Hi everyone!

This time I'd like to write about what's widely known as "Aliens trick" (as it got popularized after 2016 IOI problem called Aliens). There are already some articles about it here and there, and I'd like to summarize them, while also adding insights into the connection between this trick and generic Lagrange multipliers and Lagrangian duality which often occurs in e.g. linear programming problems.

Lagrange duality

Let $$$f : X \to \mathbb R$$$ be the objective function and $$$g : X \to \mathbb R^c$$$ be the constraint function. The constrained optimization problem

in some cases can be reduced to finding stationary points of the Lagrange function

Here $$$\lambda \cdot g(x)$$$ is the dot product of $$$g(x)$$$ and a variable vector $$$\lambda \in \mathbb R^c$$$, called the Lagrange multiplier. Mathematical optimization typically focuses on finding stationary points of $$$L(x,\lambda)$$$. However, in our particular case we're more interested in the function

which is called the Lagrange dual function. If $$$x^*$$$ is the solution to the original problem, then $$$t(\lambda) \leq L(x^*,\lambda)=f(x^*)$$$.

This allows to introduce the Lagrangian dual problem $$$t(\lambda) \to \max$$$. Note that $$$t(\lambda)$$$, as a point-wise infimum of concave (specifically, linear) functions, is always concave, even when $$$X$$$ is, e.g., discrete. If $$$\lambda^*$$$ is the solution to the dual problem, the value $$$f(x^*) - t(\lambda^*)$$$ is called the duality gap. We're specifically interested in the case when it equals zero, which is called the strong duality.

Typical example here is Slater's condition, which says that strong duality holds if $$$f(x)$$$ is convex and there exists $$$x$$$ such that $$$g(x)=0$$$.

Change of domain

In competitive programming, the set $$$X$$$ in definitions above is often weird and very difficult to analyze directly, so Slater's condition is not applicable. As a typical example, $$$X$$$ could be the set of all possible partitions of $$$\{1,2,\dots, n\}$$$ into non-intersecting segments.

To mitigate this, we define $$$h(y)$$$ as the minimum value of $$$f(x)$$$ subject to $$$g(x)=y$$$. In this notion, the dual function is written as

where $$$Y=\{ g(x) : x \in X\}$$$. The set $$$Y$$$ is usually much more regular than $$$X$$$, as just by definition it is already a subset of $$$\mathbb R^c$$$. The strong duality condition is also very clear in this terms: it holds if and only if $$$0 \in Y$$$ and there is a $$$\lambda$$$ for which $$$y=0$$$ delivers infimum. Geometrically it means that epigraph of $$$h(y)$$$ has a supporting plane that touches it in $$$y=0$$$.

Competitive programming problems typically assume that $$$y$$$ is a variable given in the input, so this condition should actually hold for all $$$y \in Y$$$ which is essentially equivalent to $$$h(y)$$$ being convex on $$$Y$$$, which is a sufficient and almost always necessary condition here.

Interpreting lambda

If $$$h(y)$$$ is continuously differentiable and convex, the minimum for specific $$$\lambda$$$ is obtained with $$$y=g(x)$$$ such that $$$\nabla h(y) = \lambda$$$. This property allows to find $$$\lambda$$$ that corresponds to $$$y$$$ with nested ternary search.

Problem examples

References

• Duality (optimization) — English Wikipedia
• The Trick From Aliens — Serbanology
• My Take on Aliens' Trick — Mamnoon Siam's Blog
• Comment on Codeforces by _h_
• Part of the article was once revealed to me in a dream