P vs NP: Vertex Cover and Subset Sum

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Questions and Answers

What does it mean for a Vertex Cover S in a graph G = (V, E) to be considered minimal?

  • _S_ contains all vertices from _V_.
  • _S_ contains the maximum possible number of vertices from _V_.
  • _S_ contains the fewest possible vertices such that every edge in _E_ is incident to at least one vertex in _S_. (correct)
  • _S_ contains a random selection of vertices from _V_.

Why is the Vertex Cover problem considered NP-hard?

  • Because it is impossible to verify a solution.
  • Because it is easy to verify a solution, but difficult to find one.
  • Because it can be reduced to other NP-complete problems. (correct)
  • Because it can be solved in polynomial time.

A graph G = (V, E) has a Vertex Cover of size k. What can be inferred about the size of the largest independent set in G?

  • The size of the largest independent set is unrelated to _k_.
  • The largest independent set has a size of _k_.
  • The largest independent set has a size of at most |_V_| - _k_. (correct)
  • The largest independent set has a size of greater than _k_.

Which of the following problems can be reduced to the VERTEX-COVER problem?

<p>All of the above (D)</p> Signup and view all the answers

Given an instance of the SUBSET-SUM problem with numbers $x_1, ..., x_n$ and a target value T, what is the goal?

<p>To find a subset <em>S</em> of the numbers such that the sum of the numbers in <em>S</em> equals <em>T</em>. (A)</p> Signup and view all the answers

What is the significance of showing that VERTEX-COVER $\leq_p$ SUBSET-SUM?

<p>It proves that SUBSET-SUM is at least as hard as VERTEX-COVER. (A)</p> Signup and view all the answers

In the reduction from VERTEX-COVER to SUBSET-SUM, suppose we have a vertex VC corresponding to a 'hnúta og leggi'. What does this vertex cover?

<p>It covers all edges incident to the vertex. (B)</p> Signup and view all the answers

During the reduction from VERTEX-COVER to SUBSET-SUM, what question needs to be answered to determine if there is a suitable reduction?

<p>Is there a subset of vertices with a sum equal to <em>T</em>? (B)</p> Signup and view all the answers

When reducing VERTEX-COVER to SUBSET-SUM, what is the significance of finding a 'hunts pelja' (vertex cover) of size $\leq k$ for a graph G = (V, E)?

<p>It implies that the SUBSET-SUM instance has a solution. (B)</p> Signup and view all the answers

In the described reduction from VERTEX-COVER to SUBSET-SUM, each edge e in the graph is assigned a numerical value. What is the range of these values?

<p>0 to |<em>E</em>| - 1 (C)</p> Signup and view all the answers

When constructing the numbers $a_v$ and $b_i$ for the reduction from VERTEX-COVER to SUBSET-SUM, what base is used for representing these numbers?

<p>Base 4 (C)</p> Signup and view all the answers

In the VERTEX-COVER to SUBSET-SUM reduction, what does the term $4^{|E|}$ in the value of $a_v$ represent?

<p>A unique identifier for each vertex. (C)</p> Signup and view all the answers

During the reduction of VERTEX-COVER to SUBSET-SUM, how is the target value T calculated using k (the size of the vertex cover) and |E| (the number of edges)?

<p>$T = k + \sum_{i=0}^{|E|-1} 2 \cdot 4^i$ (B)</p> Signup and view all the answers

What does $S' = {a_v | v \in S} \cup {b_i | i \text{ senhetr niku. I enlepotis }}$ represent in the context of reducing VERTEX-COVER to SUBSET-SUM?

<p>A candidate solution to the SUBSET-SUM problem based on the vertex cover <em>S</em>. (A)</p> Signup and view all the answers

If the sum of $x_i$ over all elements in $S'$ equals a constant T, what does this imply about the set S in the VERTEX-COVER problem?

<p><em>S</em> is a vertex cover of the graph. (B)</p> Signup and view all the answers

What is the significance of ensuring that there is no 'carry' during the summation when solving the SUBSET-SUM problem resulting from a VERTEX-COVER reduction?

<p>It guarantees that each bit position corresponds uniquely to either a vertex or an edge. (D)</p> Signup and view all the answers

In the context of the reduction from VERTEX-COVER to SUBSET-SUM, what does $V'$ and $E'$ represent?

