Martin Milanič

Transcription

1 1 / 75 Algorithmic Graph Theory Part I - Review of Basic Notions in Graph Theory, Algorithms and Complexity Martin Milanič martin.milanic@upr.si University of Primorska, Koper, Slovenia Dipartimento di Informatica Università degli Studi di Verona, March 2013

2 What we ll do 1 / 75 1 Overview. 2 Basic Graph Theoretic Definitions. 3 Algorithms - Basic Definitions. 4 Graph Representations. 5 Basics of Complexity.

3 Routing Problems 2 / 75 Source:

4 Routing Problems 2 / 75 Source:

5 Traffic Flow Modeling 3 / 75 Source:

6 Shortest Paths 4 / 75 Source:

7 Social Networks 5 / 75 Source:

8 Social Networks 5 / 75 Source:

9 Computer Networks 5 / 75 Source:

10 World Wide Web 6 / 75 Source:

11 Protein Networks 7 / 75 Source:

12 Metabolic Networks Source: 8 / 75

13 Phylogenetic Networks 9 / 75 Source:

14 Map Drawing 10 / 75 Source:

15 Boolean Circuits 11 / 75 Source:

16 What we ll do Week 1 12 / 75 1 Tue March 5: Review of basic notions in graph theory, algorithms and complexity 2 Wed March 6: Graph colorings 3 Thu March 7: Perfect graphs and their subclasses, part 1 4 Fri March 8: Perfect graphs and their subclasses, part 2

17 What we ll do Week 2 13 / 75 1 Tue March 19: Further examples of tractable problems, part 1 2 Wed March 20: Further examples of tractable problems, part 2 Approximation algorithms for graph problems 3 Thu March 21: Lectio Magistralis lecture, Graph classes: interrelations, structure, and algorithmic issues

18 BASIC GRAPH THEORETIC DEFINITIONS. 13 / 75

19 Graphs Graph: G = (V, E) ordered pair, where V = finite set of vertices, E ( V 2) (2-element subsets of V ) set of edges Notation: V(G) = V, E(G) = E. Example: V = {1, 2, 3, 4}, E = {{1, 2},{1, 3},{2, 3},{3, 4}} / 75

20 Graphs 14 / 75 e = {x, y} E: we will write e = xy (= yx) x, y endpoints of the edge e x y Petersen graph

21 Multigraphs multigraphs: loop parallel edges (E is a multisubset of ( V 2) V ) notation e = xy can be ambiguous for multigraphs; if necessary we use an incidence function: ψ(e) = xy some authors use different terminology: graph = multigraph simple graph = graph (without loops and parallel edges) 15 / 75

22 Digraphs 16 / 75 directed graphs (digraphs): D = (V, A) V : vertices, A: directed edges A V V A digraph is simple if it has no loops and no directed parallel edges.

23 Neighborhoods and Degrees 17 / 75 We write x y, x is adjacent to y, if xy E (x and y are neighbors). Neighborhood of vertex x: For X V we write N(x) = N G (x) := {y : y x}. N(X) := N(x)\X x X X N(X)

24 Handshaking Lemma The degree of vertex x: d(x) = d G (x) := N G (x). A vertex v is isolated if d(v) = 0. Handshaking Lemma: For every graph G = (V, E), d(x) = 2 E. Proof: x d(x) = x x V y x 1 = 2 E. Corollary: The number of vertices of odd degree in a graph is even. 18 / 75

25 Degrees in Digraphs 19 / 75 If D = (V, A) is a digraph, we define: N + (x) := {y V : xy A} successors of vertex x N (x) := {y V : yx A} predecessors of vertex x out-degree of vertex x: d + (x) := N + (x) in-degree of vertex x: d (x) := N (x)

26 Isomorphisms 20 / 75 An isomorphism of graphs G 1 = (V 1, E 1 ) and G 2 = (V 2, E 2 ) is a bijective mapping ϕ : V 1 V 2 that preserves adjacencies: uv E 1 ϕ(u)ϕ(v) E 2. If there exists an isomorphism of graphs G 1 and G 2, we say that the graphs are isomorphic. We write G 1 = G2. Example: x a ϕ(c) ϕ(x) ϕ(a) y b = z c ϕ(z) ϕ(b) ϕ(y)

27 Isomorphisms 20 / 75 Example: x a ϕ(c) ϕ(x) ϕ(a) y b = z c ϕ(z) ϕ(b) ϕ(y) Isomorphism is an equivalence relation. Typically, we will not distinguish pairwise isomorphic graphs.

