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    Area of Science:

    • Theoretical Computer Science
    • Algorithm Analysis
    • Combinatorial Optimization

    Background:

    • The Maximum Cut (MAX-CUT) problem is a fundamental NP-complete problem in graph theory.
    • While evolutionary algorithms (EAs) show empirical success for MAX-CUT, theoretical understanding of their performance is limited.

    Purpose of the Study:

    • To theoretically investigate the performance of simple evolutionary algorithms, specifically the (1+1) EA and (1+1) EA*, on the MAX-CUT problem.
    • To provide theoretical bounds on the approximation quality and runtime of these EAs for MAX-CUT.

    Main Methods:

    • Analysis of the (1+1) EA and (1+1) EA* on the MAX-CUT problem.
    • Derivation of theoretical performance guarantees, including approximation ratios and expected runtimes.
    • Comparison with existing local search algorithms.

    Main Results:

    • The (1+1) EA and (1+1) EA* achieve approximation solutions of (m/2)+(1/4)s(G) and (m/2)+(1/2)(√{8m+1}-1) respectively.
    • The (1+1) EA* finds a cut of size at least k in expected runtime O(nm+1/δ(4k)) and a cut of size at least (m/2)+k in expected runtime O(n(2)m+1/δ((64/3)k(2))).
    • These EAs demonstrate superior performance over some local search algorithms in specific instances, but may not be efficient in all cases.

    Conclusions:

    • Simple evolutionary algorithms like (1+1) EA and (1+1) EA* offer theoretically grounded efficient approximation strategies for the MAX-CUT problem.
    • The theoretical analysis provides insights into the strengths and limitations of EAs for solving MAX-CUT.
    • Further research can explore more complex EAs and problem instances to fully leverage their potential for MAX-CUT.