CS 6515 Algos Test 3 with verified answers
Search Problem - correct answer ✔A search problem is specified by an algorithm C that
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takes two inputs, an instance I and a proposed solution S, and runs in time polynomial in
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|I|. We say S is a solution to I if and only if C(I, S) = true
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Steps for an NP Proof - correct answer ✔1. Demonstrate that problem B is in the class of
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NP problems
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2. Demonstrate that problem B is at least as hard as a problem believed to be NP-
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Complete*. |!|
This is done via reduction from a known problem A (A→B)
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2a. Show how an instance of A is converted to B in polynomial time
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2b. Show how a solution to B can be converted to a solution for A, again in polynomial
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time
2c. Show that a solution for B exists if-and-only-if (IFF) a solution to A exists. You must
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prove both directions parts:
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|!| - If you have a solution to B you have a solution to A
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If there is no solution for B, then no solution exists for A. This may be done via the
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contra-positive approach, showing that if a solution exists for A then a solution for B must
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exist
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Reduction - correct answer ✔A reduction from search problem A to search problem B is a
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polynomial-time algorithm f that transforms any instance I of A into an instance f(I) of B,
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together with another polynomial-time algorithm h that maps any solution S of f(I) back
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into a solution h(S) of I; see the following diagram. If f(I) has no solution, then neither
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does I |!|
Independent Set Definition - correct answer ✔Given a graph G = (V, E) and an integer k, is
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there a subset
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of vertices S ⊆ V such that | S | ≥ k, and for each edge at most one of its endpoints is
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in S ?
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,Independent Set Input/Output - correct answer ✔Input: An undirected graph, G = (V, E). A
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goal g. |!|
Output: The independent set S with size |S| ≥ g, if one exists.
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Clique definition - correct answer ✔find a set of g ver- tices such that all possible edges
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between them are present |!| |!| |!|
Clique Input / Output - correct answer ✔Input: A graph G = (V, E) and a goal, g.
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Output: A subset of vertices, S ⊆ V where S is a clique whose size meets the goal, so |S|
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≥ g.
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Vertex Cover definition - correct answer ✔find b vertices that cover (touch) every edge.
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This is a special case of set cover
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Vertex Cover Input/Output - correct answer ✔Input: A graph, G = (V, E) and a budget, b
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Output: A set of vertices S ⊂ V that form a vertex cover with a size that is under-budget,
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so |S| ≤ b.
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Rudrata Cycle Definition - correct answer ✔given a graph, find a cycle that visits each
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vertex exactly once—or report that no such cycle exists
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Rudrata Cycle Input/Output - correct answer ✔Input: Graph G = (V, E)Output: A cycle that
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goes through every vertex of the graph exactly once
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Rudrata Path Definition - correct answer ✔given a graph, find a path that visits each vertex
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exactly once—or report that no such path exists
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Rudrata (s, t)-Path Input/Output - correct answer ✔Input: Graph G = (V, E), Start and End
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Vertices (s, t) |!| |!|
Output: A path that starts at s, ends at t, and goes through every vertex of the graph
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exactly once |!|
, Zero-One (0-1) Integer Linear Programming (ILP): Definition - correct answer ✔an approach
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used for problems that can be set up as linear programs with the additional requirement
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that some or all of the decision recommendations be integer values
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The unknowns are binary, and only the restrictions must be satisfied.
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Note: General ILP is NP-HARD
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if ILP includes an objective function then it would classify as NP-Hard but not NP-Complete,
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as we have no known polynomial-time algorithm for validating the solution. By removing
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the objective function we essentially make it a search problem -- is there a set of values
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for the variables which satisfy all the constraints? That is easily validated in linear time.
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Integer Linear Programming (ILP) Input/Output - correct answer ✔Input: Ax <= b where A is
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a m x n matrix, b is an m-Vector
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Output: Non-Negative Integer Vector x satisfying the inequality Ax <= b
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SAT Definition - correct answer ✔given a Boolean formula in conjunctive normal form,
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either find a satisfying truth assignment or else report that none exists.
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A satisfying truth assignment is an assignment of false or true to each variable so that
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every clause contains a literal whose value is true.
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SAT Input/Output - correct answer ✔Input: n clauses with any number of literals in each
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clause
Output: An assignment of literals that satisfy all the clauses of the input
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3SAT definition - correct answer ✔given a Boolean formula in conjunctive normal form with
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at most 3 variables in each clause, either find a satisfying truth assignment or else report
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that none exists.