28 Subgraphs A subgraph of a graph G = (V, E): a graph G = (V, E ) such that V V and E E (Note: in order for G to be a graph, we must have E E ( V 2).) Example: a V b c d e V g f A subgraph is: induced if E = E ( V 2). Notation: G = G[V ]. spanning if V = V. 21 / 75

29 Walks, Trails and Paths 22 / 75 A walk of length l in a graph G = (V, E) is a sequence x 0 e 1 x 1 e l x l, where x i V and e i = x i x i 1 E. Remarks: l = number of edges we usually write just x 0 x 1 x k (omitting the e i s) (not for multigraphs) A trail is a walk with all edges distinct. A path is a walk with all vertices distinct. Sometimes we consider a path as a subgraph: P = ({x 0, x 1,..., x l },{e 1,..., e l }).

30 Walks, Trails and Paths 23 / 75 Proposition x-y path in G x-y walk in G. Proof: ( ): trivial ( ): Let x = x 0 e 1 x 1 e l x l = y be a shortest x-y walk. We claim that this is a path. If not, then i < j: x i = x j. But then x = x 0 e 1 x 1 e i x i (= x j )e j+1 x j+1 e l x l = y is a shorter x-y walk, a contradiction.

31 Cycles 24 / 75 A walk is closed if x l = x 0. A cycle: a closed trail such that x 0 x 1 x l 1 is a path. l 3, except for multigraphs, where we can also have l = 1 (loop) and l = 2 (a pair of parallel edges). When defining paths and cycles in digraphs we take into account directions of edges.

32 Distances 25 / 75 The distance between two vertices x and y: d(x, y)(= d G (x, y)) = length of a shortest x-y path (= if x-y path) G x y G d G (x,y) = 1,d G (x,y) = 2

33 Components 26 / 75 A graph is connected if for every two vertices u, v V(G) there exists a path between them. [x is a path of length 0] Components of a graph G: maximal connected subgraphs

34 Euler Tours 27 / 75 An Euler tour (in G) is a closed trail passing through every edge exactly once. Source: the 7 bridges of Königsberg source: wikipedia

35 Euler Tours 28 / 75 Theorem (Euler, 1736) A multigraph G without isolated vertices has an Euler tour G is connected and all vertices are of even degree. Hence, the obvious necessary condition is also sufficient.

36 Hamiltonian Cycles 29 / 75 Hamiltonian cycle: a cycle going through every vertex exactly once. Can we give a simple necessary and sufficient condition for the existence of a Hamiltonian cycle in a graph? Most likely not, since the decision problem whether a given graph is Hamiltonian is NP-complete.

37 Operations on Graphs 30 / 75 The complement of a graph G = (V, E) is the graph G = (V, ( V 2) \ E). (This is the graph with the same vertex set as G, in which two vertices are adjacent if and only if they are not adjacent in G.) Example: v 3 v 3 v 2 v 4 v 2 v 4 v 1 G v 5 v 1 G v 5

38 Operations on Graphs 31 / 75 v V : G v = the subgraph of G induced by the vertex set V \{v}. Similarly, we have G U = G[V \ U] for U V. G e = (V, E \{e}) for e E (spanning subgraph) Similarly, G F = (V, E \ F) for F E.

