Graph Classes: from Practice to Theory

Transcription

1 Graph Classes: from Practice to Theory Martin Milanič University of Primorska, Koper, Slovenia Fifth Workshop on Graph Theory and its Applications İstanbul Center for Mathematical Sciences 28 November 2015

2 What is this talk about? Aims of the talk: To present a selection of problems arising from practical applications sharing the common feature that they can be efficiently modeled and solved using a graph theoretic approach. To discuss the well-known class of perfect graphs and some of its subclasses. Roughly, these will be the topics for the first part of the talk. The graph classes we will see will have either combinatorial or geometric definitions.

3 In the second part of the talk, we will turn from combinatorics/geometry to algebra: We will outline a general framework for defining graph classes numerically and connect it with the well-known knapsack problem. We will discuss the class of circulant graphs.

4 Algorithmic graph problems and graph classes

5 Algorithmic graph problems Many algorithmic graph problems are of the following form: Given an input graph G = (V, E), find a substructure of G satisfying certain criteria and/or optimizing a given objective function (or determine that such a substructure does not exist). substructure could be any of the following: a subset of vertices, a subset of edges, a partition of the vertex set, a partition of the edge set, etc. Typically, every search problem of the above form also has a corresponding decision problem.

6 Examples CLIQUE Input: Graph G = (V, E), integer k Question: Does G contain a clique of size k? clique: a set of pairwise adjacent vertices

7 Examples INDEPENDENT SET Input: Graph G = (V, E), integer k Question: Does G contain an independent set of size k? independent set: a set of pairwise non-adjacent vertices

8 Examples VERTEX COVER Input: Graph G = (V, E), integer k Question: Does G contain a vertex cover of size k? vertex cover: a set of vertices hitting all edges

9 Examples DOMINATING SET Input: Graph G = (V, E), integer k Question: Does G contain a dominating set of size k? dominating set: a set of vertices such that every vertex not in the set has a neighbor in it

10 Examples COLORABILITY Input: Question: Graph G = (V, E), integer k Does G admit a k-coloring? k-coloring: a partition of V into k (pairwise disjoint, possibly empty) independent sets

11 Complexity All these problems CLIQUE INDEPENDENT SET VERTEX COVER DOMINATING SET COLORABILITY (as well as hundreds of others) are NP-complete.

12 Coping with intractability There are several approaches to coping with the intractability of NP -hard problems: parameterized complexity, polynomial algorithms for particular input instances, approximation algorithms (in general or for particular input instances), heuristics, local optimization, efficient exponential algorithms (e.g., 1.5 n instead of 2 n ), randomized algorithms, etc. A useful framework of studying particular input instances is given by the notion of graph classes.

13 Coping with intractability There are several approaches to coping with the intractability of NP -hard problems: parameterized complexity, polynomial algorithms for particular input instances, approximation algorithms (in general or for particular input instances), heuristics, local optimization, efficient exponential algorithms (e.g., 1.5 n instead of 2 n ), randomized algorithms, etc. A useful framework of studying particular input instances is given by the notion of graph classes.

14 Coping with intractability There are several approaches to coping with the intractability of NP -hard problems: parameterized complexity, polynomial algorithms for particular input instances, approximation algorithms (in general or for particular input instances), heuristics, local optimization, efficient exponential algorithms (e.g., 1.5 n instead of 2 n ), randomized algorithms, etc. A useful framework of studying particular input instances is given by the notion of graph classes.

15 Graph classes A graph class = a set of graphs closed under isomorphism. Examples: Planar graphs Connected graphs Trees Forests Bipartite graphs Interval graphs 3-colorable graphs Perfect graphs Cayley graphs vertex-transitive graphs

16 Main questions In the study of graph classes the following questions are of central interest: Computational complexity of a given problem within a particular class. Typically either polynomial or NP-hard. Prompts development of algorithmic techniques.

17 Main questions Inclusion relationships between different graph classes. Useful for transfer of: positive results: If X Y and a problem Π is polynomially solvable for graphs in Y, then Π is also polynomially solvable for graphs in X. negative results: If X Y and a problem Π is NP -hard for graphs in X, then Π is also NP -hard for graphs in Y.

