On the Approximability of NP-complete Optimization Problems Viggo Kann Department of Numerical Analysis and Computing Science Royal Institute of Technology S-100 44 Stockholm Sweden [email protected] Dissertation, May 1992 Akademisk avhandling f¨or teknisk doktorsexamen vid Kungl. Tekniska H¨ogskolan, Stockholm Maj 1992 c Viggo Kann 1992 ISBN 91-7170-082-X TRITA-NA-9206 • ISSN 1101-2250 • ISRN KTH/NA–9206 Repro Print AB, Stockholm mcmxcii i Abstract This thesis deals with the polynomial approximability of combinatorial opti- mization problems which are NP-complete in their decision problem version. Different measures of approximation quality are possible and give rise to differ- ent approximation preserving reductions and different approximation classes of problems. We present definitions and characteristics of existing measures, re- ductions and classes and make comparisons. The main contributions comprise the following four subjects. • We show that the definition of the approximability class Max SNP due to Panconesi and Ranjan differs from the original definition due to Pa- padimitriou and Yannakakis and that the same problem may or may not be included in the class depending on how it is encoded. A new definition of Max SNP without this limitation is proposed. • We prove that maximum three dimensional matching is Max SNP-hard using a reduction constructing a structure of rings of trees. Using similar techniques we can show that several other matching and packing problems are Max SNP-hard, for example the maximum H-matching problem (the problem of determining the maximum number of node-disjoint copies of a fixed graph H contained in a variable graph). Most of the problems are Max SNP-complete when bounding the degree of the input structure. • Some versions of the maximum common subgraph problem are studied and approximation algorithms are given. The maximum bounded com- mon induced subgraph problem is shown to be Max SNP-hard and the maximum unbounded common induced subgraph problem is shown to be as hard to approximate as the maximum independent set problem. The maximum common induced connected subgraph problem is still harder to approximate and is shown to be NPO PB-complete, i.e. complete in the class of optimization problems with optimum value bounded by a polynomial. • An algorithm which solves the Euclidean travelling salesperson problem in two dimensions√ optimally is given. The worst case time bound for the algorithm is 2O( n log n). As an appendix there is a list of NP optimization problems including their definitions, approximation properties and references. Keywords: computational complexity, approximability, NP optimization prob- lems, combinatorial problems, graph problems, optimization, approximation, approximation algorithms, approximation preserving reduction, approximabil- ity classes, relative error, performance ratio, Max SNP-completeness, maxi- mum three dimensional matching, maximum H-matching, maximum common subgraph, travelling salesperson problem. ii Sammanfattning Denna avhandling handlar om hur v¨al NP-fullst¨andiga kombinatoriska opti- meringsproblem kan approximeras i polynomisk tid. Approximerbarhet kan m¨atas p˚afleras¨att, vilket har lett till att det finns flera olika approximer- barhetsbevarande reduktioner och flera komplexitetsklasser av problem. I av- handlingen ges definitioner av, egenskaper hos och j¨amf¨orelser mellan olika approximerbarhetsm˚att, reduktioner och klasser. F¨oljande fyra punkter sam- manfattar huvudsakligen avhandlingens nya resultat. • Skillnader mellan Panconesis och Ranjans definition av komplexitetsklas- sen Max SNP och Papadimitrious och Yannakakis ursprungliga defini- tion p˚avisas. T ex kan det h¨anda att samma problem antingenar ¨ med eller inte i klassen beroende p˚a hur det kodas. En ny definition av Max SNP som inte har denna brist l¨aggs fram. • Det tredimensionella matchningsproblemet visas vara Max SNP-sv˚art med hj¨alp av en reduktion som konstruerar en speciell struktur av tr¨ad- ringar. Samma teknik kan anv¨andas f¨or att visa att flera andra match- nings- och packningsproblemar ¨ Max SNP-sv˚ara, bl a maximal H-match- ning (att hitta det st¨orsta antalet nodskilda kopior av en fix graf H i en variabel graf). Dessa problem visas vara Max SNP-fullst¨andiga f¨or inmatningar med begr¨ansat gradtal. • N˚agra varianter av problemet att hitta den st¨orsta gemensamma del- grafen i tv˚agraferbeskrivs.N¨ar grafernas gradtalar ¨ begr¨ansat visas problemet vara Max SNP-sv˚art och n¨ar gradtaletar ¨ obegr¨ansat visas problemet vara ungef¨ar lika sv˚art att approximera som problemet att hitta den st¨orsta oberoende m¨angden h¨orn i en graf. Att approximera problemet att hitta den st¨orsta sammanh¨angande gemensamma delgrafen visas vara NPO PB-fullst¨andigt, dvs