arXiv subjects

Bochuan Lyu

Publications and source records attributed to Bochuan Lyu.

4 recordsLinked to original sources

arXiv2023

Building Formulations for Piecewise Linear Relaxations of Nonlinear Functions

We study mixed-integer programming formulations for the piecewise linear lower and upper bounds (in other words, piecewise linear relaxations) of nonlinear functions that can be modeled by a new class of combinatorial disjunctive constraints (CDCs), generalized $n$D-ordered CDCs. We first introduce a general formulation technique to model piecewise linear lower and upper bounds of univariate nonlinear functions concurrently so that it uses fewer binary variables than modeling bounds separately. Next, we propose logarithmically sized ideal non-extended formulations to model the piecewise linear relaxations of univariate and higher-dimensional nonlinear functions under the CDC and independent branching frameworks. We also perform computational experiments for the approaches modeling the piecewise linear relaxations of univariate nonlinear functions and show significant speed-ups of our proposed formulations. Furthermore, we demonstrate that piecewise linear relaxations can provide strong dual bounds of the original problems with less computational time in order of magnitude.

math.OC↗

arXiv2023

Maximal Clique and Edge-Ranking Bounds of Biclique Cover Number

The biclique cover number $(\text{bc})$ of a graph $G$ denotes the minimum number of complete bipartite (biclique) subgraphs to cover all the edges of the graph. In this paper, we show that $\text{bc}(G) \geq \lceil \log_2(\text{mc}(G^c)) \rceil \geq \lceil \log_2(χ(G)) \rceil$ for an arbitrary graph $G$, where $χ(G)$ is the chromatic number of $G$ and $\text{mc}(G^c)$ is the number of maximal cliques of the complementary graph $G^c$, i.e., the number of maximal independent sets of $G$. We also show that $\lceil \log_2(\text{mc}(G^c)) \rceil$ could be a strictly tighter lower bound of the biclique cover number than other existing lower bounds. We can also provide a bound of $\text{bc}(G)$ with respect to the biclique partition number ($\text{bp}$) of $G$: $\text{bc}(G) \geq \lceil \log_2(\text{bp}(G) + 1) \rceil$ or $\text{bp}(G) \leq 2^{\text{bc}(G)} - 1$ if $G$ is co-chordal. Furthermore, we show that $\text{bc}(G) \leq χ_r'(T_{{K}^c})$, where $G$ is a co-chordal graph such that each vertex is in at most two maximal independent sets and $χ_r'({T}_{{K}^c})$ is the optimal edge-ranking number of a clique tree of $G^c$.

math.CO↗

arXiv2023

Finding Biclique Partitions of Co-Chordal Graphs

The biclique partition number $(\text{bp})$ of a graph $G$ is referred to as the least number of complete bipartite (biclique) subgraphs that are required to cover the edges of the graph exactly once. In this paper, we show that the biclique partition number ($\text{bp}$) of a co-chordal (complementary graph of chordal) graph $G = (V, E)$ is less than the number of maximal cliques ($\text{mc}$) of its complementary graph: a chordal graph $G^c = (V, E^c)$. We first provide a general framework of the ``divide and conquer" heuristic of finding minimum biclique partitions of co-chordal graphs based on clique trees. Furthermore, a heuristic of complexity $O[|V|(|V|+|E^c|)]$ is proposed by applying lexicographic breadth-first search to find structures called moplexes. Either heuristic gives us a biclique partition of $G$ with size $\text{mc}(G^c)-1$. In addition, we prove that both of our heuristics can solve the minimum biclique partition problem on $G$ exactly if its complement $G^c$ is chordal and clique vertex irreducible. We also show that $\text{mc}(G^c) - 2 \leq \text{bp}(G) \leq \text{mc}(G^c) - 1$ if $G$ is a split graph.

math.CO↗

arXiv2022

Modeling Combinatorial Disjunctive Constraints via Junction Trees

We introduce techniques to build small ideal mixed-integer programming (MIP) formulations of combinatorial disjunctive constraints (CDCs) via the independent branching scheme. We present a novel pairwise IB-representable class of CDCs, CDCs admitting junction trees, and provide a combinatorial procedure to build MIP formulations for those constraints. Generalized special ordered sets ($\text{SOS} k$) can be modeled by CDCs admitting junction trees and we also obtain MIP formulations of $\text{SOS} k$. Furthermore, we provide a novel ideal extended formulation of any combinatorial disjunctive constraints with fewer auxiliary binary variables with an application in planar obstacle avoidance.

math.OC↗