18 Answering such types of questions often leads to a more or less precise mapping of the boundary of the computational complexity of a given problem (P vs. NP-complete) Example: Complexity of the efficient domination problem line graphs of planar cubic bipartite bipartite planar bipartite chordal bipartite interval bigraphs AT-free convex bipartite perfect cocomparability trapezoid permutation bipartite permutation chordal split circular arc interval NP-c dually chordal strongly chordal block P distancehereditary tree

19 Main questions Characterizations of graphs in a given class. by minimal forbidden induced subgraphs (in case of hereditary graph classes) by minimal forbidden minors (in the case of minor-closed graph classes), etc. decomposition theorems Computational complexity of recognizing graphs in a given class.

20 Recognition problems For a given graph class X we can define the following decision problem: RECOGNITION OF GRAPHS IN X Input: Graph G. Question: Is G X? Examples: If X = the class of all 3-colorable graphs, the recognition problem is NP-complete. If X = the class of planar graphs, the recognition problem is solvable in linear time.

21 Recognition problems If a graph class arises from a practical application, the graph typically comes equipped with a certificate of membership in the class. Example: Intersection graphs of geometric objects such as intervals, disks, etc. Adapted from: Kammer and Tholey, Algorithmica 68 (2014) In such cases, hardness of the recognition problem is not a real issue.

22 Perfect graphs and their subclasses

23 χ and ω χ(g): chromatic number of G = the smallest k such that G admits a k-coloring ω(g): clique number of G = the maximum size of a clique in G.

24 Perfect graphs Trivially: χ(g) ω(g). Definition A graph G is perfect, if χ(h) = ω(h) holds for every induced subgraph H of G. The class of perfect graphs is hereditary (that is, closed under vertex deletions).

25 Perfect graphs Examples of non-perfect graphs: odd cycles of order at least 5: C 5, C 7, C 9,... their complements: C 5, C 7, C 9,... Theorem (Lovász, 1972, Perfect Graph Theorem) A graph G is perfect if and only if its complement G is perfect.

26 Berge graphs Berge graph: a {C 5, C 7, C 7, C 9, C 9,...}-free graph. Claude Berge, , a French mathematician Source: rg/m532.html

27 The Strong Perfect Graph Theorem Berge graph: a {C 5, C 7, C 7, C 9, C 9,...}-free graph. Clearly, every perfect graph is Berge. Conjecture (Berge, 1963) A graph G is perfect if and only if it is Berge. Strong Perfect Graph Theorem (Chudnovsky, Robertson, Seymour, Thomas, 2002) A graph G is perfect if and only if it is Berge. Total length of the proof 150 pages.

28 Algorithmic aspects of perfect graphs The following problems are solvable in polynomial time for perfect graphs: COLORABILITY, INDEPENDENT SET, CLIQUE. These results are due to Grötschel, Lovász, Schrijver (1984) and are not combinatorial. They are based on semidefinite programming and the ellipsoid method. Existence of combinatorial algorithms for the COLORABILITY, INDEPENDENT SET, and CLIQUE problems on perfect graphs is an open problem.

29 Recognizing perfect graphs Theorem (Chudnovsky, Cornuéjols, Liu, Seymour, Vušković, 2005) There is a polynomial-time algorithm for recognizing Berge graphs. O( V 9 ) 36 pages independent of the proof of SPGT

30 Hasse diagram of some classes of perfect graphs perfect line graphs of bipartite graphs comparability co-comparability chordal distance- hereditary bipartite chordal bipartite trapezoid split strongly chordal planar bipartite permutation interval block cograph bipartite permutation threshold tree

31 Hasse diagram of some classes of perfect graphs perfect line graphs of bipartite graphs comparability co-comparability chordal distance- hereditary bipartite chordal bipartite trapezoid split strongly chordal planar bipartite permutation interval block cograph bipartite permutation threshold tree

32 Chordal graphs

33 Chordal graphs Definition A graph is chordal if every cycle on at least 4 vertices contains a chord. chord: an edge connecting two non-consecutive vertices of the cycle. Figure: A cycle with four chords.

