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Local Search (Approximation cont.)

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Local

  • A feasible set of neighborhoods
  • Local optimum
  • Start with feasible solution, and find a better one.
  • Stop when no (or little) improvement

Vertex cover

  • Start with all vertices
  • Delete a potential vertex, which makes the remaining set also feasible
  • May be add a chance to add a vertex (\(e^{-\delta_{cost}/kT}\)), \(T\) keeps cool down

Simulated Annealing

Hopfield Neural Networks

  • Assign states of each node

  • Edge are constraints and reward

    image-20240515164613416

    All edge constraints may not be satisfied, but all nodes are stable.

  • A stable node

    \(\sum w(e)_{good} \geq \sum w(e_{bad})\)

  • We flip a node if it's not stable

    A flip will surely increase the total gain.

    Total gain has upper bound.

    There always a solution that all nodes are stable.

    Local search is not polynomial. It has at most \(\sum w_i\) iterations.

Maximum cut problem

Special case of hopfield neural networks, with positive edge weight. (only nodes belong to different group, we gain the reward)

  • 2-Approximation (may not polynomial)

    image-20240515165143226

  • \(2+\epsilon\) approximation

​ Only flip when gain is at least \((2\epsilon/|V|) * w(A,B)\)

​ \((2+\epsilon)w >= w^*\)

​ \(O((n/\epsilon) logW)\) flips

\(f'(k) = 1/Nln(N/k) + -1/N = 1/N*(ln(N/k) - 1)\)

\(N/k = e\), \(k = N/e\)