<p>The chosen vertices and edges that verify the vertex cover. (B)</p> Signup and view all the answers

When transforming a VERTEX-COVER problem into a SUBSET-SUM problem, if you find that $V'$ is a vertex cover, what can you say about the corresponding SUBSET-SUM instance?

<p>The SUBSET-SUM instance has a solution where the sum equals <em>T</em>. (B)</p> Signup and view all the answers

Assuming VERTEX-COVER $\leq_p$ SUBSET-SUM and SUBSET-SUM is NP-hard, what conclusion can be drawn about VERTEX-COVER?

<p>VERTEX-COVER is NP-hard. (D)</p> Signup and view all the answers

In the context of algorithm analysis, what does 'Kült bestor leyir Subset-son!' suggest about solving the Subset-Sum problem?

<p>It suggests that the Subset-Sum problem can be solved efficiently using dynamic programming. (A)</p> Signup and view all the answers

When using dynamic programming to solve the Subset-Sum problem, what does $A_{ij}$ represent?

<p>Whether there is a subset of the first <em>i</em> numbers that sums to <em>j</em>. (D)</p> Signup and view all the answers

In the dynamic programming solution for SUBSET-SUM, what is the recurrence relation for determining $A_{ij}$?

<p>$A_{ij} = \max(A_{i-1, j-x_i}, A_{i-1, j})$ (A)</p> Signup and view all the answers

What is the time complexity for solving the Subset-Sum problem using dynamic programming, where n is the number of items and T is the target sum?

<p>$O(n \cdot T)$ (D)</p> Signup and view all the answers

What relationship best describes P, NP, and NP-complete?

<p>NP-complete is a subset of NP, and P is a subset of NP (A)</p> Signup and view all the answers

In computational complexity theory, what is the generally accepted relationship between P and NP?

<p>P ⊆ NP (B)</p> Signup and view all the answers

If a problem is NP-complete, what does this imply about its relationship to all other problems in NP?

<p>Any problem in NP can be reduced to it in polynomial time. (C)</p> Signup and view all the answers

What is the significance of a problem being classified as NP-opluk?

<p>The significance is unmentioned in the text. (B)</p> Signup and view all the answers

What is the relationship between NP-hard and NP-complete problems?

<p>All NP-complete problems are also NP-hard. (A)</p> Signup and view all the answers

What is the class 'co-NP'?

<p>The set of problems that are complements of NP problems. (B)</p> Signup and view all the answers

What is the practical implication of a problem being NP-hand?

<p>The significance is unmentioned in the text. (D)</p> Signup and view all the answers

What is the significance of problems being within P-space?

<p>It is unmentioned in the text in relation to the illustration. (A)</p> Signup and view all the answers

Besides exact algorithms, what are some strategies used in practice for addressing computationally hard problems?

<p>Approximation algorithms (A)</p> Signup and view all the answers

What is the purpose of using 'Nálganir (materoour laugκης med heiltök bestun)' techniques?

<p>To solve the problem effectively (C)</p> Signup and view all the answers

What does the term 'SAT-solers' refer to?

<p>A type of algorithm that solves the Boolean satisfiability problem (C)</p> Signup and view all the answers

Which algorithm can have a simple 2-approximation algorithm

<p>Vertex cover (C)</p> Signup and view all the answers

What is the time complexity of this version Subset-Sum dynamic program?

<p>$O(nT = n2^t)$ (C)</p> Signup and view all the answers

Flashcards

Vertex Cover (Hnutþekja)

Given a graph G=(V,E), find the smallest subset S of vertices, such that all edges in E are incident to at least one endpoint in S.

Vertex-Cover NP-erfitt

Determining if a vertex cover of size k exists in a graph is an NP-hard problem.

NP-hard

A problem is NP-hard if an algorithm to solve it can be converted to solve any NP-problem.

Subset-Sum

Given a set of numbers X1...Xn and a target value T, determine if there exists a subset S of the numbers such that the sum of the numbers in S equals T.

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Problem Reduction

A reduction from problem A to problem B shows that if B can be solved efficiently, then A can also be solved efficiently.

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Cook-Levin Theorem

The Cook-Levin theorem states that the Boolean satisfiability problem (SAT) is NP-complete.

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Vertex Cover and Subset-Sum

A vertex cover of size k exists in G if and only if a subset-sum solution exists.

