Lectures 17 18 NPhard problems: Introduction

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Lectures 17 - 18 NP-hard problemsIntroduction

• COMP 523 Advanced Algorithmic Techniques
• Lecturer Dariusz Kowalski

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Overview

• Previous lectures
• Network flows
• Applications
• largest matching in bipartite graphs
• largest number of disjoint paths
• This lecture
• NP-hard problems

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P versus NP

• Decision problem problem for which the answer
  must be either YES or NOT
• Polynomial time algorithm there is a constant c
  and the algorithm solving the problem in time
  O(nc) for every input of size n
• P (polynomial time) - class of decision problems
  for which there is a polynomial time
  deterministic algorithm solving the problem
• NP (nondeterministic polynomial time) - class of
  decision problems for which there is a certifier
  which can check a witness in polynomial time

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Certifying in polynomial time

• A decision problem set of inputs which are
  correct (and should be answered YES, while others
  should be answered NOT)
• An efficient certifier B for problem X
• B is in P
• s is in X iff B(s,t) YES for some t of size
  polynomial of s
• Example
• Problem is there a clique of size k in a given
  graph with n nodes?
• Certifier if a given graph has a clique of size
  k then given
• this clique as the second parameter we can
  answer YES

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Computing an efficient certifier

• How to compute witnesses for an efficient
• certifier for a given problem?
• Fact
• If the witnesses for the efficient certifier can
  be
• found in polynomial time then the problem is in
  P.
• Conclusion
• P is included in NP
• Open question
• P NP ?

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Polynomial reductions

• Example decision problem if there exists a
  perfect
• matching in bipartite graphs can be reduced to
• network flow problem in polynomial time (by
• adding source, target and directing the edges)
• Other problems for undirected graphs (in NP and
• not known to be in P)
• Independent Set of nodes
• Vertex cover
• Set Cover

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Polynomial reductions

• Definition
• Problem X is polynomial-time reducible to problem
  Y, or
• problem Y is at least as hard as problem X, iff
• problem X can be solved by the algorithm which
  works in
• polynomial time and uses polynomial number of
  calls to the
• black box solving problem Y.
• Notation X ?P Y
• Transitivity Property if X ?P Y and Y ?P Z then
  X ?P Z

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Independent Set to Vertex Cover

• Independent Set given graph G of n nodes and
  parameter k,
• is there a set of k nodes such that none two of
  them are connected by
• an edge?
• Vertex Cover given graph G of n nodes and
  parameter k,
• is there a set of k nodes such that every edge
  has at least one
• end selected?
• Polynomial Reduction
• Solve Vertex-Cover(n-k) for the same graph
• Proof
• Set S of size n - k is a vertex cover set in G
  iff
• There is no edge between remaining k nodes iff
• Set of k remaining nodes is independent in G

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Vertex Cover to Set Cover

• Vertex Cover given graph G of n vertices and
  parameter k,
• is there a set of k vertices such that every edge
  has at least one
• end selected?
• Set Cover given n nodes, m sets which cover the
  set of nodes, and parameter k,
• is there a family of k sets which covers all n
  nodes?
• Polynomial Reduction
• Let each edge from the graph correspond to the
  node for SC system
• For each vertex in the graph create a set of
  incident edges to SC system
• Solve Set-Cover(m,n,k) for the created SC system
  with m nodes and n sets
• Proof
• Each node-edge is covered by at least one
  set-vertex since each node is covered
• This covering is minimal

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NP-completeness and NP-hardness

• NP-complete class of problems X such that every
• problem from NP is polynomial-time reducible to
  X.
• Optimization problems problems where the
• answer is a number (maximum/minimum possible)
• Each optimization problem has its decision
  version, e.g.,
• Find a maximum Independent Set
• Is there an Independent Set of size k?
• NP-hard class of optimization problems X such
  that its
• decision version is NP-complete.
• Example having solution for decision version of
• Independent Set problem, we can probe a parameter
  k,
• starting from k 1 , to find a size of a maximum
  independent set

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Approximation algorithms

• Having an NP-hard problem, we do not know in this
  
• moment any polynomial-time algorithm solving the
• problem (exact solution)
• How to find almost optimal solution?
• Approximation algorithm with ratio a gt 1 gives a
  solution
• A such that
• A ? a ? OPT for a min-optimization problems
• OPT ? a ? A for a max-optimization problems
• where OPT is an optimal solution.

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2-approximation for VC

• Minimum Vertex Cover - NP-hard problem
• (maximum is trivially n)
• Algorithm
• Initialize set C to empty set
• While there are remaining edges
• Choose an edge v,w with the largest degree,
  where degree of an edge is a sum of degrees of
  its ends v,w in a current graph G
• Put v,w to C
• Remove all the edges adjacent to nodes v,w from
  graph G
• Output witness set C and its size

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Analysis of 2-approximation for VC

• Correctness
• Each edge is removed only after one of its ends
  is chosen
• to set C, so each edge is covered
• Termination
• In each iteration we remove at least one edge
  from the
• graph, and there are less than n2 edges
• Approximation ratio 2
• For each edge v,w selected at the beginning of
  an iteration at least
• one end must be in min-VC, and we selected two,
  so set C is at most
• twice bigger than the min-VC
• Time complexity O(m n)
• Exercise

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Approximation for SC

• Minimum Set Cover - NP-hard problem
• (maximum is trivially m)
• Greedy Algorithm
• Initialize set C to empty set
• While there are uncovered nodes
• Choose a set F which covers the largest number of
  uncovered nodes
• Put F to C
• Remove all nodes covered by F
• Output witness set C and its size

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Analysis of approximation for SC

• Correctness
• Each node is marked as covered when we put the
  set covering it to set C.
• The algorithm stops when all nodes are covered.
• Termination
• In each iteration we cover at least one node, and
  there are n nodes.
• Approximation ratio log n
• Let Si be the set selected to C in ith iteration,
  and denote by si the number of uncovered nodes
  covered by Si OPT be the minimum covering set
• Let cv 1/ si for each node v which was covered
  by Si for the first time
• The following holds C ?v cv
• For every set S Si ?v?S cv ? H(S)
  (H(i)?j?i1/j denotes harmonic number)
• C ?v cv ? ?S?OPT ?v?S cv ? H(n) ?S?OPT1
  H(n) OPT ? OPT log n
• Time and memory complexities
• O(M n), where M is the sum of cardinalities of
  sets
• Exercise

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Conclusions

• Decision problems P and NP-complete
• Polynomial-time reduction
• Optimization problems in NP-hard
• Maximum Independent Set
• Minimum Vertex Cover
• Minimum Set Cover
• Approximation algorithms - polynomial time
• Min-VC with ratio 2
• Min-SC with ratio log n

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Exercises

• Analyse the time and memory complexities of
  min-VC approximation algorithm with ratio 2 and
  min-SC algorithm
• Consider a modification of min-VC algorithm
  choose a node which covers the largest number of
  uncovered edges. Is it 2-approximation algorithm?
• Having the 2-approximation algorithm for min-VC,
  is it easy to modify it to be a 2-approximation
  algorithm for max-IS (since there is a simple
  polynomial-time reduction between these two
  problems)?