponential-time algorithms to solve combinatorial optimization problems exactly. It divides the entire solution space into independent search branches using predetermined branching rules, and ignores the search on suboptimal branches to reduce the time complexity. The complexity of a branching algorithm is primarily determined by the branching rules it employs, which are often designed by human experts. In this paper, we show how to automate this process with a focus on the maximum independent set problem. The main contribution is an algorithm that efficiently generate optimal branching rules for a given sub-graph with tens of vertices. Its efficiency enables us to generate the branching rules on-the-fly, which is provably optimal and significantly reduces the number of branches compared to existing methods that rely on expert-designed branching rules. Numerical experiment on 3-regular graphs shows an average complexity of O(1.0441^n) can be achieved, better than any previous methods.

Figures (8)

• Figure 1: Applying a MIS branching strategy on a sub-graph $R$ of the parent graph $G$. (a) A sub-graph $R$, with boundary vertices $\partial R = \{a, b, c\}$, where the dashed lines indicate their connections to vertices in the environment $G \backslash R$. (b) The left side shows the boundary-grouped MISs $\mathcal{S}_R = \{S_{000}, S_{001}, S_{010}, S_{111}\}$ of the sub-graph $R$. Each row represents a set $S_{\boldsymbol{s}_{\partial R}}$ of relevant configurations with the boundary configuration $\boldsymbol{s}_{\partial R}$. The right side shows the clauses in the resulting optimal branching $\mathcal{D} = \delta(R)$.
• Figure 1: An example subgraph, where the $j$, $k$, $l$ are the boundary vertices. It satisfies the condition of domination rule, where $N[v] \subseteq N[w]$.
• Figure 1: The average number of branches produced by various algorithms on different graphs as functions of the graph size.
• Figure 1: An example of the fixed point iteration of $\gamma$ in \ref{['alg:opt_branching']}, the inset subplot shows the iteration near the fixed point. The blue line represents the value of $f(\gamma)$, the black line represents $y = x$, and the red dashed lines and points represent the iteration steps.
• Figure 1: The largest number of branches produced by various algorithms on different graphs as functions of the graph size.

Theorems & Definitions (18)

• Definition 2.1: Branching strategy for bitstring searching
• Definition 2.2: Branching complexity of MIS
• Definition 3.1: $\alpha$-tensor Liu2024
• Definition 3.2: reduced $\alpha$-tensor
• Definition 3.3: boundary-grouped MISs
• Theorem 3.4
• Proof 1
• Definition 3.5: Valid branching rule
• Definition 3.6: Weighted minimum set covering problem
• Theorem 3.7