[Paper Review] Analysis and Optimization of Graph Decompositions by Lifted Multicuts

Andrea Horňáková, Jan-Hendrik Lange+1 more|arXiv (Cornell University)|Mar 12, 2015

Advanced Graph Theory Research15 references13 citations

TL;DR

This paper introduces lifted multicuts as a unified framework to analyze and optimize graph decompositions, enabling efficient handling of must-join and must-cut constraints, metric-based comparison of clusterings, and exact optimization via branch-and-cut. The key contribution is a polyhedral characterization of lifted multicut polytopes with efficient separation procedures, enabling exact solution of minimum-cost lifted multicut problems, which generalize correlation clustering with applications in computer vision and segmentation.

ABSTRACT

We study the set of all decompositions (clusterings) of a graph through its characterization as a set of lifted multicuts. This leads us to practically relevant insights related to the definition of a class of decompositions by must-join and must-cut constraints and related to the comparison of clusterings by metrics. To find optimal decompositions defined by minimum cost lifted multicuts, we establish some properties of some facets of lifted multicut polytopes, define efficient separation procedures and apply these in a branch-and-cut algorithm.

Motivation & Objective

• To characterize all graph decompositions via lifted multicuts, enabling a unified framework for clustering problems.
• To study the consistency and maximal specificity of must-join and must-cut constraints in defining decomposition classes.
• To define and compute metrics for comparing decompositions and constraint-defined classes of decompositions.
• To optimize graph decompositions via minimum-cost lifted multicuts using branch-and-cut with efficient separation.
• To establish theoretical properties of lifted multicut polytopes and derive valid inequalities for exact optimization.

Proposed method

• Lifted multicuts extend standard multicuts to a supergraph to make path-connectedness explicit, enabling full component connectivity information.
• The paper defines a one-to-one correspondence between graph decompositions and lifted multicuts in an extended edge set.
• It introduces a polyhedral model of lifted multicuts, deriving facet-defining inequalities and characterizing their structure.
• Separation procedures are designed to identify violated inequalities efficiently during branch-and-cut optimization.
• A branch-and-cut algorithm is implemented using these separation procedures to solve the minimum-cost lifted multicut problem.
• Metrics on decompositions and constraint-defined classes are defined via partial lifted multicuts, enabling comparison under different lifting schemes.

Experimental results

Research questions

• RQ1 Can a consistent set of must-join and must-cut constraints be efficiently checked for feasibility?
• RQ2 Is a given consistent set of constraints maximally specific, i.e., no further constraint can be added without changing the solution set?
• RQ3 How can meaningful metrics be defined to compare decompositions and classes of decompositions defined by constraints?
• RQ4 What are the structural properties of lifted multicut polytopes that enable efficient optimization?
• RQ5 Can the minimum-cost lifted multicut problem be solved exactly using branch-and-cut with effective separation?

Key findings

• The decision problem for consistency of must-join and must-cut constraints is NP-complete in general, but solvable in polynomial time for a subclass of constraints.
• The problem of maximal specificity of constraints is NP-hard in general, but efficiently solvable for a subclass.
• Computing the metric between decompositions or constraint-defined classes is NP-hard in general, but efficient for a subclass of constraints.
• The paper identifies necessary and sufficient conditions for lifted multicut inequalities to define facets of the lifted multicut polytope.
• Separation procedures for these facet-defining inequalities are designed and integrated into a branch-and-cut algorithm for exact optimization.
• The minimum-cost lifted multicut problem is a generalization of correlation clustering and is NP-hard, but solvable exactly using the proposed framework.