NP-hard problems and Approximation algorithms 1

Lectures on NP-hard problems and

Approximation algorithms

COMP 523: Advanced Algorithmic Techniques

Lecturer: Dariusz Kowalski

NP-hard problems and Approximation algorithms 2

Overview

Previous lectures:

• Greedy algorithms

• Dynamic programming

• Network flows

These lectures:

• NP-hard problems

• Approximation algorithms

NP-hard problems and Approximation algorithms 3

P versus NPDecision problem: problem for which the answer

must be either YES or NOPolynomial time algorithm: there is a constant c

such that the algorithm solves 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

NP-hard problems and Approximation algorithms 4

Certifying in polynomial timeRepresentation of decision problem: set of inputs which

are correct (and should be answered YES, while others should be answered NO)

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 in sExample: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

NP-hard problems and Approximation algorithms 5

Computing an efficient certifierHow 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 NPOpen question:

P = NP ?

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Polynomial reductionsExample: decision problem

if there exists a perfect matching in a bipartite graph 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 reductionsDefinition:Problem X is polynomial-time reducible to problem Y, orproblem Y is at least as hard as problem X, iffproblem X can be solved by an 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

NP-hard problems and Approximation algorithms 8

Independent Set to Vertex CoverIndependent Set: given a 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 a 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 graphProof:Set S of size n - k is a vertex cover set in G iff

There is no edge between remaining k nodes iffSet of k remaining nodes is independent in G

NP-hard problems and Approximation algorithms 9

Vertex Cover to Set CoverVertex Cover: given a graph G of n nodes and parameter k, is there a set of k nodes such that every edge has at least one end among selected nodes?

Set Cover: given n nodes, m sets which cover the set of nodes, and parameter k, is there a family of k sets that 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.

NP-hard problems and Approximation algorithms 10

NP-completeness and NP-hardnessNP-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 the size of the maximum independent set

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Approximation algorithmsHaving an NP-hard problem, we do not know at this moment any polynomial-time algorithm solving the problem (exact solution)How to find an almost optimal solution?Approximation algorithm with ratio a > 1 gives a solution A such that

OPT A a OPT for a min-optimization problems

(1/a) OPT A OPT for a max-optimization problems

where OPT is the optimal solution.

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2-approximation for VCMinimum Vertex Cover - NP-hard problem(maximum is trivially n)Algorithm:Initialize set C to an empty setWhile 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 GOutput: witness set C and its size

NP-hard problems and Approximation algorithms 13

Analysis of 2-approximation for VCCorrectness: Each edge is removed only after one of its ends is chosen to set C, so each edge is coveredTermination:In each iteration we remove at least one edge from the graph, and there are less than n2 edgesApproximation 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-VCTime complexity: O(m + n)Exercise

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Approximation for SCMinimum Set Cover - NP-hard problem(maximum is trivially m)Greedy Algorithm:Initialize set C to empty setWhile 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 SCCorrectness: 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 new 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 : vS cv H(|S|) (H(i)=ji1/j denotes harmonic

number)• |C| = v cv SOPT vS cv H(n) SOPT1 = H(n) |OPT| |OPT| log nTime and memory complexities:

O(M + n), where M is the sum of cardinalities of setsExercise

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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

NP-hard problems and Approximation algorithms 17

Textbook and QuestionsREADING:• Chapters 8 and 11, Sections 8.1, 8.2, 8.3, 11.3, 11.4EXERCISES: • What is 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 a 2-approximation algorithm?

• Having a 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)?

NP-hard problems and Approximation algorithms 18

Overview

Previous lectures:

• NP-hard problems and approximation algorithms– Graph problems (IS, VC)– Set problem (SC)

This lecture:

• NP-hard numerical problems and their approximation– Numerical Knapsack problem– Weighted Independent Set

NP-hard problems and Approximation algorithms 19

Knapsack problemInput: set of n items, each represented by its

weight wi and value vi ; thresholds W and VDecision problem: is there a set of items of total

weight at most W and total value V ?Optimization problem: find a set of items with

– total weight at most W , and – maximum possible value

Assumptions: – weights and values are positive integers– each weight is at most W

NP-hard problems and Approximation algorithms 20

NP-hardness of knapsackKnapsack is NP-hard problem, butthere exists pseudo-polynomial algorithm

