Fabrizio Frati

1. Tecniche Algoritmiche per Grafi e Reti Fabrizio Frati Dipartimento di Informatica e Automazione Università degli Studi Roma Tre

2. 3 CREDITI • I° periodo: • Lunedì 9:45-11:15 aula N13 • II° periodo: • Venerdì 8:00-9:45 aula N7 • Contatti: • {angelini,gdb,frati,ratm} @dia.uniroma3.it • Ricevimento: mercoledìdalle 17 alle 19

3. Struttura del Corso • ~ 8 seminari • algoritmi, strutturedati, combinatorica, graph drawing… • progetti a gruppidi 2 - 4 persone • problemadiricerca • MaterialeDidattico • slides • G. Di Battista, P. Eades, R. Tamassia, I. G. Tollis, GraphDrawing: Algorithmsfor the VisualizationofGraphs, 1999. Prentice Hall.

4. ACM Computing Classification System • General Literature • Hardware • Computer System Organization • Software • Data • Theory of Computation • Mathematics of Computing • Information Systems • Computing Methodologies • Computer Applications

5. ACM Computing Classification System • C. Computer System Organization • Processor Architectures • Computer-Communication Networks • Application-Based Systems • Performance of Systems • Computer System Implementations

6. ACM Computing Classification System • C. Computer System Organization • 2. Computer-Communication Networks • 2.1 Network Architectures • 2.2 Network Protocols • 2.3 Network Operations • 2.4 Distributed Systems • 2.5 LAN & WAN • 2.6 Internetworking

7. ACM Computing Classification System • General Literature • Hardware • Computer System Organization • Software • Data • Theory of Computation • Mathematics of Computing • Information Systems • Computing Methodologies • Computer Applications

8. ACM Computing Classification System • E. Data • Data Structures • Data Storage Representations • Data Encryption • Coding and Information Theory • Files

9. ACM Computing Classification System • E. Data • 1. Data Structures • 1.1 Arrays • 1.2 Distributed Data Structures • 1.3 Graphs and Networks • 1.4 Lists, Stacks, and Queues • 1.5 Records • 1.6 Tables • 1.7 Trees

10. ACM Computing Classification System • General Literature • Hardware • Computer System Organization • Software • Data • Theory of Computation • Mathematics of Computing • Information Systems • Computing Methodologies • Computer Applications

11. ACM Computing Classification System • F. Theory of Computation • Computation by Abstract Devices • Analysis of Algorithms and Problem Complexity • Logics and Meanings of Programs • Mathematical Logic and Formal Languages

12. ACM Computing Classification System • F. Theory of Computation • 1. Computation by Abstract Devices • 1.1 Models of Computation • 1.2 Complexity Measures and Completeness • 1.2.1 Complexity Hierarchies • 1.2.2 Machine-Independent Complexity • 1.2.3 Reducibility and Completeness • 1.2.4 Relations among Complexity Classes

13. ACM Computing Classification System • F. Theory of Computation • 2. Analysis of Algorithms and Problem Complexity • 2.1 Numerical Algorithms • 2.2 Non-numerical Algorithms and Problems • 2.2.1 Complexity of Proof Procedures • 2.2.2 Computations on Discrete Structures • 2.2.3 Geometrical Problems • 2.2.4 Pattern Matching • 2.2.5 Routing and Layout • 2.2.6 Sequencing and Scheduling • 2.2.7 Sorting and Searching

14. ACM Computing Classification System • General Literature • Hardware • Computer System Organization • Software • Data • Theory of Computation • Mathematics of Computing • Information Systems • Computing Methodologies • Computer Applications

15. ACM Computing Classification System • G. Mathematics of Computing • Numerical Analysis • Discrete Mathematics • Probability and Statistics • Mathematical Software

16. G. Mathematics of Computing ACM Computing Classification System • 2. Discrete Mathematics • 2.1 Combinatorics • 2.2 Graph Theory • 2.3 Applications • 2.1.1 Combinatorial Algorithms • 2.1.2 Counting Problems • 2.1.3 Generating Functions • 2.1.4 Permutations and Combinations • 2.1.5 Recurrences and Difference Equations • 2.2.1 Graph Algorithms • 2.2.2 Graph Labeling • … • 2.2.5 Path and Circuit Problems • 2.2.6 Trees

17. a world FULLofNP-hardproblems Problem:I wanttotravelamong a set ofcitiesdriving the fewestpossiblenumberofKMs. IT’S DIFFICULT!! Problem: I wantto put a set ofobjectsinto some bags, knowingthateach bag can notafford more than 10 KGs and tryingtouseasfewbagsaspossible. IT’S DIFFICULT!!