39 Operations on Graphs 32 / 75 union of graphs: G H: graph (V(G) V(H), E(G) E(H)) intersection of graphs: G H: graph (V(G) V(H), E(G) E(H)) G+H: disjoint union of graphs G and H (Formally: V(G + H) = (V(G) {1}) (V(H) {2}), E(G + H) = {(u, 1)(v, 1) : uv E(G)} {(u, 2)(v, 2) : uv E(H)}) G H join of graphs G and H: disjoint union of graphs G and H together with all possible edges between V(G) {1} and V(H) {2}

40 Complete and Complete Bipartite Graphs 33 / 75 A complete graph: K V := (V, ( V 2) ), we usually write just Kn where n = V ng:= disjoint union of n copies of a graph G K 2 2K 2 Complete bipartite graph: K m,n : graph G = (V, E) such that V = A B, A B =, A = m, B = n and E = {ab : a A, b B}.

41 Paths and Cycles 34 / 75 Path P n : graph G = (V, E) with V = {v 1,..., v n } and E = {v 1 v 2,..., v n 1 v n }. Cycle C n : graph G = (V, E) with V = {v 1,..., v n } and E = {v 1 v 2,..., v n 1 v n, v n v 1 }.

42 Forests and Trees 35 / 75 a forest = a graph without cycles (acyclic graph) a tree = a connected forest Proposition Let T = (V, E) be a graph with n vertices. The following are equivalent: 1 T is a tree. 2 T is a connected graph and E = n 1. 3 E = n 1 and T is acyclic. 4 Every pair of vertices in graph T is joined by a unique path. 5 T is connected and for every edge e E(T) the graph T e is not connected. 6 T is an acyclic graph, but if we add an arbitrary edge to T, we obtain exactly one cycle.

43 ALGORITHMS - BASIC DEFINITIONS. 35 / 75

44 Algorithms 36 / 75 An algorithm is every well defined sequence of rules with which we compute something or solve some problem. A mathematical formalization of an algorithm is given by the notion of a Turing machine. An algorithm transforms input data into output data. Example: Algorithm that solves a system of linear equations Ax = b: Gaussian elimination with partial pivoting Input: A (an invertible matrix of order n), b (a vector of order n) Output: x (sought solution) if the decomposition of matrix A succeeded, or a message that due to rounding errors decomposition did not succeed

45 Algorithms 37 / 75 We require from an algorithm that it stops after a finite number of calculation steps. When developing an algorithm for a given problem, we should also provide: an analysis of the time complexity of the algorithm; a proof of correctness. The algorithms can be described either in the natural language or in pseudocode.

46 Algorithms Example: An algorithm that computes the product C of two square matrices A and B of order n: MATRIX MULTIPLICATION Input: Real matrices A and B of size n n. Output: Matrix C = A B. for i = 1,...,n do for j = 1,...,n do C[i, j] := A[i, 1] B[1, j]; for k = 2,...,n do C[i, j] := C[i, j]+a[i, k] B[k, j]; end for end for end for return C; The algorithm makes n 3 multiplications and n 2 (n 1) additions. Proof of correctness is obvious. 38 / 75

47 Instances and Their Sizes 39 / 75 We are given an (optimization, decision,...) problem P. Examples: Compute the product of two given matrices of order n. Find the smallest number among n given rational numbers. Determine whether the given graph G has a Hamiltonian cycle. instance: concrete input data for problem P size of the instance: the number of bits needed to store the instance in the computer. We need to choose an appropriate way of representing the instance.

48 Representing the Instances 40 / 75 Examples: We usually store a positive integer n with log 2 n +1 bits (sometimes also with n bits as 11 1 (n ones)). We can store a matrix of n 2 positive integers with at most n 2 log 2 M bits where M is the biggest number in the matrix. Or: with a set of triples {(i, j, a ij ) : a ij 0}. This representation is particularly suitable for matrices with many zero elements. We can represent a graph in several ways (more about this later).

49 Time Complexity of Algorithms 41 / 75 Let A be an algorithm that solves a problem P. Running time of algorithm A = number of basic calculation steps performed by A (additions, subtractions, multiplications, comparison of two numbers...). The time complexity of algorithm A is the function T A (n) that measures the running time of A in the worst case: T A : n largest running time of A of input instances of size n.