(complexity is polynomial in terms of values)Typical numerical polynomial algorithm:

polynomial in logarithm from the maximum values (longest representation)

Existence of pseudo-polynomial solution often yields very good approximation schemes

NP-hard problems and Approximation algorithms 21

Dynamic pseudo-polynomial optimization algorithm

Let v* be the maximum (integer) value of an item.Consider any order of objects.Let OPT(i,v) denote the minimum possible total weight of

a subset of items 1,2,…,i which has total value vDynamic formula for i = 0,1,…,n-1 and v = 0,1,…,nv* :OPT(i+1,v) =

= min{ OPT(i,v) , wi+1 + OPT(i,max{0,v-vi+1})}Formula OPT does not provide direct solution for our

problem, but can be easily adapted: maximum value of knapsack is the maximum value v such that OPT(n,v) W

NP-hard problems and Approximation algorithms 22

Dynamic algorithmInitialize array M[0…n,0…nv*] for storing OPT(i,v)Fill positions M[,0] and M[0,] with zerosFor i = 0,1,…,n-1

For v = 0,1,…,nv* M[i+1,v] :=

= min{ M[i,v] , wi+1 + M[i,max{0,v-vi+1}] }

Go through the whole array M and find the maximum value v such that M[n,v] W

NP-hard problems and Approximation algorithms 23

ComplexitiesTime: O(n2v*)• Initializing array M : O(n2v*)• Iterating loop: O(n2v*)• Searching for maximum v : O(n2v*)Memory: O(n2v*)

NP-hard problems and Approximation algorithms 24

Polynomial approximation algorithmAlgorithm:• Fix b = (/(2n)) v*• Set (by rounding up) xi = [vi/b]• Solve knapsack problem for values xi and

weights wi using dynamic program• Return set of computed items and its total value

in terms of the sum of vi’s

NP-hard problems and Approximation algorithms 25

AnalysisPTAS: polynomial time approximation scheme - for any fixed

positive it produces (1+)-approximation in polynomial time (but is hidden in big Oh)

Time: O(n2x*) = O(n3/)

Approximation: (1+)

NP-hard problems and Approximation algorithms 26

Analysis of approximation ratioRecall notation:• b = (/(2n)) v*• xi = [vi/b]

Approximation: (1+)Let S denote the set of items returned by the algorithm

• vi bxi vi + b iS bxi - b|S| iS vi

iS bxi v* = 2nb/ (2/ -1)nb iS vi

iOPT vi iOPT bxi iS bxi b|S|+iS (bxi - b)

b(2/ -1)n + iS vi iS vi + iS vi = (1+) iS vi

NP-hard problems and Approximation algorithms 27

Weighted Independent SetOptimization problem:Weighted Independent Set: given graph G of n valued

nodes, find an independent set of maximum value (set of nodes such that none two of them are connected by an edge)

Even for values 1 problem remains NP-hard, which is not the case for knapsack problem! WIS problem is an example of strong NP-hard problem, and no PTAS is known for it

NP-hard problems and Approximation algorithms 28

Conclusions

• Optimization numerical problem in NP-hard– Maximum Knapsack– Weighted Independent Set

• PTAS in time O(n3) for Knapsack, based on dynamic programming

NP-hard problems and Approximation algorithms 29

Textbook and Questions

• Chapters 6 and 11, Sections 6.4, 11.8

• Is it possible to design an efficient Knapsack algorithm based on dynamic programming for the case where weights are small (values can be arbitrarily large)

• How to implement arithmetical operations: + - * / and rounding, each in time proportional to at most square of the length of the longest number? What are the complexity formulas?

NP-hard problems and Approximation algorithms 30

Overview

Previous lectures:• NP-hard problems • Approximation algorithms

– Greedy (VC and SC)– Dynamic Programming (Knapsack)

This lecture:

• Approximation through integer programming

NP-hard problems and Approximation algorithms 31

Vertex CoverWeighted Vertex Cover: (weights are in nodes)• Decision problem:

– given weighted graph G of n nodes and parameter k,

– is there a set of nodes with total weight k such that every edge has at least one end in this set?

• Optimization problem: – given weighted graph G of n nodes, what is the minimum

total weight of a set such that every edge has at least one end in this set?