18. Howto Deal withNP-hardproblems? • Exact Algorithms • Randomized Algorithms • Approximation Algorithms • Fixed-Parameter Tractable Algorithms • …

19. ApproximationAlgorithms

20. ApproximationAlgorithms • We want a solution “close” to the optimal one. • Given a minimization problem Π, analgorithmisanα-approximationforΠif, foreveryinstance I ofΠ, itoutputs a solution SOL(I) suchthat • SOL(I)/OPT(I) ≤α

21. Vertex Cover • Problem: Given a graph G(V,E), find a minimum vertex cover, that is, a set V’ V such that every edge e E has an endvertex in V’. NP-hard [Karp]

22. HowtoGuaranteeanApproximation? • We want an approximation algorithm for Vertex Cover. • The cost of the solution produced by the algorithm should be compared with the cost of an optimal solution. • Butcomputing the costofanoptimalsolutionis NP-HARD!! LOWER BOUNDS!

23. LowerBoundforVertex Cover • Given a graph G(V,E), a matching M is a set of edges M E such that no two edges share an endvertex. • A matching is maximal if no matching M’ exists such that M M’. • The sizeof a matchingis a lowerbound on the sizeofanoptimalsolutiontoVertex Cover!

24. An ApproximationAlgorithmforVertex Cover • Algorithm: Find a maximal matching M and output the set S of matched vertices

25. An ApproximationAlgorithmforVertex Cover • Theorem: The previous algorithm is a 2-approximation algorithm for Vertex Cover • Proof: • S is a vertex cover, otherwhise the matching would not be maximal. • OPT≥M, where M is the size of the output maximal matching. • SOL=2M, as the number of matched vertices is twice the size of the matching.

26. ApproximationAlgorithms:muchmore… • Approximation schemes: (1+ε)-approx. • Inapproximability results. • Complexity classes, e.g., APX.

27. Fixed-ParameterTractableAlgorithms

28. Easy or Hard? Problem (optimization):Let G be a graph. Which is the minimum number of vertices whose deletion makes G planar? NP-hard [Lewis-Yannakakis]

29. Easy or Hard? Problem (decision):Let G be a graph and k be an integer. Is there a set of k vertices whose deletion makes G planar? Polynomial • Easy O(nk+1) time algorithm for solving the problem: • Consider every set of k vertices. Remove such vertices. Test the planarity of the resulting graph. • There are O(nk) such sets. Testing the planarity of an n-vertex graph takes O(n) time. Then, • T(n,k)= O(nk) O(n) = O(nk+1)

30. Easy or Hard? Whereis the trick? In the decisionversion, k is a constantparameter part of the input An O(nk+1)-time algorithm, with k constant, is a polynomial-time algorithm. But it is very slow 

31. Fixed-ParameterTractability A problemisfixed-parametertractableif it can be solved in f(k) nO(1) time, where k is a parameter of the problem, f is an arbitrary function, and nO(1) is a polynomial (not depending on k).

32. FPT algorithmforVertex Cover Theorem[Melhorn]:There is an O(2k n)-timealgorithmforVertex Cover. Proof: Consideraninstance (G,k). • There is a vertex cover with k=0 if and only if G has no edge. • Consider any edge e=(u,v). Either u or v belongs to any vertex cover S. • Consider both the case in which u S and the case in which v S.