50 Time Complexity of Algorithms 42 / 75 Remarks: Besides time complexity, space complexity of an algorithm might also be important (how much computer memory the algorithm needs). If we have a probability distribution on the input instances, we can also estimate the expected time complexity of the algorithm.

51 Big O Notation 43 / 75 A function f : N N is of order (at most) O(g), if there exists a constant C > 0 such that f(n) C g(n) for all n N. Notation: f = O(g) If f = O(g), we also write g Ω(f) and say that g is of order (at least) Ω(f). Two functions f and g are of the same order if f = O(g) and f = Ω(g). Notation: f = Θ(g).

52 Big O Notation 44 / 75 Some properties of O: 1 We can ignore the constant factor: For all k > 0, kf = O(f). 2 Higher powers grow faster than lower ones: n r = O(n s ) if r s. 3 The speed of growth of a sum is the speed of the fastest growing summand: If f = O(g), then f + g = O(g). (Example: 6n 3 + 9n 2 = O(n 3 ).) 4 The order of a polynomial is equal to the order of the leading term: A polynomial of degree d is of order O(n d ). 5 Transitivity: If f = O(g) and g = O(h), then f = O(h).

53 Big O Notation 45 / 75 6 Exponential functions grow faster than power functions: For all k 0, b > 1, it holds n k = O(b n ). (Example: n 4 = O(2 n ), n 4 = O(e n ), n 4 = O( n ).) 7 Logarithms grow slower than power functions: For all k > 0, b > 1, it holds log b n = O(n k ). (Example: log 2 n = O(n 1/2 ).) 8 Logarithms are of the same order: For all b, d > 1, it holds log b n = O(log d n). 9 If f = O(g) and h = O(r), then fh = O(gr). (Example: if f = O(n 2 ) and g = O(log n), then fg = O(n 2 log n).)

54 Typical Time Complexities 46 / 75 Typical time complexities of algorithms: linear Θ(n) finding a maximum element in a table inner product of two vectors quadratic Θ(n 2 ) matrix addition, matrix transposition, multiplication of a matrix and a vector (a matrix is of order n n, a vector has n components; we take n for the measure of size) cubic Θ(n 3 ) typical matrix multiplication polynomial O(p(n)), where p is a polynomial exponential O(2 p(n) ), where p is a polynomial of degree 1 with positive leading coefficient

55 Polynomial Algorithms 47 / 75 An algorithm A is polynomial if its time complexity T A (n) is of the order O(n k ) for some k N. Polynomial algorihtms are also said to be efficient. Example: The following table shows the amount of time an algorithm of time complexity f(n) would need on a computer that performs million operations per second. f(n) n = 50 n = 100 n = 200 n s 10 4 s s n s 0.01 s 0.04 s n s 1 s 8 s 1.1 n s s 190 s 2 n 35.7 years years years

56 Polynomial Algorithms 48 / 75 On a 1000 times faster computer: f(n) n = 50 n = 100 n = 200 n s 10 7 s s n s 10 5 s s n s 10 3 s s 1.1 n s s 0.19 s 2 n 13 days years years

57 GRAPH REPRESENTATIONS. 48 / 75

58 Graph Representations 49 / 75 How to represent a graph in a computer? This depends on what graphs we will work with and what operations we want to preform on them. Let G = (V, E) where V = {v 1,..., v n }, E = {e 1,...,e m }.

59 Adjacency Matrix A(G) 50 / 75 Adjacency matrix of a graph G: A(G) = a 11 a a 1n a 21 a a 2n a n1 a n2... a nn, a ij = { 1, if vi v j ; 0, otherwise. Order of magnitude of this representation: O(n 2 )

60 Adjacency Matrix A(G) 51 / 75 Example: v 4 G v 3 A(G) v 1 v 2 v 3 v 4 v 1 v 2 v 3 v v 1 v 2 The definition of adjacency matrix can also be generalized to digraphs and multigraphs. (How?)

