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Fixed-Parameter Tractability of the (1+1) Evolutionary Algorithm on Random Planted Vertex Covers

Jack Kearney, Frank Neumann, Andrew M. Sutton

TL;DR

It is shown that if the planted cover is at most logarithmic, restarting the (1+1) EA every O(n log n) steps will find a cover at least as small as the plantedCover in polynomial time for sufficiently dense random graphs p > 0.71.

Abstract

We present the first parameterized analysis of a standard (1+1) Evolutionary Algorithm on a distribution of vertex cover problems. We show that if the planted cover is at most logarithmic, restarting the (1+1) EA every $O(n \log n)$ steps will find a cover at least as small as the planted cover in polynomial time for sufficiently dense random graphs $p > 0.71$. For superlogarithmic planted covers, we prove that the (1+1) EA finds a solution in fixed-parameter tractable time in expectation. We complement these theoretical investigations with a number of computational experiments that highlight the interplay between planted cover size, graph density and runtime.

Fixed-Parameter Tractability of the (1+1) Evolutionary Algorithm on Random Planted Vertex Covers

TL;DR

It is shown that if the planted cover is at most logarithmic, restarting the (1+1) EA every O(n log n) steps will find a cover at least as small as the plantedCover in polynomial time for sufficiently dense random graphs p > 0.71.

Abstract

We present the first parameterized analysis of a standard (1+1) Evolutionary Algorithm on a distribution of vertex cover problems. We show that if the planted cover is at most logarithmic, restarting the (1+1) EA every steps will find a cover at least as small as the planted cover in polynomial time for sufficiently dense random graphs . For superlogarithmic planted covers, we prove that the (1+1) EA finds a solution in fixed-parameter tractable time in expectation. We complement these theoretical investigations with a number of computational experiments that highlight the interplay between planted cover size, graph density and runtime.
Paper Structure (6 sections, 7 theorems, 13 equations, 6 figures, 2 algorithms)

This paper contains 6 sections, 7 theorems, 13 equations, 6 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Small $k$
  4. Large $k$
  5. Computational Experiments
  6. Conclusion

Key Result

Theorem 1

Let $(X_t)_{t \in \mathbb{N}}$ be a stochastic process over $\mathbb{R}$, $x_{\min} > 0$ and let $T \coloneqq \min\{t : X_t < x_{\min}\}$. Suppose that $X_0 \ge x_{\min}$ and, for all $t \le T$, it holds that $X_t \ge 0$, and there exists some $\delta > 0$ such that, for all $t < T$, $\mathop{\mathr

Figures (6)

  • Figure 1: Runtime dependence on $n$ for $k=\ln n$ and $k= \sqrt{n}$. Error bars denote standard deviation.
  • Figure 2: Runtime dependence on $p$ for fixed $n$ varying $k=10,\ldots,100$. Error bars denote standard deviation.
  • Figure 3: Runtime dependence on $k$ ($p$ aggregated). Error bars denote standard deviation.
  • Figure 4: Runtime dependence on both $k$ and $p$ for fixed $n$.
  • Figure 5: Proportion of runs in which the planted $k$-core was recovered.
  • ...and 1 more figures

Theorems & Definitions (15)

  • Theorem 1: Multiplicative Drift DoerrGoldberg2010DriftAnalysisTailKoetzingKrejca2019firsthitting
  • Theorem 2
  • proof
  • Definition 1
  • Definition 2
  • Lemma 1
  • proof
  • Theorem 3
  • proof
  • Corollary 1: to Theorem \ref{['thm:small-k']}
  • ...and 5 more