Partial Optimality in the Preordering Problem

TL;DR

This work addresses the NP-hard preordering problem by developing partial optimality tools that certify certain pairwise relations in an optimal preorder without solving the full problem. It introduces improving maps built from dicut and join operations, plus three families of partial optimality conditions (cut, join, fixation) and corresponding polynomial-time tests and algorithms. The approach yields substantial fractions of pairs $ab$ for which it can be efficiently decided that $a \not\lesssim b$ or $a \lesssim b$, validated through synthetic and social-network experiments. Overall, the methods prune the search space and enhance existing solvers for transitive-digraph based relaxations, with practical impact in clustering, ordering, and related domains.

Abstract

Preordering is a generalization of clustering and partial ordering with applications in bioinformatics and social network analysis. Given a finite set $V$ and a value $c_{ab} \in \mathbb{R}$ for every ordered pair $ab$ of elements of $V$, the preordering problem asks for a preorder $\lesssim$ on $V$ that maximizes the sum of the values of those pairs $ab$ for which $a \lesssim b$. Building on the state of the art in solving this NP-hard problem partially, we contribute new partial optimality conditions and efficient algorithms for deciding these conditions. In experiments with real and synthetic data, these new conditions increase, in particular, the fraction of pairs $ab$ for which it is decided efficiently that $a \not\lesssim b$ in an optimal preorder.

Figures (9)

• Figure 1: An instance of the preordering problem with the set $V = \{i,j,k,l,m\}$ and the values $c$ is defined by the table on the left. An optimal solution, i.e. a maximum value preorder $\lesssim$, is shown on the right with arrows from $a$ to $b$ indicating $a \lesssim b$. The symmetric subset of the preorder is an equivalence relation and thus defines a partition, or clustering, of $V$ (gray). The anti-symmmetric subset well-defines a partial order over the clusters.
• Figure 2: In the example depicted above, must-join constraints (solid black arrows) and must-cut constraints (dashed black arrow) defined by a partial characteristic function of preorders, $\tilde{x}\in \tilde{X}_V$, imply additional must-join constraints (solid gray arrows) as well as additional must-cut constraints (dashed gray arrows). In this case, the closure $\textnormal{cl}_{V}\,\tilde{x}$ is strictly more specific than $\tilde{x}$.
• Figure 3: The map $\sigma_{ij} \circ \sigma_{\delta(V\setminus U, U)} \circ \sigma_{\delta(U', V\setminus U')}$ transforms the preorder depicted on the left to the preorder depicted on the right.
• Figure 4: Depicted above is an instance of the preordering problem with $V = \{p, q, r, s\}$, with non-zero values $c$ shown as directed edges, and with zero values not shown. It witnesses the fact that \ref{['proposition:subset-U-maximizer']} is not subsumed by \ref{['corollary:boecker-condition-weak']}; see \ref{['example:subset-U-maximizer']}.
• Figure 5: Shown above are the percentage of fixed variables (Row 1) and runtimes (Row 2) for applying \ref{['proposition:edge-cut-condition']}, \ref{['corollary:directed-cut-condition']}, \ref{['proposition:edge-join-condition']}, \ref{['proposition:subset-U-maximizer']} (with \ref{['proposition:tractable-U']}) and \ref{['corollary:boecker-condition-strong']} for $b\in \{0, 1\}$ individually to instances of the synthetic dataset with respect to $\alpha\in \left[0, 1\right]$, $\lvert V\rvert = 40$ and $p_E \in \{0.25, 0.50, 0.75\}$.

Theorems & Definitions (34)

• Definition 1.1: wakabayashi1998
• Definition 3.1
• Proposition 3.2
• Corollary 3.3
• Definition 4.1
• Proposition 4.2
• Definition 4.3
• Proposition 4.4
• Definition 5.1
• Lemma 5.2