fullst¨andigt i den klass av optime- ringsproblem vars optimum ¨ar begr¨ansat av ett polynom i indatas storlek. • En algoritm som l¨oser handelsresandeproblemet i√ det euklidiska planet beskrivs. Tidskomplexiteten f¨or algoritmenar ¨ 2O( n log n). I ett appendix finns en lista ¨over optimeringsproblem. F¨or varje problem ges definitionen, approximationsegenskaper och referenser. Nyckelord: komplexitetsteori, approximerbarhet, NP-optimeringsproblem, approximering, approximeringsalgoritmer, approximerbarhetsbevarande reduk- tioner, relativt fel, approximerbarhetsfaktor, Max SNP-fullst¨andighet, max- imal tredimensionell matchning, maximal H-matchning, st¨orsta gemensamma delgraf, handelsresandeproblemet. Contents 1 Introduction 1 1.1Background............................. 1 1.2Approximatesolutions....................... 2 1.3 Degrees of approximability . 3 1.4 Organization of the presentation . 4 1.5 Approximability of other types of problems . 5 1.6 Approximation using other types of algorithms . 6 1.7Approximationbylocalsearch.................. 6 2 Definitions 7 2.1Basicnotation........................... 7 2.2 NP, NPO and PO ......................... 7 2.3Howapproximationismeasured.................. 8 2.4Approximationschemes...................... 11 2.5 NPO PB, polynomially bounded NP optimization problems . 12 2.6 Why do NP-complete problems have different approximability? 12 3 Reductions 14 3.1 Reductions preserving approximability . 14 3.2Reencoding of optimization problems . 15 3.3Strictreduction........................... 16 3.4Relativeerrorpreservingreductions............... 16 3.4.1 A-reduction......................... 16 3.4.2P-reduction......................... 17 3.4.3 R-reduction......................... 18 3.4.4 L-reduction......................... 18 3.4.5 F-reduction......................... 19 3.4.6 Transitivity......................... 20 3.5Ratiopreservingreduction..................... 20 3.6Parameterdependentreductions................. 24 3.7Non-constructivereductions.................... 25 3.8Structurepreservingreduction.................. 26 iii iv Contents 4 Approximability classes 29 4.1Introduction............................. 29 4.2 Fptas ................................ 30 4.3 Ptas ................................. 32 4.4 Fptas∞ and Ptas∞ ........................ 33 4.5 Rptas ................................ 33 4.6 Apx ................................. 34 4.7 NPO ................................. 35 4.8 NPO problems with polynomially bounded optimum . 35 4.9Theexistenceofintermediatedegrees.............. 37 4.10 Classes defined by logical formulas . 37 4.10.1 Syntactic Max NP ................... 38 4.10.2 Syntactic Max SNP .................. 40 4.10.3 Closures of Max NP and Max SNP .......... 42 4.10.4 Relations between the classes Syntactic Max SNP and Max SNP ......................... 44 4.10.5 New definitions of Max SNP and Max NP ....... 45 4.10.6 The hierarchies Max Πi, Max Σi, Min Πi and Min Σi ........................... 46 4.10.7 Closures of these classes . 48 4.10.8 Other syntactically defined classes . 49 4.11 The Max Ind Set class...................... 52 5 Matching and packing problems 54 5.1Introduction............................. 54 5.2Maximum three dimensional matching . 54 5.3Maximumthree-setpacking.................... 61 5.4 Maximum k-setpacking...................... 62 5.5 Maximum k-dimensionalmatching................ 63 5.6Maximumtrianglepacking..................... 63 5.7MaximumH-matching....................... 64 6 Maximum common subgraph problems 74 6.1Introduction............................. 74 6.2Maximumcommoninducedsubgraph.............. 74 6.2.1 Approximation of maximum bounded common induced subgraph.......................... 75 6.2.2 Approximation of maximum unbounded common induced subgraph.......................... 80 6.3 Maximum common edge subgraph . 82 6.3.1 Approximation of maximum bounded common edge sub- graph............................ 82 6.3.2Approximation of maximum unbounded common edge subgraph.......................... 85 6.4 Maximum common induced connected subgraph . 86 Contents v 7 The travelling salesperson problem 87 7.1Introduction............................. 87 7.2General Tsp ............................ 88 7.3 Tsp with triangle inequality . 88 7.3.1 Approximation algorithm by Christofides . 88 7.3.2 Tsp withdistancesoneandtwo............. 89 7.3.3 2-dimensional Euclidean Tsp ............... 90 8 How problems are related approximability-wise 100 8.1Masterandslaveproblems..................... 100 8.2Problemsintheplane....................... 100 8.3 Summary of how all problems are related . 102 8.3.1 Problems with constant absolute error guarantee . 102 8.3.2Problems with approximation schemes . 103 8.3.3 Problems which can be approximated within a constant