61 Adjacency List Representation 52 / 75 Adjacency list representation: A collection of unordered lists, one for each vertex in the graph. Each list describes the set of neighbors of its vertex (in arbitrary order). Example: v 4 G v 3 Adjacency lists: v 1 v 2 v 3 v 4 v 2 v 1 v 3 v 1 v 4 v 2 v 3 v 3 v 1 v 2 Order of magnitude of this representation: n n [1+d(v i )] = n+ d(v i ) = O(n+m) i=1 i=1

62 Adjacency List Representation 53 / 75 This representation is particularly useful for sparse graphs (graphs with O(n) edges). Many useful graphs are sparse : In a country there are several 1000 cities, but only a small number of roads goes out of each city. In Italy there are about 60 million people, but each person has only several 10 or several 100 acquaintances. The adjacency list representation can also be generalized to digraphs and multigraphs. (How?)

63 Comparison of Representations 54 / 75 With adjacency matrix representation each of the following operations takes constant time O(1): 1 check if v i v j, 2 remove an edge, 3 add an edge. With adjacency list representation we need the following time for the three operations: 1 O(d(v i )), since we need to traverse the list of neighbors of v i (or v j ), 2 O(max{d(v i ), d(v j )}), since we need to traverse both lists of neighbors (of v i and of v j ), 3 O(1), since we need to add vertex v i on the list of neighbors of v j and vice versa

64 Comparison of Representations 55 / 75 The space complexity of the adjacency matrix representation is Θ(n 2 ), independently of the number of edges. The space complexity of the adjacency list representation is O(n+m), which is O(n) for sparse graphs. We say that a graph problem is solvable in linear time if it can be solved by an algorithm of time complexity O(n+m) (where n = V, m = E ). We assume adjacency list representation.

65 BASICS OF COMPLEXITY. 55 / 75

66 Decision Problems, Classes P, NP, and co-np 56 / 75 Decision problem: a problem in which the set of instances divides into two sets depending on whether the answer is YES or NO. We define three classes of decision problems: P is the set of decision problems that can be solved by a polynomial algorithm Intuitively: P is the set of problems that can be solved efficiently.

67 Decision Problems, Classes P, NP, and co-np 56 / 75 Decision problem: a problem in which the set of instances divides into two sets depending on whether the answer is YES or NO. We define three classes of decision problems: NP is the set of decision problems with the following property: If the answer is YES then there exists a certificate that enables us to verify this fact in polynomial time.

68 Decision Problems, Classes P, NP, and co-np 56 / 75 Decision problem: a problem in which the set of instances divides into two sets depending on whether the answer is YES or NO. We define three classes of decision problems: NP is the set of decision problems with the following property: If the answer is YES then there exists a certificate that enables us to verify this fact in polynomial time. Intuitively: NP is the set of problems for which we can quickly verify a positive answer if we are given a solution.

69 Decision Problems, Classes P, NP, and co-np Decision problem: a problem in which the set of instances divides into two sets depending on whether the answer is YES or NO. We define three classes of decision problems: NP is the set of decision problems with the following property: If the answer is YES then there exists a certificate that enables us to verify this fact in polynomial time. Intuitively: NP is the set of problems for which we can quickly verify a positive answer if we are given a solution. (Formally: NP is the set of languages recognizable by some Turing machine in polynomial time.) 56 / 75

70 Decision Problems, Classes P, NP, and co-np 56 / 75 Decision problem: a problem in which the set of instances divides into two sets depending on whether the answer is YES or NO. We define three classes of decision problems: co-np is the set of decision problems with the following property: If the answer is NO then there exists a certificate that enables us to verify this fact in polynomial time.