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3SAT Input/Output - correct answer ✔Input: n clauses with at most 3 unique literals each
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Output: An assignment if literals that satisfy all the clauses of the input
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Search Problem - correct answer ✔A search problem is specified by an algorithm C that
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takes two inputs, an instance I and a proposed solution S, and runs in time polynomial in
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|I|. We say S is a solution to I if and only if C(I, S) = true
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Steps for an NP Proof - correct answer ✔1. Demonstrate that problem B is in the class of
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NP problems
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2. Demonstrate that problem B is at least as hard as a problem believed to be NP-
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Complete*. |!|
This is done via reduction from a known problem A (A→B)
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2a. Show how an instance of A is converted to B in polynomial time
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2b. Show how a solution to B can be converted to a solution for A, again in polynomial
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time
2c. Show that a solution for B exists if-and-only-if (IFF) a solution to A exists. You must
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prove both directions parts:
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|!| - If you have a solution to B you have a solution to A
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If there is no solution for B, then no solution exists for A. This may be done via the
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contra-positive approach, showing that if a solution exists for A then a solution for B must
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exist
|!|
Reduction - correct answer ✔A reduction from search problem A to search problem B is a
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polynomial-time algorithm f that transforms any instance I of A into an instance f(I) of B,
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together with another polynomial-time algorithm h that maps any solution S of f(I) back
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into a solution h(S) of I; see the following diagram. If f(I) has no solution, then neither
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does I |!|
Independent Set Definition - correct answer ✔Given a graph G = (V, E) and an integer k, is
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there a subset
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of vertices S ⊆ V such that | S | ≥ k, and for each edge at most one of its endpoints is
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in S ?
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,Independent Set Input/Output - correct answer ✔Input: An undirected graph, G = (V, E). A
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goal g. |!|
Output: The independent set S with size |S| ≥ g, if one exists.
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Clique definition - correct answer ✔find a set of g ver- tices such that all possible edges
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between them are present |!| |!| |!|
Clique Input / Output - correct answer ✔Input: A graph G = (V, E) and a goal, g.
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Output: A subset of vertices, S ⊆ V where S is a clique whose size meets the goal, so |S|
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≥ g.
|!| |!|
Vertex Cover definition - correct answer ✔find b vertices that cover (touch) every edge.
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This is a special case of set cover
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Vertex Cover Input/Output - correct answer ✔Input: A graph, G = (V, E) and a budget, b
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Output: A set of vertices S ⊂ V that form a vertex cover with a size that is under-budget,
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so |S| ≤ b.
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Rudrata Cycle Definition - correct answer ✔given a graph, find a cycle that visits each
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vertex exactly once—or report that no such cycle exists
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Rudrata Cycle Input/Output - correct answer ✔Input: Graph G = (V, E)Output: A cycle that
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goes through every vertex of the graph exactly once
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Rudrata Path Definition - correct answer ✔given a graph, find a path that visits each vertex
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exactly once—or report that no such path exists
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Rudrata (s, t)-Path Input/Output - correct answer ✔Input: Graph G = (V, E), Start and End
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Vertices (s, t) |!| |!|
Output: A path that starts at s, ends at t, and goes through every vertex of the graph
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exactly once |!|
, Zero-One (0-1) Integer Linear Programming (ILP): Definition - correct answer ✔an approach
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used for problems that can be set up as linear programs with the additional requirement
|!| |!| |!| |!| |!| |!| |!| |!| |!| |!| |!| |!| |!| |!| |!|
that some or all of the decision recommendations be integer values
|!| |!| |!| |!| |!| |!| |!| |!| |!| |!|
The unknowns are binary, and only the restrictions must be satisfied.
|!| |!| |!| |!| |!| |!| |!| |!| |!| |!|
Note: General ILP is NP-HARD
|!| |!| |!| |!|
if ILP includes an objective function then it would classify as NP-Hard but not NP-Complete,
|!| |!| |!| |!| |!| |!| |!| |!| |!| |!| |!| |!| |!| |!|
as we have no known polynomial-time algorithm for validating the solution. By removing
|!| |!| |!| |!| |!| |!| |!| |!| |!| |!| |!| |!| |!| |!|
the objective function we essentially make it a search problem -- is there a set of values
|!| |!| |!| |!| |!| |!| |!| |!| |!| |!| |!| |!| |!| |!| |!| |!| |!|
for the variables which satisfy all the constraints? That is easily validated in linear time.
|!| |!| |!| |!| |!| |!| |!| |!| |!| |!| |!| |!| |!| |!|
Integer Linear Programming (ILP) Input/Output - correct answer ✔Input: Ax <= b where A is
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a m x n matrix, b is an m-Vector
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Output: Non-Negative Integer Vector x satisfying the inequality Ax <= b
|!| |!| |!| |!| |!| |!| |!| |!| |!| |!|
SAT Definition - correct answer ✔given a Boolean formula in conjunctive normal form,
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either find a satisfying truth assignment or else report that none exists.
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A satisfying truth assignment is an assignment of false or true to each variable so that
|!| |!| |!| |!| |!| |!| |!| |!| |!| |!| |!| |!| |!| |!| |!| |!|
every clause contains a literal whose value is true.
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SAT Input/Output - correct answer ✔Input: n clauses with any number of literals in each
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clause
Output: An assignment of literals that satisfy all the clauses of the input
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3SAT definition - correct answer ✔given a Boolean formula in conjunctive normal form with
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at most 3 variables in each clause, either find a satisfying truth assignment or else report
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that none exists.
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3SAT Input/Output - correct answer ✔Input: n clauses with at most 3 unique literals each
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Output: An assignment if literals that satisfy all the clauses of the input
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