NP-Completeness Explained

NP-Completeness

• A language $B$ is NP-complete if $B$ is in NP and every $A$ in NP is polynomial time reducible to $B$ ($A \le_p B$).
• If an NP-complete language $B$ is in P, then P = NP.
• If B is NP-complete, C is in NP, and $B \le_p C$, then C is NP-complete.
• To prove a language is NP-complete, show it is in NP, and demonstrate that a known NP-complete language reduces to it.
• SAT language consists of satisfiable boolean formulas and was the first language to be proved NP-complete.
• 3SAT consists of satisfiable boolean formulas in CNF with 3 literals per clause and is NP-complete.
• Literals are variables or their negations.

Showing a Language B is NP-Complete

• Show that language $B$ belongs to NP.
• Show that some NP-complete language $A$ is polynomial-time reducible to $B$.
• A mapping reduction involves a function $f: \Sigma^* \to \Sigma^*$.
  • $w \in A \iff f(w) \in B$: the function maps strings in A to strings in B.
  • Compute $f$ in polynomial time.

CLIQUE

• A clique in an undirected graph is a subgraph where every two nodes are connected.
• Clique Size is determined by the number of nodes it contains.
• $CLIQUE = { | G$ is an undirected graph with a $k$-clique$}$.
• CLIQUE is in NP because a certificate of $k$ nodes can be checked in polynomial time to determine if they form a $k$-clique.

Proof That CLIQUE is NP-Complete

• 3SAT $\le_p$ CLIQUE: Maps a 3SAT formula to a graph and a number such that the formula is satisfiable if and only if the graph has a clique of the given size.
• Given a 3SAT formula $\phi$ with $k$ clauses ($\phi = C_1 \land C_2 \land... \land C_k$), a graph $G$ constructed with $3k$ nodes is created, where each literal in each clause becomes a node.
• Add an edge between each pair of literals if they are in different clauses and are not negations of each other
• The graph $G$ has a $k$-clique if and only if $\phi$ is satisfiable.
• If $\phi$ is satisfiable, one true literal is picked from each clause ($k$ literals total), come from different clauses, literals cannot be negations of each other, $k$ literals are connected and for a $k$-clique.
• If $G$ has a $k$-clique, assigning each literal to true satisfies $\phi$, $k$ nodes in $k$-clique must all come from different clauses (because there are no edges between literals in the same clause.), the $k$ nodes in the $k$-clique are not negations of each other.
• Constructing the graph takes polynomial time, with $3k$ nodes and $O(k^2)$ edges.

VERTEX-COVER

• A vertex cover of a graph $G = (V, E)$ is a subset of nodes $V' \subseteq V$ such that for every edge $(u, v) \in E$, either $u \in V'$, $v \in V'$, or both.
• $VERTEX-COVER = { | G$ is an undirected graph that has a vertex cover of size $k}$.
• VERTEX-COVER is NP-Complete.
• VERTEX-COVER is in NP because a set of $k$ nodes can be verified in polynomial time to check if it's a vertex cover.
• Reduction: CLIQUE $\le_p$ VERTEX-COVER.

Proof That VERTEX-COVER is NP-Complete

• Demonstrates that any graph with a k-clique can be transformed into another graph that has a vertex cover of size $|V| - k$.
• Given a graph $G$ and number $k$, a graph $G'$ (complement of $G$) and a number $k' = |V| - k$ is constructed.
• Therefore, $G$ has a $k$-clique if and only if $\overline{G}$ has a vertex cover of size $|V| - k$.
• If G has a k-clique, then $\overline{G}$ has a vertex cover of size $|V| - k$, set $S = V - C$ covers all edges in $\overline{G}$, where $C \subseteq V$ be a $k$-clique in $G$.
• If $\overline{G}$ has a vertex cover $S$ of size $|V| - k$, then $G$ has a $k$-clique, every edge in $\overline{G}$ must be covered by $S$. So $|C|=k$, which is not in $\overline{G}$.
• The reduction takes polynomial time (complementing a graph takes polynomial time).