34 Chordal graphs Example: chordal not chordal

35 Properties of chordal graphs A graph is chordal if and only if it is {C 4, C 5,...}-free. Theorem (Gavril, 1974) Chordal graphs are precisely the vertex-intersection graphs of subtrees in a tree. Example: T T 1 G T 4 T 6 T 5 T 4 T 6 T5 T 2 T 3 T 1 T 2 T 3

36 Chordal graphs: structural properties A cutset: a set of vertices X V such that the graph G X is disconnected. Theorem (Dirac, 1961) Every minimal cutset in a chordal graph is a clique. cutset

37 Chordal graphs: structural properties A vertex is simplicial if its neighborhood is a clique. Corollary Let G be a chordal graph. Then, G is either complete or it contains a pair of non-adjacent simplicial vertices. In particular, G contains a simplicial vertex. perfect elimination ordering: a linear ordering (v 1,..., v n ) of the vertices such that each v i is a simplicial vertex of G[{v i,...,v n }]. Theorem A graph is chordal if and only if it has a perfect elimination ordering. (Fulkerson, Gross, 1965) A perfect elimination ordering of a chordal graph can be computed in linear time. (Rose, Lueker, Tarjan, 1976)

38 Chordal graphs: algorithmic aspects Theorem Every chordal graph contains a simplicial vertex. If G is chordal and v V(G) then G v is chordal. With iterative deletion of simplicial vertices, it is easy to develop polynomial-time algorithms for the following problems on chordal graphs: CLIQUE, COLORABILITY, INDEPENDENT SET. On the other hand, the DOMINATING SET problem is NP-complete on chordal graphs.

39 And now, for something completely different... Example application of chordal graphs: A problem from computational biology

40 The Haplotype Inference Problem How to infer biologically meaningful haplotype data from the genotype data of a population? Human DNA is organized in pairs of (almost identical) chromosomes. one copy in each pair is inherited from the mother...accgt GAT C... and one from the father...agcgaact G... Ideally, we would like to know the exact sequence of bases on both chromoseme copies. However, experimentally it is much easier and less expensive to collect a mixed description of the two chromosomes.

41 The Haplotype Inference Problem Original sequences: A C C G T G A T C A G C G A A C T G A {C, G} C G {T, A} {G, A} {A, C} T {C, G} Mixed sequence: A {C, G} C G {T, A} {G, A} {A, C} T {C, G} The original sequences are called haplotypes. The mixed sequences are called genotypes. Hence, the task is to search for a haplotypical resolution of genotypical information.

42 The Haplotype Inference Problem HAPLOTYPE INFERENCE PROBLEM: Given: a set G of genotypes, Task: determine a (biologically meaningful) set of haplotypes such that each genotype g G is explained by two haplotypes from the set. The problem is being extensively studied, under many objective functions, one of which is Parsimony.

43 Parsimony Haplotyping PARSIMONY HAPLOTYPING: Given: a set of n vectors in {0, 1, 2} m (genotypes). Task: find a minimum-sized set of vectors in {0, 1} m (haplotypes) such that every genotype can be expressed as the sum of two haplotypes from the set. Addition rules: 0+0 = 0, 1+1 = 1, 0+1 = 1+0 = 2

44 Parsimony Haplotyping haplotypes genotypes

45 Compatibility graph Compatibility graph G: the graph with V(G) = {genotypes} and E(G) = {gg : r such that {g r, g r} = {0, 1}}. that is, two genotypes are adjacent if some haplotype can be used for expressing both genotypes

46 k-bounded instances A parsimony haplotyping instance is k-bounded if in every coordinate, at most k genotypes contain a 2. k = 1 k = 2 k 3 Complexity of k-bounded PARSIMONY HAPLOTYPING polynomial? NP-hard k = 1: van Iersel, Keijsper, Kelk, Stougie, 2008 k 3: Sharan, Halldórsson, Istrail, 2006 k = 2: interesting open case; partial results are known

47 1-bounded case Polynomial time algorithm for 1-bounded parsimony haplotying by van Iersel et al. (2008) is based on structural properties of the resulting compatibility graphs. Compatibility graphs of 1-bounded PH instances = block graphs: graphs every block of which is complete. The algorithm is based on a simplicial elimination ordering of the compatibility graph.

