There are well-known families of approximation algorithms.

We can imagine some clustering problem with $k$-centers. For $G = (V,E)$ with distance $d_{ij} \ge 0$ between each pair of vertices $i,j \in V$. We assume that $d_{ii} = 0, d_{ij} = d_{ji}, d_{ik} + d_{kj} \ge d_{ij}$ for each $i,j,k \in V$. Now, problem is find $S \subset V$ which $|S| = k$. Which makes $max(d(j,S))$ smallest. It’s like making clusters efficiently.

Problem is like below. There is a set of items $I = \{1,2,\cdots,n$}, where each item $i$ has a value $v_i \gt 0$ and a size $s_i \gt 0$. Now, knapsack problem is a “Find the maximum possible value with capacity $B$”.

CUDA is a parallel computing platform and programming API for NVIDIA GPUs. It provides multiple levels of abstraction for GPU programming.

The Kronecker product is a mathematical tool for enumerating all possible combinations of elements. It is useful for many combinatorial problems, such as the Traveling Salesman Problem.

Matrix representation

There are many formats to represent matrices.

Recap $\operatorname{SET COVER}$ can be LP-relaxed to “Minimize the $\sum\limits_{j=1}^{m} w_j x_j$ such that $\sum\limits_{j:e_i \in S_j} x_j \ge 1$, $x_j \ge 0$ then choose $x_j$ to $1$ which $x_j \ge \frac{1}{f}$ and otherwise to 0”. Which is $f$-approximation algorithm. Which $e_i$ denotes each elements, $S_j$ denotes each subsets, $w_j$ denotes costs of each subsets, $x_j$ is the chocie of each subset. With $f$ is the maximum number of existance of elements.

Recap $\operatorname{SET COVER}$ can be LP-relaxed to “Minimize the $\sum\limits_{j=1}^{m} w_j x_j$ such that $\sum\limits_{j:e_i \in S_j} x_j \ge 1$, $x_j \ge 0$ then choose $x_j$ to $1$ which $x_j \ge \frac{1}{f}$ and otherwise to 0”. Which is $f$-approximation algorithm. Which $e_i$ denotes each elements, $S_j$ denotes each subsets, $w_j$ denotes costs of each subsets, $x_j$ is the chocie of each subset. With $f$ is the maximum number of existance of elements.

An optimization problem is a problem of finding the minimum or maximum possible solution. For example, $\operatorname{SET COVER}$ is a well-known optimization problem. For a universe set $U = \{e_1,e_2,e_3,\cdots,e_n$} and subsets of $U$, $S_j(1 \le j \le n)$. $\operatorname{SET COVER}$ is a problem to find a collection of subsets which minimize the sum of costs of subset $W_j$ and it contains every element in $U$. For example, if $U = \{e_1,e_2,e_3,e_4,e_5$}, $S_1 = \{e_1,e_2$}, $S_2 = \{e_3,e_4$}, $S_3 = \{e_5$}, $S_4 = \{e_2,e_3$}, $S_5 = \{e_1$}, $S_6 = \{e_4$}, $W_1=10$, $W_2=10$, $W_3=1$, $W_4=15$, $W_5=1$, $W_6=1$. Optimal solution would be $W_3 + W_4 + W_5 + W_6 = 1 + 15 + 1 + 1 = 18$. Notice that $S_3 \cup S_4 \cup S_5 \cup S_6 = U$. However, $\operatorname{SET COVER}$ is known to be NP-hard, meaning it requires more than polynomial time unless P $=$ NP.

The ABA problem is a well-known issue in CAS-based lock-free data structures. C++ supports std::atomic for arbitrary data types, but it is not always lock-free. You can check it with below.