MY-ASSIGNMENTEXPERT™可以为您提供stanford.edu MS&E310 Linear Programming线性规划课程的代写代考和辅导服务！
这是斯坦福大学 线性规划课程的代写成功案例。
MS&E310课程简介
Topics include: Problem formulation of standard (conic) linear programming models, the theory of polyhedral and conic convex sets, linear inequalities, alternative theorems and duality, sensitivity analyses and economic interpretations, and relaxations of harder optimization problems. Algorithms include the simplex method, interior-point methods, and ADMM and other (first-order) iterative methods. Complexity and/or computation efficiency analysis for linear programming. Applications include dynamic resource allocation, on-line mechanism design, algorithmic game-theory, SVM/data-classification, and MDP/reinforced learning.
Prerequisites
The field of optimization is concerned with the study of maximization and minimization of mathematical functions. Very often the arguments of (i.e., variables in) these functions are subject to side conditions or constraints. By virtue of its great utility in such diverse areas as applied science, engineering, economics, finance, medicine, and statistics, optimization holds an important place in the practical world and the scientific world. Indeed, as far back as the Eighteenth Century, the famous Swiss mathematician and physicist Leonhard Euler (1707-1783) proclaimed that … nothing at all takes place in the Universe in which some rule of maximum or minimum does not appear. The subject is so pervasive that we even find some optimization terms in our everyday language.
MS&E310 Linear Programming HELP（EXAM HELP， ONLINE TUTOR）
Let $G=(V, E)$ be a connected, undirected graph, and suppose $T_s=(V, F)$ is a tree on $G$ created by exploring $G$ using depth-first search starting from vertex $s$. Vertex $s$ is the root of tree $T_s$. Argue that $s$ has more than one child in $T_s$ if and only if removing $s$ from $G$ breaks $G$ into several disconnected parts.
The diameter of a connected, undirected graph $G=(V, E)$ is the length (in number of edges) of the longest shortest path between two nodes. (a) Show that if the diameter of a graph is $d$ then there is some set $S \subseteq V$ with $|S| \leq n /(d-1)$ such that removing the vertices in $S$ from the graph would break it into several disconnected pieces. (b) Give an efficient algorithm to find $S$.
Let $G=(V, E)$ be a connected, undirected graph with positive edge weights $d(u, v)$, and suppose $|E|=\Omega\left(|V|^2\right)$. That is, $|E|$ is perhaps $\left(\begin{array}{c}|V| \ 2\end{array}\right) / 100$. Give an implementation of Dijkstra’s shortest path algorithm that is asymptotically faster on such dense graphs than the heap-based version we saw in class.
Suppose $G=(V, E)$ is a directed, acyclic graph (a DAG) with positive weights $d(u, v)$ on each edge. Let $s$ be a vertex of $G$ with no incoming edges and such that every other node is reachable from $s$ through some path.
(a) Give an $O(|V|+|E|)$-time algorithm to compute the shortest paths from $s$ to all other vertices in $G$. Note that this is faster than Dijkstra’s algorithm in general.
(b) Give an efficient algorithm to compute the longest paths from $s$ to all other vertices. (Interestingly, this is a hard problem in general, non-DAG graphs.)
Give an $O(|E|)$-time algorithm to check whether a given tree $T$ is a shortest-path tree of an undirected, connected graph $G=(V, E)$ with positive weights $d(u, v)$ on each edge.
Hint: check whether the edges that are not in $T$ are correctly excluded.
MY-ASSIGNMENTEXPERT™可以为您提供STANFORD.EDU MS&E310 LINEAR PROGRAMMING线性规划课程的代写代考和辅导服务！