NP-hard problems and Approximation algorithms 32

Approximation algorithmsHaving 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 > 1 gives a solution A such that

OPT A a OPT for a min-optimization problemsOPT/a A OPT for a max-optimization problems

where OPT is an optimal solution.

NP-hard problems and Approximation algorithms 33

2-approximation for VCMinimum Vertex Cover - NP-hard problem even for all weights = 1(maximum is trivially n)Algorithm: (for all weights equal to 1)Initialize set C to empty setWhile 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 GOutput: witness set C and its size

NP-hard problems and Approximation algorithms 34

Integer Programming

• Represent the problem as Integer Programming

• Relax the problem to Linear Programming

• Solve Linear Programming

• Round the solution to get integers

NP-hard problems and Approximation algorithms 35

Integer and linear programsSet of constraints (linear equations):

x1 , x2 0

x1 + 2x2 6

2x1 + x2 6Function to minimize (linear):

4x1 + 3x2

Linear programming: – variables are real numbers– there are polynomial time algorithms solving it (e.g., interior point

method - by N. Karmarkar in 1984); simplex method is not polynomial

Integer programming: – variables are integers– problem is NP-hard

NP-hard problems and Approximation algorithms 36

VC as Integer ProgramSet of constraints :

xi {0,1} for every node i

xi + xj 1 for every pair {i,j} EFunction to minimize:

i xiwi

Example: x1 , x2 , x3 , x4 {0,1}

x1 + x3 1 , x1 + x4 1 , x2 + x4 1 , x2 + x3 1

Minimize: x1 + x2 + x3 + x4 x1

x2

x3

x4

NP-hard problems and Approximation algorithms 37

Relaxation to Linear ProgramSet of constraints :

yi [0,1] for every node i

yi + yj 1 for every pair {i,j} EFunction to minimize:

i yiwi

Example: y1 , y2 , y3 , y4 [0,1]

y1 + y3 1 , y1 + y4 1 , y2 + y4 1 , y2 + y3 1

Minimize: y1 + y2 + y3 + y4 y1

y2

y3

y4

NP-hard problems and Approximation algorithms 38

Rounding the linear program solutionObtained exact Linear Program solution yi [0,1] for every node isatisfying

yi + yj 1 for every pair {i,j} EHow to obtain a (approximate?) solution for Integer Program?Rounding: for every node i

xi = 1 iff yi 1/2 (otherwise xi = 0)

Example: y1 , y2 , y3 , y4 = 1/2

x1 , x2 , x3 , x4 = 1 Optimum solution (minimum) e.g.:

x1 , x2 = 1, x3 , x4 = 0

x1

x2

x3

x4

NP-hard problems and Approximation algorithms 39

AnalysisCorrectness: since each xi {0,1} and each edge is guarded by

constraint xi + xj 1 which is satisfied also after roundingTime: time for solving linear program plus O(m+n)Approximation:

Each xi is at most twice as large as yi hence the weighted sum of xi is also at most twice bigger than the weighted sum of yi

Example: y1 , y2 , y3 , y4 = 1/2

x1 , x2 , x3 , x4 = 1 Optimum solution (minimum) e.g.:

x1 , x2 = 1, x3 , x4 = 0

x1

x2

x3

x4

NP-hard problems and Approximation algorithms 40

Conclusions• Decision problems P and NP-complete

– Polynomial-time reduction

• Optimization problems in NP-hard– Maximum Independent Set– Minimum Vertex Cover– Minimum Set Cover– Maximum Knapsack

• Approximation algorithms - polynomial time– Greedy (VC, SC)– Dynamic program (Knapsack)– Integer and Linear programs (weighted VC)

NP-hard problems and Approximation algorithms 41

Textbook and QuestionsREADING:

Chapter 11, Section 11.6

EXERCISES:

• Could we solve Weighted VC by modification of greedy algorithm solving (pure) VC?

• What approximation we get if we apply randomized rounding, i.e.,xi = 1 with probability yj (otherwise xi = 0)

• Traveling Salesman Problem : Section 8.5

• TSP can not be approximated with a constant unless P=NP

• Constant approximation of TSP problem under the assumption that the weights satisfy metric conditions (symmetric weights satisfying triangle inequality)