Chun-Yan Zhao | Searcharxiv

arXiv2026

Long-range frustration in minimal vertex cover problem on random graphs

A vertex cover on a graph is a set of vertices in which each edge of the graph is adjacent to at least one vertex in the set. The minimal vertex cover (MVC) problem concerns finding vertex covers with the smallest cardinality, which is a typical computationally hard problem among combinatorial optimization on graphs. Here, we follow the idea of the long-range frustration (LRF) in MVC configurations proposed in [\textsl{Physical Review Letters} \textbf{94} (2005) 217203]. We correct its analytical framework and further extend it from Erdös-Rényi random graphs to general random graphs. We formulate the framework of LRF into a percolation model, and analytically estimate the energy density of MVCs on uncorrelated random graphs only with their degree distributions. We test our framework on some typical random graph models along with other methods, such as a hybrid algorithm of greedy leaf removal (GLR) procedure combined with survey propagation-guided decimation (SPD) algorithm and an analytical theory based on the GLR procedure which ignores LRF effect. We show that, when there is a percolation of LRF effect, the above three predictions of energy density, say $x_{\rm LRF}$, $x_{\rm GLR + SPD}$, and $x_{\rm GLR}$, follow a scenario as $x_{\rm LRF} > x_{\rm GLR+SPD} > x_{\rm GLR}$ in most cases and $x_{\rm GLR+SPD} > x_{\rm LRF} > x_{\rm GLR}$ in the other cases, and $x_{\rm LRF}$ is much closer to $x_{\rm GLR+SPD}$ than $x_{\rm GLR}$ as $|x_{\rm LRF} - x_{\rm GLR+SPD} | < x_{\rm GLR+SPD} - x_{\rm GLR}$. Our results show that LRF is a proper mechanism for the formation of complex energy landscape in the MVC problem and a theoretical framework of LRF helps to characterize its ground-state properties.

cond-mat.stat-mech↗

arXiv2022

A residual-based message passing algorithm for constraint satisfaction problems

Message passing algorithms, whose iterative nature captures well complicated interactions among interconnected variables in complex systems and extracts information from the fixed point of iterated messages, provide a powerful toolkit in tackling hard computational tasks in optimization, inference, and learning problems. In the context of constraint satisfaction problems (CSPs), when a control parameter (such as constraint density) is tuned, multiple threshold phenomena emerge, signaling fundamental structural transitions in their solution space. Finding solutions around these transition points is exceedingly challenging for algorithm design, where message passing algorithms suffer from a large message fluctuation far from convergence. Here we introduce a residual-based updating step into message passing algorithms, in which messages varying large between consecutive steps are given high priority in the updating process. For the specific example of model RB, a typical prototype of random CSPs with growing domains, we show that our algorithm improves the convergence of message updating and increases the success probability in finding solutions around the satisfiability threshold with a low computational cost. Our approach to message passing algorithms should be of value for exploring their power in developing algorithms to find ground-state solutions and understand the detailed structure of solution space of hard optimization problems.

cond-mat.dis-nn↗