33. FPT algorithmforVertex Cover u v v • u S -> S is a vertex cover of G if and only if S-{u} is a vertex cover of G-{u} -> solve the instance (G-{u}, k-1) • v S -> S is a vertex cover of G if and only if S-{v} is a vertex cover of G-{v} -> solve the instance (G-{v}, k-1) u

34. FPT algorithmforVertex Cover • Time complexity T(n,k): • T(n,0) = O(n) • T(n,k) = 2 T(n-1, k-1) +O(n) ≤ • 2 T(n, k-1) +O(n) ≤ • 2 (2 T(n, k-2) +O(n)) +O(n) ≤ • 2 (2 (2 T(n, k-3) +O(n)) +O(n)) +O(n) ≤ • 2 (2 (2 (…(2 T(n, 0)+O(n)) + … +O(n)) +O(n)) +O(n) ≤ 2k O(n) + (2k-1 + 2k-2 + 2k-3 + ... + 1) O(n) = 2k O(n) + (2k - 1) O(n) = O(2k n)

35. ReductionRules Reduction rule : a rule (that is, a polynomial time algorithm) that transforms an instance (I,k) to an "equivalent and simpler” instance (I', k’). Equivalent: (I,k) is a YES instance if and only if (I', k') is a YES instance. Simpler: |I'|<|I| or k'<k or I' has fewer occurrences of a particular substructure.

36. Another FPT algorithmforVertex Cover Theorem:There is an O(1.6181k n2)-timealgorithmforVertex Cover. Proof: Consideraninstance (G,k). • We apply the following two reduction rules: • If G has a vertex u of degree 0, then let (G',k')=(G-{u},k). • If G has a vertex u of degree 1, then let N(u) denote the set of neighbors of u (here |N(u)|=1). Add N(u) to S and let (G',k')=(G-{u,N(u)},k-1).

37. Another FPT algorithmforVertex Cover • If G has a vertex u of degree 0, then let (G',k')=(G-{u},k). Correctness: if S is a vertex cover of G and u is in S, then S-{u} is also a vertex cover of G, as u has no incident edge. • If G has a vertex u of degree 1, then let N(u) denote the set of neighbors of u (here |N(u)|=1). Add N(u) to S and let (G',k')=(G-{u, N(u)},k-1). Correctness: if S is a vertex cover of G and u is in S, then S-{u} N(u) is also a vertex cover of G, as u has no other incident edge.

38. Another FPT algorithmforVertex Cover • If neither of the two rules can be applied, then every vertex has degree at least 2. • Pick a vertex u. Any vertex cover of G contains either u or N(u). Thus, (G,k) is a YES instance if and only if (G-{u},k-1) or (G-N(u),k-|N(u)|) is.

39. FPT algorithmforVertex Cover • Time complexity T(n,k): • Since |N(u)|\geq 2, • T(n,k) ≤ T(n,k-1)+ T(n,k-2)+O(n^2). • Fibonacci series: every number is the sum of the previous two xk =xk-1 +xk-2 +c. • The k-th number of the Fibonacci series tends to the golden ratio to the power of k. • T(n,k)=Φk O(n2)=((1+√5)/2)k O(n2) = O(1.6181k n2).

40. Kernelization • Sometimes the reduction rules work till the size of the problem is reduced really a lot. That is, after the reduction rules have been applied, the problem has size g(k), for some function k. In this case the problem has a g(k)-kernel. • Kernel: given a problem and an instance (I,k), a kernel is an algorithm that outputs in polynomial time an instance (I',k') such that (a) (I',k') is a YES instance if and only if (I,k) is; (b) |I'|<g(k), and k'<g(k).

41. Kernelization • Theorem [Niedermaier]: Every fixed-parameter tractable problem has a g(k)-kernel. • Proof: Suppose that there exists a FPT algorithm with running time f(k) nc, for some function f and some constant c. Consider an instance (I,k) with |I|=n. • If n>f(k), we run the decision algorithm in time f(k) nc < nc+1. If it returns YES (NO), the kernelization algorithm outputs a constant size YES (resp. NO) instance. • If n≤f(k), the thekernelization algorithm returns (I,k).

42. Kernelization • The last theorem implies that every problem that is FPT has a kernel. • However, the goal is to obtain kernels that are "small", that is, polynomial in k, or even constant.

43. FPT: muchmore… • Some problems are not FPT (e.g., graph coloring). • Complexity classes: W[1] -> Can a non-deterministic Turing machine accept an unary string s in at most k steps? • Independent Set is W[1]-hard. • Reductions to W[1]-hard problems give new W[1]-hard problems. • Lower bounds for FPT tractability • Upper and lower bounds for the size of the kernels

44. FPT: muchmore… • Theorem [Robertson and Seymour]: Every graph problem with parameter k whose YES istances are closed under taking minors can be solved in O(f(k) n3)