48 The general case The compatibility graphs of general instances are characterized in the following. Theorem (Cicalese, M., 2012) The compatibility graphs of k-bounded PH instances are exactly the graphs in which every two non-adjacent vertices are connected by at most k internally disjoint paths.

49 Back to graph classes: Interval graphs

50 Definition A graph is an interval graph if its vertices can be put into one-to-one correspondence with a set of intervals on the real line such that two vertices are connected by an edge if and only if their corresponding intervals have nonempty intersection. Source: graph

51 Algorithmic aspects Theorem (Booth, Lueker, 1976) Interval graphs can be recognized in linear time. Other algorithmic problems on interval graphs: COLORABILITY: In P. CLIQUE: In P. INDEPENDENT SET: In P. DOMINATING SET: In P.

52 Coloring interval graphs A given interval graph can be optimally colored by the greedy algorithm: 1. Sort the vertices according to the left-to-right order of the right endpoint of the corresponding intervals. This gives a perfect elimination ordering of G. 2. Color the vertices greedily in the opposite order. Optimality can be proved by showing that only ω(g) colors are used.

53 Applications Interval graphs have applications in: genetics (Benzer, 1959), archeology (Kendall, 1969), population biology (Cohen, 1968), bioinformatics (Zhang et al., 1994), scheduling (Bar-Noy et al., 2001).

54 Example applications of coloring interval graphs: Scheduling

55 Example 1: A tourist agency is organizing excursions for n groups. For each group we know the starting and finishing time of their excursion. At least how many tourist guides does the company need? Example 2: An airline company needs to schedule n flights from the central hub and back with known departure and arrival times. What is the smallest number of aircrafts the company needs? The resulting conflict graphs are interval graphs.

56 What can we hope for? The examples we have seen so far represent a best possible scenario: the problem arising from a practical application can be solved optimally in polynomial (or even in linear) time. What would be a second best scenario? The problems arising from practical applications often remain NP-hard. however, they can have better approximability properties than in general.

57 Two examples (with a little detour from perfect graphs)

58 Example 1: Coloring unit disk graphs

59 Assignment of radio frequencies Assume that we have a number of radio stations, identified by x and y coordinates in the plane. We have to assign a frequency to each station, but due to interferences, stations that are close to each other have to receive different frequencies. Such problems arise in frequency assignment of base stations in cellular phone networks. Source: The resulting conflict graph is a unit disk graph (intersection graph of unit disks in the plane).

60 Coloring unit disk graphs Bad news: Theorem (Clark, Colbourn, Johnson, 1990) Computing the chromatic number is NP-hard for unit disk graphs. Good news: Every unit disk graph has (G) 6ω(G) 6. Therefore, the greedy coloring algorithm gives a 6-approximation to the chromatic number.

61 Coloring unit disk graphs Even better news: Theorem (Peeters, 1991; Gräf, Stumpf, Weißenfels, 1998) Chromatic number is 3-approximable for unit disk graphs. The algorithm by Gräf et al. is based on the geometric structure of unit disk graphs... but uses also network flow and matching techniques. Is this indeed good news?

62 Coloring in general It is! In general graphs, the chromatic number is very hard to approximate: Theorem (Zuckerman, 2007) For all ǫ > 0, approximating the chromatic number within a factor of n 1 ǫ on n-vertex graphs is NP-hard.

63 Example 2: Maximum weight independent set problem in rectangle intersection graphs

64 Map labeling problem Suppose we want to draw a map of a city. We have a set of important locations in the city (given by their x and y coordinates) that we would like to label on the map. Labels that we place on the map must be pairwise disjoint. We also assume that each location is weighted according to its importance. The resulting conflict graph is an intersection graph of axis-parallel rectangles, and the map labeling problem becomes the maximum weight independent set problem in rectangle intersection graphs.