SUBSET-SUM

• $SUBSET-SUM = {(S, t) | S = {x_1,..., x_k}$ which is a set of integers, $\Sigma_{x_i \in S'} x_i = t}$, which means there must be $S' \subseteq S$.
• SUBSET-SUM is NP-complete.
• To check in polynomial time whether $S'$ equals $t$, it becomes SUBSET-SUM $\in$ NP.
• Reduction VERTEX-COVER $\le_p$ SUBSET-SUM where a set $S$ and target $t$ are constructed such that $G$ has a vertex cover of size $k$ if and only if $S$ has a subset summing to $t$.

Algorithmic Trading and Order Execution

• Course run by Tucker Balch
• Recommended texts are Derman's "Algorithmic Trading", Chan's "Quantitative Trading", and McKinney's "Python for Data Analysis."
• Course objectives include understanding mechanics of trading, trading tactics, strategy simulation, order execution, market microstructure, and portfolio management.

Topics Covered

• Market Mechanics
• Trading Tactics and Strategies
• Simulation and Evaluation
• Order Execution
• Market Microstructure
• Portfolio Management

Course Project Details

• involves groups of 3-4
• Trading strategy is to be designed, implemented and evaluated
• Project requires a presentation and a final report.

Grading Breakdown

• Homework: 20%
• Quizzes: 20%
• Project: 40%
• Final Exam: 20%

Academic Honesty

• All work must be original; academic dishonesty will be reported.

Accommodations

• Available for students with documented disabilities (contact Disability Services Program).

Partial Differential Equations (PDEs)

• Problems in physics and engineering lead to PDEs, involving functions of multiple variables and their partial derivatives.

Examples of PDEs

• Heat equation: $\frac{{\partial u}}{{\partial t}} = k\frac{{\partial^2 u}}{{\partial x^2}}$.
• Wave equation: $\frac{{\partial^2 u}}{{\partial t^2}} = c^2\frac{{\partial^2 u}}{{\partial x^2}}$.
• Laplace equation: $\frac{{\partial^2 u}}{{\partial x^2}} + \frac{{\partial^2 u}}{{\partial y^2}} = 0$.

Definitions Related to PDEs

• A partial differential equation involves partial derivatives of an unknown function with two or more variables.
• Order is determined by the order of the highest derivative in the equation.
• Linearity means the function and derivatives appear linearly.
• Homogeneity is when a linear PDE is satisfied when terms with the unknown function are set to zero

Types of PDEs

Solving PDEs Involves

• General solutions: use arbitrary functions.
• Boundary and initial conditions determine a unique solution.
• Methods include separation of variables, transform methods, and numerical methods.

Common Solution Methods

• General solutions for PDEs include arbitrary functions. For $\frac{{\partial^2 u}}{{\partial x \partial y}} = 0$, $u(x, y) = F(x) + G(y)$ is the general solution.
• Define Conditions to determine a particular solution, boundary conditions specify values on the domain's boundary, and initial conditions specify the function's value at an initial time.

Methods to Solving PDEs

• Separation of Variables: assumes solutions in the form of $u(x, t) = X(x)T(t)$
• Transform Methods: Uses Fourier or Laplace transforms to simplify PDEs.
• Numerical Methods: Approximates solutions using finite difference, finite element, or finite volume methods.

Examples and Applications of PDEs

• Heat Equation describes heat distribution over time, with $u(x, t)$ as temperature and $k$ as thermal diffusivity.
• Wave equation describes wave propagation, with $u(x, t)$ as the wave's displacement and $c$ as wave speed.
• Laplace Equation describes steady-state phenomena such as electrostatics, fluid dynamics, and heat conduction.

Advanced Topics of PDEs

• Nonlinear PDEs: lacks general solutions and requires special techniques.
• Systems of PDEs: multiple coupled PDEs used in fluid dynamics and electromagnetism.
• Numerical Solutions: includes the finite difference, finite element, and finite volume method.