Clause Learning and New Bounds for Graph Coloring

2019 ◽

Author(s):

Emmanuel Hebrard ◽

George Katsirelos

Keyword(s):

Lower Bound ◽

Graph Coloring ◽

Upper Bounds ◽

Complete Graphs ◽

Scheduling Problems ◽

Original Graph ◽

Branch And Bound Search ◽

Clause Learning ◽

Free Tree ◽

Different Color

Graph coloring is a major component of numerous allocation and scheduling problems. We introduce a hybrid CP/SAT approach to graph coloring based on exploring Zykov’s tree: for two non-neighbors, either they take a different color and there might as well be an edge between them, or they take the same color and we might as well merge them. Branching on whether two neighbors get the same color yields a symmetry-free tree with complete graphs as leaves, which correspond to colorings of the original graph. We introduce a new lower bound for this problem based on Mycielskian graphs; a method to produce a clausal explanation of this bound for use in a CDCL algorithm; and a branching heuristic emulating Brelaz on the Zykov tree. The combination of these techniques in a branch- and-bound search outperforms Dsatur and other SAT-based approaches on standard benchmarks both for finding upper bounds and for proving lower bounds.

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Backjumping is Exception Handling

Theory and Practice of Logic Programming ◽

10.1017/s1471068420000435 ◽

2020 ◽

pp. 1-20

Author(s):

ED ROBBINS ◽

ANDY KING ◽

JACOB M. HOWE

Keyword(s):

Graph Coloring ◽

Exception Handling ◽

Control Flow ◽

Specific Point ◽

Clause Learning ◽

Sat Solver

Abstract ISO Prolog provides catch and throw to realize the control flow of exception handling. This pearl demonstrates that catch and throw are inconspicuously amenable to the implementation of backjumping. In fact, they have precisely the semantics required: rewinding the search to a specific point and carrying of a preserved term to that point. The utility of these properties is demonstrated through an implementation of graph coloring with backjumping and a backjumping SAT solver that applies conflict-driven clause learning.

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Study on coloring method of airport flight-gate allocation problem

Journal of Mathematics in Industry ◽

10.1186/s13362-019-0068-5 ◽

2019 ◽

Vol 9 (1) ◽

Author(s):

Hongyan Li ◽

Xianfeng Ding ◽

Jiang Lin ◽

Jingyu Zhou

Keyword(s):

Graph Coloring ◽

Programming Model ◽

Weight Vector ◽

Multi Objective ◽

Average Value ◽

Two Factors ◽

Minimum Number ◽

First Come First Serve ◽

Airport Terminals ◽

Single Objective

Abstract With the development of economy, more and more people travel by plane. Many airports have added satellite halls to relieve the pressure of insufficient boarding gates in airport terminals. However, the addition of satellite halls will have a certain impact on connecting flights of transit passengers and increase the difficulty of reasonable allocation of flight and gate in airports. Based on the requirements and data of question F of the 2018 postgraduate mathematical contest in modeling, this paper studies the flight-gate allocation of additional satellite halls at airports. Firstly, match the seven types of flights with the ten types of gates. Secondly, considering the number of gates used and the least number of flights not allocated to the gate, and adding the two factors of the overall tension of passengers and the minimum number of passengers who failed to transfer, the multi-objective 0–1 programming model was established. Determine the weight vector $w=(0.112,0.097,0.496,0.395)$ w = ( 0.112 , 0.097 , 0.496 , 0.395 ) of objective function by entropy value method based on personal preference, then the multi-objective 0–1 programming model is transformed into single-objective 0–1 programming model. Finally, a graph coloring algorithm based on parameter adjustment is used to solve the transformed model. The concept of time slice was used to determine the set of time conflicts of flight slots, and the vertex sequences were colored by applying the principle of “first come first serve”. Applying the model and algorithm proposed in this paper, it can be obtained that the average value of the overall tension degree of passengers minimized in question F is 35.179%, the number of flights successfully allocated to the gate maximized is 262, and the number of gates used is minimized to be 60. The corresponding flight-gate difficulty allocation weight is $\alpha =0.32$ α = 0.32 and $\beta =0.40$ β = 0.40 , and the proportion of flights successfully assigned to the gate is 86.469%. The number of passengers who failed to transfer was 642, with a failure rate of 23.337%.

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