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Subset-Sum Time Complexity

The time complexity of solving subset-sum using dynamic programming is O(n*T), where n is the number of items and T is the target sum.

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NP vs. P

Problems in NP (Nondeterministic Polynomial time) are verifiable in polynomial time, while problems in P (Polynomial time) are solvable in polynomial time.

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NP-complete

NP-complete problems are the hardest problems in NP, meaning that if you find a polynomial-time algorithm for one NP-complete problem, you can solve all NP problems in polynomial time.

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Study Notes

  • The text is about algorithm analysis, specifically focusing on the concepts of P and NP. It's dated 2025 and attributed to Páll Melsted

Vertex Cover

  • The goal of the Vertex Cover problem, given a graph G=(V,E), is to find the smallest set S of vertices such that every edge in E is incident to at least one endpoint in S.
  • The Vertex Cover problem is classified as NP-hard
  • A solution S constitutes a vertex cover if it contains k vertices
  • The complement of a vertex cover, known as the Independent Set, has a size of |V|-k.

Reductions

  • Circuitsat reduces to 3-SAT, 3-SAT reduces to Max Independent Set, Max Independent set reduces to vertex cover, and vertex cover reduces to Subset Sum

Subset-Sum

  • In the Subset-Sum problem, given a set of numbers x1 to xn and a target value T, the question is whether there exists a subset S of the numbers that sums exactly to T
  • Showing that Vertex Cover is reducible to Subset-Sum

Vertex Cover Reduction to Subset Sum

  • The Vertex Cover problem involves vertices and edges, where the aim is to find a subset of vertices that covers all edges
  • The Subset-Sum problem involves selecting certain numbers that sum to a value T.
  • To transform an instance of Vertex Cover to Subset Sum you need to decide what numbers to use and what the sum T should be
  • Given an instance of Vertex Cover with a graph G = (V, E), the task is to determine if there exists a vertex cover of size less than or equal to k.
  • Each edge is assigned a unique index from 0 to |E|-1.
  • Numbers are represented in base 4, where each number ai corresponds to a vertex and has a base value of 4 raised to the power of |E|, plus an additional 4j if vertex Vi is touching the jth edge
  • Each number bi corresponds to an edge and has a base value of 4j
  • Target sum T = k * 4|E| + sum from i=0 to |E|-1 of 2 * 4i

Proof of Correctness for Vertex Cover to Subset Sum

  • If S is a vertex cover of size k, then S' = {av | v ∈ S} ∪ {bi | i is index of an edge} constitutes a solution
  • The sum of xi over i in S' is equal to the sum of av over v in S, plus the sum of bi
  • The sum of av is equal to k ∗ 4|E| + sum of 4j
  • Taking the sum of all bi values, from j=0 to |E|-1 of 2 * 4j, gives k ∗ 4|E| + sum from i=0 to |E|-1 of 2 * 4i = T
  • There are at most three numbers in s' that have significant digits in the sort j
  • Let V' and E' represent the vertices and edges, respectively
  • Then, summation of av over v in V' plus summation of bj over j in E’ = T
  • If an edge (u,v) belongs to E, then either u or v belongs to V
  • If any u is a member of V', or v is a member of V', then adding j to 0 in the first set of values gives a sum of av over v in V', plus 1
  • If vertex cover reduces to Subset Sum, and Subset Sum is NP-Difficult

Subset Sum Algorithm

  • Subset Sum is in NP
  • Given X1...Xn, T
  • A is an n x T array
  • A(i,j) is equal to 1 if some of the first i elements can sum to j
  • Otherwise it is zero
  • A(i,j) = max {A(i-1, j - xi), A(i-1, j)}
  • A(i, 0) = 1
  • A(0, 0) = 1
  • A(0, j) = 0 for j > 0
  • log2(T) bits are used to file/save T
  • t = log2(T)
  • n.T = n * 2t

Complexity Classes

  • A Venn diagram shows the relationship between complexity classes P, NP, co-NP, and NP-complete
  • P-space
  • Log-space

Practical Considerations and Approaches

  • A few real-world approaches given the computational complexity
  • Ensure you understand the type of problem (first!)
  • Vertex cover has some simple 2-approximation algorithms
  • Use "meta-heuristic" algorithms (simulated annealing)
  • SAT-solvers are known

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