65 Bad news: Theorem (Formann, Wagner, 1991) The independent set problem is NP-hard for rectangle intersection graphs.

66 Good news: Theorem The maximum weight independent set problem in rectangle intersection graphs admits: a O(log n/ log log n)-approximation (Chan, Har-Peled, 2012) a (1 + ǫ)-approximation with quasipolynomial running time 2 poly(log(n/ǫ)) (Adamaszek, Wiese, 2013) A PTAS for a relaxed model where we are allowed to shrink the rectangles by a tiny bit (Adamaszek, Chalermsook, Wiese, 2015) Techniques used: local search and LP relaxation dynamic programming algorithms + geometric structure

67 Are these news indeed good news? They are! In general graphs, the independence number is very hard to approximate: Theorem (Zuckerman, 2007) For all ǫ > 0, approximating the independence number within a factor of n 1 ǫ on n-vertex graphs is NP-hard.

68 Back to graph classes

69 Hasse diagram of some classes of perfect graphs perfect line graphs of bipartite graphs comparability co-comparability chordal distance- hereditary bipartite chordal bipartite trapezoid split strongly chordal planar bipartite permutation interval block cograph bipartite permutation threshold tree

70 Algorithmic Graph Theory and Perfect Graphs Source:

71 Threshold Graphs and Related Topics Source:

72 Graph Classes Source:

73 Source:

74 A statistic On November 25, 2015, the MathSciNet database contained the following numbers of papers having the name of one of the following graph classes in the title: 1. Perfect graphs: Chordal graphs: Interval graphs: 422 For comparison: Minimum spanning tree: 396 Travel(l)ing salesman: over 1200

75 Threshold graphs

76 Definition A graph G = (V, E) is threshold if there exist positive integer vertex weights w(v) for all v V and a threshold t N such that for every set X V, X is independent if and only if w(x) t.

77 Definition A graph G = (V, E) is threshold if there exist positive integer vertex weights w(v) for all v V and a threshold t N such that for every set X V, X is independent if and only if w(x) t. 4 2 t =

78 Characterizations Theorem (Chvátal, Hammer 1977) The following properties are equivalent for a graph G: 1. G is threshold. 2. G is {2K 2, C 4, P 4 }-free. 2K 2 C 4 P 4 3. G can be constructed from the one-vertex graph by repeated applications of the following two operations: Addition of an isolated vertex. Addition of a universal vertex.

79 Algorithmic aspects Threshold graphs can be recognized in linear time. Other algorithmic problems on threshold graphs: COLORABILITY? In P. CLIQUE? In P. INDEPENDENT SET? In P. DOMINATING SET? In P.

80 In the rest of the talk, we will turn from combinatorics/geometry to algebra: We will outline a general framework for defining graph classes numerically. The framework captures (in several ways) the class of threshold graphs, as well as several other graph classes. It is also connected to the well-known knapsack problem. We will discuss the class of circulant graphs.

81 A general framework for defining graph classes numerically

82 A general framework Let P denote some property meaningful for vertex or edge subsets of a graph. For example, P could be any of the following: In case of vertex subsets: a clique, a maximal clique, an independent set, a dominating set, a total dominating set, etc.

83 Total dominating sets total dominating set: a set of vertices such that every vertex has a neighbor in it

84 A general framework P could also be a property of edge subsets, for example: a matching, a maximal matching, an edge dominating set, a spanning tree, a Hamiltonian cycle, etc.

85 Definition A P-separator of a graph G = (V, E) is a pair (w, T) where w is a non-negative integer weight function on vertices of G (or on its edges, depending on P) and T is a set of integers (called a target set) such that for every S V (resp., S E): P(S) holds if and only if w(s) T. Given a P-separator, one can determine whether S has the desired property just by computing its total weight, w(s), and checking if w(s) T.

86 The P-separator problem: Given a graph G, does G admit a P-separator? Several graph classes can be defined as the set of graphs having a P-separator for a suitable choice of property P and some natural restriction on the target set T.