71 Some Polynomial Graph Problems 57 / 75 The following problems are all in P: TOPOLOGICAL SORT: find an ordering of the vertices of a given digraph such that the first endpoint of each edge will precede the last endpoint in the order BREADTH-FIRST SEARCH (BFS): systematically check everything reachable from a given starting vertex DEPTH-FIRST SEARCH (DFS): like BFS, but in a different order SHORTEST PATHS: every edge has a length, find a shortest path between two vertices

72 Some Polynomial Graph Problems (cont d) 57 / 75 The following problems are all in P: MINIMUM SPANNING TREE: every edge has a length, find a set of edges with minimum total length such that every vertex is covered by an edge MAXIMUM FLOW: every (directed) edge has a capacity, find the maximum amount of flow from a source to a sink so that the conservation of flow is preserved MAXIMUM MATCHING: find a largest set of pairwise disjoint edges in a graph

73 Example of a Problem in NP Example: SATISFIABILITY Input: Boolean variables x 1,...,x n, clauses C 1,..., C m over x 1,..., x n [clause = a disjunction of literals Question: (variables or their negations)] Is there a satisfying truth assignment? SATISFIABILITY is in NP : If the answer is YES then every satisfiable truth assignment is a certificate: we can verify in polynomial time whether this is indeed a satisfying assignment. It is widely believed that SATISFIABILITY is not in P and not in co-np, though nobody knows for sure. 58 / 75

74 Decision Problems, Classes P, NP, and co-np Proposition P NP co-np. Proof: P NP : certificate is empty. We can verify a positive answer in polynomial time by solving the problem completely! Similarly for P co-np. Conjecture: P NP. One of the most important mathematical open questions. Clay Mathematics Institute offers million $ for a correct solution to this problem. Conjecture: NP co-np. (If we are able to quickly verify a positive answer, there should be no reason why we should be able to quickly verify a negative answer as well.) Conjecture: P = NP co-np. 59 / 75

75 Decision Problems, Classes P, NP, and co-np 60 / 75 Jack Edmonds believes P = NP co-np : Source:

76 NP-hard Problems 61 / 75 A problem Π is NP-hard if the existence of a polynomial algorithm for Π would imply the existence of a polynomial algorithm for every problem in NP. In other words: Π is NP-hard If Π is solvable in polynomial time then P = NP. Intuitively: if we could solve efficiently just one NP-hard problem Π, then we would be able to solve efficiently every problem whose solution we can verify quickly, by means of an algorithm for problem Π. NP-hard problems are at least as hard as an arbitrary problem in NP.

77 NP-complete Problems 62 / 75 A problem is NP-complete if it is NP-hard and it belongs to NP. These are the hardest problems in NP. If there exists a polynomial time algorithm for just one NP-complete problem, then all NP-complete problems are polynomially solvable. Thousands of NP-complete problems are known. A polynomial algorithm for either of them is very unlikely.

78 NP-complete Problems 63 / 75 NP-hard co-np NP NP-complete P Figure: Most likely relations between classes NP, co-np, NP -complete and NP -hard problems The existence of NP-complete problems is not immediately evident. Theorem (Cook, Levin) SATISFIABILITY is NP-complete.

79 Polynomial Reductions Π 1, Π 2 decision problems Definition Problem Π 1 is polynomially reducible to Π 2, if for every instance I for Π 1 we can construct in polynomial time an instance J = J(I) for problem Π 2 such that the answer to Π 1 given I is the same as the answer to Π 2 given J. Notation: Π 1 Π / 75

80 Polynomial Reductions 65 / 75 To show that a problem Π NP is NP -complete, we reduce a known NP -complete problem to Π. Proposition Suppose that for a problem Π NP there exists an NP -complete problem Π 1 such that Π 1 Π. Then Π is NP-complete.

81 Examples of NP-complete Problems 66 / 75 INDEPENDENT SET Input: Graph G = (V, E), k N Question: Does G contain an independent set of size k? independent set: a subset I V such that u, v I uv E Proposition The INDEPENDENT SET problem is NP -complete. Proof: 1. INDEPENDENT SET NP : we can verify in polynomial time whether I is an independent set of size k.

