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| Original file line number | Diff line number | Diff line change |
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| --- | ||
| date: '2023-12-01' | ||
| lang: 'en' | ||
| slug: '/2023-12-01' | ||
| --- | ||
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| - [[Discrete Mathematics]] |
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| --- | ||
| lang: 'en' | ||
| slug: '/21FE92' | ||
| --- | ||
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| ## Dynamic Programming | ||
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| - **Optimal substructure**. The optimal solution to a problem consists of optimal solutions to subproblems. | ||
| - **Overlapping subproblems**. few subproblems in total, many recurring instances of each. | ||
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| ## Bellman-Ford | ||
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| shortest path, negative possible. | ||
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| $$ | ||
| \text{OPT}[v,k] = \min(\text{OPT}[v, k-1], \text{OPT}[u, v] + w(u,v)) | ||
| $$ | ||
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| ## Chain Matrix Multiplication | ||
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| $$ | ||
| \text{OPT}[i, j] = \text{OPT}[i, k] + \text{OPT}[k+1, j] + \text{combine} | ||
| $$ | ||
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| ## Min Cut | ||
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| The set of vertices reachable from source s in the residual graph is one part of the [[partition]]. Cut capacity $\text{cap}(A, B) = \sum_{\text{out A}} c(e)$. | ||
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| $$ | ||
| |f| = \sum_{e~\text{out of X}} f(e) - \sum_{e~\text{into X}} f(e) | ||
| $$ | ||
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| - maxflow = Min $\text{cap}(A, B)$. | ||
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| ## Ford-Fulkerson | ||
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| Given $(G, s, t, c \in \mathbb{N}+)$, start with $f(u,v)=0$ and $G_f =G$. While an augmenting path is in $G_f$, find a bottleneck. Augment the flow along this path and update the residual graph $G_f$. $\mathcal{O}(|f| (V+E))$. | ||
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| ## Edmond-Karp | ||
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| Ford-Fulkerson, but choose the shortest augmenting path. | ||
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| For any flow $f$ and any $(A,B)$ cut, $|f| ≤ \text{cap}(A,B)$. For any flow $f$ and any $(A,B)$ cut, $$|f| = \sum_f (s, v) = \sum_{u \in A,~v \in B} f(u, v) - \sum_{u \in A,~v \in B} f(v, u)$$ | ||
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| ## Solving by Reduction | ||
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| To reduce a problem $Y$ to a problem $X$ ($Y \leq_{p} X$) we want a function $f$ that maps $Y$ to $X$ such that $f$ is a polynomial time computable and $\forall y \in Y$ is solvable if and only if $f(y) \in X$ is solvable. | ||
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| ## Reduction to NF | ||
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| Describe how to construct a flow network. Claim "This is feasible if and only if the max flow is …". Prove both directions. | ||
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| ## Bipartite to Max Flow | ||
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| - Max Matching ⟹ Max Flow. If k edges are matched, then there is a flow of value k. | ||
| - Max matching ⟸ Max flow. If there is a flow f of value k, there is a matching with k edges. | ||
| - $\mathcal{O}(|f| (E' + V'))$ | ||
| - $|f| = V$ | ||
| - $V' = 2V + 2$ | ||
| - $E' = E + 2V$ | ||
| - $\mathcal{O}(V (E + 2V + 2V + 2)) = \mathcal{O}(V E + V^2) = \mathcal{O}(V E)$ | ||
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| ## Circulation | ||
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| - $d(v) > 0$ if demand | ||
| - $d(v) < 0$ if supply | ||
| - Capacity Constraint $0 \leq f(e) \leq c(e)$ | ||
| - Conservation Constraint $f^{\text{in}} (v) - f^{\text{out}} (v) = d(v)$. | ||
| - For every feasible circulation $\sum_{v \in V} d(v) = 0$, there is a feasible circulation with demands $d(v)$ in $G$ if and only if the maximum $s-t$ flow in $G'$ has value $D = \sum_{d(v)>0} d(v)$. | ||
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| ## Circulation with Demands and Lower Bounds. | ||
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| - Capacity Constraint $l(e) \leq f(e) \leq c(e)$ | ||
| - Conservation Constraint $f^{\text{in}} (v) - f^{\text{out}} (v) = d(v)$ | ||
| - Given $G$ with lower bounds, we subtract the lower bound $l(e)$ from the capacity of each edge. | ||
| - Subtract $f_0^{\text{in}}(v) − f_0^{\text{out}}(v) = L(v)$ from the demand of each node. | ||
| - Solve the circulation problem on this new graph to get a flow $f$. | ||
| - Add $l(e)$ to every $f(e)$ to get a flow for the original graph. | ||
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| ## Existence of Linear Programming Solution. | ||
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| - Given an Linear Programming problem with a feasible set and an objective function $P$. | ||
| - If $S$ is empty, Linear Programming has no solution.`` | ||
| - If $S$ is unbounded, Linear Programming may or may not have the solution. | ||
| - If $S$ is bounded, Linear Programming has one or more solutions. | ||
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| ## Standard Form Linear Programming | ||
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| - $\max (c_1x_1 + \cdots + c_nx_n)$ | ||
| - subject to | ||
| - $a_{11}x_1 + \cdots + a_{1n}x_n ≤ b_1$ | ||
| - all the way to | ||
| - $a_{m1}x_1 + \cdots + a_{mn}x_n ≤ b_m$. | ||
| - Also, $x_1 \geq 0, \cdots, x_n \geq 0$. | ||
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| ## Matrix Form Linear Programming | ||
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| - $\max(c^T x)$ | ||
| - subject to | ||
| - $Ax \leq b$ | ||
| - $x \geq 0$ | ||
| - $x \in \mathbb{R}$ | ||
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| ## Integer Linear Program | ||
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| - all variables $\in \mathbb{N}$ | ||
| - is NP-Hard | ||
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| ## Dual Program | ||
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| - $\min(b^T y)$ | ||
| - subject to | ||
| - $A^T y \geq c$ | ||
| - $y \geq 0$ | ||
| - $y \in \mathbb{R}$ | ||
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| ### Weak Duality | ||
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| - The optimum of the dual is an upper bound to the optimum of the primal. | ||
| - $\text{OPT}(\text{primal}) ≤ \text{OPT}(\text{dual})$. | ||
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| ### Strong duality | ||
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| - Let P and D be primal and dual Linear Programming correspondingly. If P and D are feasible, then $c^T x = b^T y$. | ||
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| If a standard problem and its dual are feasible, both are feasibly bounded. If one problem has an unbounded solution, then the dual of that problem is infeasible. | ||
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| ## Possibilities of Feasibility | ||
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| | P\D | Feasibly Bounded | Feasibly Unbounded | Infeasible | | ||
| | ------------------ | ---------------- | ------------------ | ---------- | | ||
| | Feasibly Bounded | Possible | Impossible | Impossible | | ||
| | Feasibly Unbounded | Impossible | Impossible | Possible | | ||
| | Infeasible | Impossible | Possible | Possible | | ||
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| ## [[P vs NP|P vs. NP]] | ||
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| - P = set of poly-time solvable problems. | ||
| - NP = set of poly-time verifiable problems. | ||
| - Decision Knapsack is NP-complete, whereas undecidable problems like Halting problems are only NP-Hard (not NP). | ||
| - X is NP-Hard if $\forall Y \in NP$ and $Y \leq_p X$. | ||
| - X is NP-complete if X is NP-Hard and $X \in NP$. | ||
| - If P = NP, then P = NP-complete. | ||
| - NP Problems can be solved in poly-time with NDTM. | ||
| - NP Problems can be verified in poly-time by DTM. | ||
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| ## Polynomial Reduction | ||
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| To reduce a decision problem Y to a decision problem X ($Y \leq_p X$), find a function $f$ that maps Y to X such that $f$ is poly-time computable and $\forall y \in Y$ is YES if and only if $f(y) \in X$ is YES. | ||
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| ## Proving NP-Complete | ||
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| Show X is in NP, Pick problem NP-complete Y, and show $Y \leq_p X$. | ||
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| ## Conjunctive Normal Form SAT | ||
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| - Conjunction of clauses. | ||
| - Literal: Variable or its negation. | ||
| - Clause: Disjunction of literals. | ||
| - [[3-SAT]]. Each clause has at most three literals. | ||
| - $(X_1 \lor \neg X_3) \land (X_1 \lor X_2 \lor X_4) \land \cdots$ | ||
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| ## Independent Set | ||
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| - Independent Set is a set of vertices in a graph, no two of which are adjacent. | ||
| - The max independent set asks for the size of the largest independent set in a given graph. | ||
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| ## Clique | ||
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| - Complete graph = every pair of vertices is connected by an edge. | ||
| - The max clique asks for the number of vertices in its largest complete subgraph. | ||
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| ## Vertex Cover | ||
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| - Vertex Cover of a graph is a set of vertices that includes at least one endpoint of every edge. | ||
| - The min vertex cover asks for the size of the smallest vertex cover in a graph. | ||
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| ## Hamiltonian Cycle | ||
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| - A cycle that visits each vertex exactly once. | ||
| - Hamiltonian Path visits each vertex exactly once and doesn't need to return to its starting point. | ||
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| ## Graph Coloring | ||
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| - Given G, can you color the nodes with $<k$ colors such that the end points of every edge are colored differently? |
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