87 Example: threshold graphs P(S) holds if and only if w(s) T. Recall: A graph G = (V, E) is threshold if there exist positive integer vertex weights w(v) for all v V and a threshold t N such that for every set X V, X is independent if and only if w(x) t. So we can take: P(X): X is independent in G T = [0, t]

88 Example: threshold graphs P(S) holds if and only if w(s) T. Recall: A graph G = (V, E) is threshold if there exist positive integer vertex weights w(v) for all v V and a threshold t N such that for every set X V, X is independent if and only if w(x) [0, t]. So we can take: P(X): X is independent in G T = [0, t] := {0, 1,...,t}

89 Further examples P(S) T introduced by resulting graph class independent set [0, t] Chvátal, Hammer, 1977 threshold graphs contains an edge [t, ) threshold graphs vertex cover [t, ) threshold graphs dominating set [t, ) Benzaken, Hammer, 1978 domishold graphs maximal ind. set {t} Payan, 1980 equistable graphs minimal dom. set {t} Payan, 1980 equidominating graphs total dom. set [t, ) Chiarelli, M., total dom. graphs connected dom. set [t, ) Chiarelli, M., 2014 connected-dom. graphs contains a maximal clique [t, ) Boros, Gurvich, M., threshold graphs perfect matching {t} all graphs

90 Algorithmic aspects Can we make use of P-separators to develop efficient algorithms? Yes, if the weights are small enough. Consider the following problem: MAXIMUM PROFIT P-SET Input: Graph G = (V, E), a profit function p : V R +. Task: Find a set with property P with maximum total profit p(s). Given a P-separator (w, T), the problem can be solved in pseudo-polynomial time by dynamic programming.

91 Algorithmic aspects Theorem (M., Orlin, Rudolf, 2011) Given a P-separator (w, T), The MAXIMUM PROFIT P-SET problem admits an O(nM) algorithm with M calls of the membership oracle for T, where n is the number of vertices of G and M is an upper bound for T. for each i {1,..., n} and each j {0, 1,..., M}, the values of p i,j = max{p(s) : S {v 1,..., v i }, w(s) = j} are computed recursively the optimal value is given by max{p n,j : j T}

92 Extensions The algorithm admits several extensions: max min, vertex subsets edge subsets, it works for hypergraphs it generalizes the standard dynamic programming algorithm for the knapsack problem The knapsack problem: Given a knapsack capacity W and a set of n items with weights w i and profits p i, find a set of items of total weight W maximizing the total profit. For property P, we take the following property of a subset S of items: all items in S fit into the knapsack T = [0, W].

93 Limitations Due to its generality, this approach also has some limitations: a P-separator has to be known; there exist graphs that require exponentially large weights in any P-separator. Example: Take P(S) = S is a maximal independent set of G, T = {t}, G = nk 2. Then in every P-separator of G. ( 2 n ) max{w(v) : v V(G)} = Ω n This is proved using a connection with the Distinct Subset Sums property of sets of numbers.

94 In some cases, faster algorithms can be obtained with a direct approach, exploiting the structure of graphs in the corresponding class. e.g., for threshold graphs or for domishold graphs But not always...

95 Example: A graph G is said to be equistable if a P-separator for: P(S) = S is a maximal independent set of G, T = {t} Given such a P-separator, the dyn. prog. approach solves the MAXIMUM WEIGHT INDEPENDENT SET problem MINIMUM WEIGHT MAXIMAL INDEPENDENT SET problem However, both problems are APX-hard (already in the unweighted case) in the class of equistable graphs, even if the graph is given together with a P-separator.

96 And now, for something completely different... The class of circulant graphs

97 Circulants A circulant graph (or simply a circulant) is a Cayley graph over a cyclic group. A circulant is a graph G = C n (D) determined by: a positive integer n, and a set of positive integers D modulo n such that d D d D, as follows: V(G) = Z n, and two vertices i, j Z n are adjacent if and only if i j (mod n) D.

98 Examples C 9 ({±1,±4})

99 Examples C 9 ({±3,±4})

100 Some graph theoretic properties of circulants Several graph theoretic properties of circulants can be easily inferred from numerical properties of n and D. For example: Proposition A circulant G = C n (D) is connected if and only if gcd(d {n}) = 1. Proposition A connected circulant G = C n (D) with at least two vertices is bipartite if and only if n is even, while every d D is odd.

101 Algorithmic aspects Theorem (Evdokimov and Ponomarenko, 2004) Circulant graphs can be recognized in polynomial time. Other algorithmic problems on circulant graphs: COLORABILITY? NP-complete. CLIQUE? NP-complete. INDEPENDENT SET? NP-complete. all due to Codenotti, Gerace, Vigna, 1998 DOMINATING SET? OPEN.

102 Questions? Thank you for your attention! Source: