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Here is another example from this paper THE COMPUTATIONAL COMPLEXITY OF RECOGNIZING CRITICAL SETS: Unique edge-partitioning into triangles is $NP$-complete Input: Tripartite graph $G$ and an edge part …

Laurent Juban in Dichotomy Theorem for the Generalized Unique Satisability Problem proved a dichotomy theorem for Another SAT defined as: Input: a propositional formula $\phi$ and a satisfying assignm …

Babai invented the fastest known algorithm which runs in quasipolynomial time $2^{(\log n)^{O(1)}}$.

There is a list of open problems in computational geometry. It is edited and maintained by Demaine, Mitchell, and O'Rourke.

The case where $\epsilon$ is a fixed constant in $(0,1)$ represents a partition that each part contains at least a constant fraction of the vertices. This was proven to be NP-complete by Wagner and Wa …

Avi Wigderson, in Knowledge, Creativity and P versus NP, argues that the philosophical question: Can creativity be automated? is equivalent to P = NP?.

The paper, Knowledge, Creativity and P versus NP by Avi Wigderson is an excellent exposition of the P vs NP problem's implications on the philosophical question of automating creativity.

Computational Complexity: If you are looking for recent complexity textbooks. The following two are must have. Computational Complexity: A Modern Approach by Sanjeev Arora and Boaz Barak (Textbook …

The actual proof of the $NP$-hardness of computing Cheeger constant ( or edge expansion) was given by Kaibel in a technical report by a reduction from MAX Cut problem (See theorem 2). The proof is an …

As for question 2, there are at least two examples of $NP$-completeness proofs that involve computer-assistant. Erickson and Ruskey provided a computer-aided proof that Domino Tatami Covering is NP- …

Ball and Trap I remains $W[1]$-hard when restricted to binary trees. Theorem 5 states: Theorem 5. Ball and Trap I remains $W[1]$-hard restricted to binary trees, the maximum number of traps per …

Another example is the efficient dominating set problem also known as 1-perfect code in graphs. The problem is to determine the existence of a dominating set $C$ in undirected graph such that the shor …

An $NP$-complete structural problem is to decide the existence of odd (even) hole in directed graphs. Anna Lubiw proved the $NP$-completeness of the above two problems. A hole is chordless cycle of l …

Have look at the Encyclopedia of Algorithms by Kao (Editor). It contains over 500 entries and many of them contain advanced algorithms.

The article, The Chaos Within Sudoku is published (11 October 2012) in Nature by the same authors you site in your question. An excerpt from the abstract: Via an exact mapping of Sudoku into a de …