82 Examples of NP-complete Problems 67 / Reduction from SATISFIABILITY. Example: The set of clauses x 1 x 2, x 2 x 3 x 4, x 1 x 3 x 4, x 1 x 2 x 3 x 4 over the variables x 1, x 2, x 3, x 4 gets mapped to (G, 4), where 4 is the number of clauses and G is the following graph: x 2 x 4 x 3 x 1 x 1 x 3 x 2 x 4 x 1 x 2 x 3 x 4

83 Examples of NP-complete Problems (cont d) CLIQUE Input: Graph G = (V, E), k N Question: Does G contain a clique of size k? clique: a subset C V such that u, v C u v uv E Proposition The CLIQUE problem is NP -complete. Proof: 1. CLIQUE NP : we can verify in polynomial time whether C is a clique of size k. 2. INDEPENDENT SET CLIQUE I = (G, k) instance for INDEPENDENT SET J(I) = (G, k) instance for CLIQUE 68 / 75

84 Examples of NP-complete Problems (cont d) VERTEX COVER Input: Graph G = (V, E), k N Question: Does G contain a vertex cover of size k? vertex cover: a subset C V such that for all e E, e C Proposition The VERTEX COVER problem is NP -complete. Proof: 1. VERTEX COVER NP : we can verify in polynomial time whether C is a vertex cover of size k. 2. INDEPENDENT SET VERTEX COVER I = (G = (V, E), k) instance for INDEPENDENT SET J(I) = (G, V k) instance for VERTEX COVER 69 / 75

85 Examples of NP-complete Problems (cont d) 70 / 75 DOMINATING SET Input: Graph G = (V, E), k N Question: Does G contain a dominating set of size k? dominating set: a set D V such that every vertex is either in D or has a neighbor in D Proposition The DOMINATING SET problem is NP -complete. Proof: 1. DOMINATING SET NP : we can verify in polynomial time whether D is a dominating set of size k.

86 Examples of NP-complete Problems (cont d) 71 / VERTEX COVER DOMINATING SET I = (G = (V, E), k) instance for VERTEX COVER J(I) = (G = (V, E ), k) instance for DOMINATING SET where: V = V E, V is a clique in G, E is an independent set in G, ve E(G ) v V, e E, v is an endpoint of e.

87 Some Further NP-complete Problems 72 / 75 The following problems are NP-complete: 3-SATISFIABILITY: just like SATISFIABILITY, except that every clause consists of exactly 3 literals (2-SATISFIABILITY P.) 3-COLORABILITY: can the vertex set of a given graph be partitioned into 3 (possibly empty) independent sets? (2-COLORABILITY P.) HAMILTIONIAN CYCLE: does a given graph have a Hamiltonian cycle?

88 Some Further NP-complete Problems (cont d) 72 / 75 The following problems are NP-complete: TRAVELING SALESMAN: does a given edge-weighted graph have a Hamiltonian cycle of total length L METRIC TRAVELING SALESMAN: like TRAVELING SALESMAN, except that the lengths satisfy triangle inequality MAX CUT: does a given graph admit a partition of its vertex set into two sets such that there are at least k edges between them?

89 How to Deal With NP -complete Problems? 73 / 75 There are several approaches on how to deal with the intractability of NP -complete problems: polynomial algorithms for particular input instances approximation algorithms heuristics, local optimization efficient exponential algorithms (e.g., 1.5 n instead of 2 n ) randomized algorithms parameterized complexity (fixed-parameter tractable (FPT) algorithms)

90 What we ll do Week 1 74 / 75 1 Tue March 5: Review of basic notions in graph theory, algorithms and complexity 2 Wed March 6: Graph colorings 3 Thu March 7: Perfect graphs and their subclasses, part 1 4 Fri March 8: Perfect graphs and their subclasses, part 2

91 What we ll do Week 2 75 / 75 1 Tue March 19: Further examples of tractable problems, part 1 2 Wed March 20: Further examples of tractable problems, part 2 Approximation algorithms for graph problems 3 Thu March 21: Lectio Magistralis lecture, Graph classes: interrelations, structure, and algorithmic issues