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Algorithms and Hardness Results for Some Valued Csps “thesis” — 2009/11/10 — 14:24 — page i — #1 Link¨oping Studies in Science and Technology Dissertation No. 1274 Algorithms and Hardness Results for Some Valued CSPs by Fredrik Kuivinen Department of Computer and Information Science Link¨opings universitet SE-581 83 Link¨oping, Sweden Link¨oping 2009 “thesis” — 2009/11/10 — 14:24 — page ii — #2 “thesis” — 2009/11/10 — 14:24 — page iii — #3 To Linnea “thesis” — 2009/11/10 — 14:24 — page iv — #4 “thesis” — 2009/11/10 — 14:24 — page v — #5 Abstract In the Constraint Satisfaction Problem (CSP) one is supposed to find an assignment to a set of variables so that a set of given constraints are satisfied. Many problems, both practical and theoretical, can be modelled as CSPs. As these problems are computationally hard it is interesting to investigate what kind of restrictions of the problems implies computational tractability. In this thesis the computational complexity of restrictions of two optimisation problems which are related to the CSP is studied. In optimisation problems one can also relax the requirements and ask for an approximatively good solution, instead of requiring the optimal one. The first problem we investigate is Maximum Solution (Max Sol) where one is looking for a solution which satisfies all constraints and also maximises a linear objective function. The Maximum Solution problem is a generali- sation of the well-known integer linear programming problem. In the case when the constraints are equations over an abelian group we obtain tight inapproximability results. We also study Max Sol for so-called maximal constraint languages and a partial classification theorem is obtained in this case. Finally, Max Sol over the boolean domain is studied in a setting where each variable only occurs a bounded number of times. The second problem is the Maximum Constraint Satisfaction Problem (Max CSP). In this problem one is looking for an assignment which max- imises the number of satisfied constraints. We first show that if the con- straints are known to give rise to an NP-hard CSP, then one cannot get arbitrarily good approximate solutions in polynomial time, unless P = NP. We use this result to show a similar hardness result for the case when only one constraint relation is used. We also study the submodular function min- imisation problem (SFM) on certain finite lattices. There is a strong link between Max CSP and SFM; new tractability results for SFM implies new tractability results for Max CSP. It is conjectured that SFM is the only reason for Max CSP to be tractable, but no one has yet managed to prove this. We obtain new tractability results for SFM on diamonds and evidence which supports the hypothesis that all modular lattices are tractable. “thesis” — 2009/11/10 — 14:24 — page vi — #6 Sammanfattning I ett villkorsprogrammeringsproblem ¨ar uppgiften att tilldela v¨arden till var- iabler s˚aatt en given m¨angd villkor blir uppfyllda. M˚anga praktiska prob- lem, s˚asom schemal¨aggning och planering, kan formuleras som villkorspro- grammeringsproblem och det ¨ar d¨arf¨or ¨onskv¨art att ha effektiva algoritmer f¨or att hitta l¨osningar till denna typ av problem. De generella varianterna av dessa problem ¨ar NP-sv˚ara att l¨osa. Detta inneb¨ar att det antagligen inte finns effektiva algoritmer f¨or problemen (om inte P = NP vilket anses vara mycket osannolikt). Av denna anledning f¨orenklar vi problemet genom att studera restriktioner av det och ibland n¨ojer vi oss med approximativa l¨osningar. I den h¨ar avhandlingen studeras tv˚avarianter av villkorsprogrammer- ingsproblemet d¨ar man inte bara ska hitta en l¨osning utan hitta en s˚abra l¨osning som m¨ojligt. I den f¨orsta varianten ¨ar m˚alet att hitta en tilldel- ning d¨ar samtliga villkor uppfylls och att en viktad summa av variablerna maximeras. Detta problem kan ses som en generalisering av det v¨alk¨anda linj¨ara heltalsprogrammeringsproblemet. I den andra varianten ¨ar m˚alet att hitta en tilldelning som uppfyller s˚am˚anga villkor som m¨ojligt. D˚adet g¨aller den f¨orsta varianten, d˚aman ska hitta en l¨osning som upp- fyller samtliga villkor som ocks˚amaximerar summan av variablerna, presen- teras nya resultat f¨or ett antal specialfall. De s˚akallade maximala villkors- m¨angderna studeras och komplexiteten f¨or ett antal av dessa best¨ams. Vi studerar ocks˚aen variant av problemet ¨over den Boolska dom¨anen d˚aantal variabelf¨orekomster ¨ar begr¨ansat. I detta fall ger vi en partiell klassifikation ¨over vilka villkorsm¨angder som ¨ar hanterbara och vilka som inte kan l¨osas effektivt. F¨or den andra varianten, d˚aman ska uppfylla s˚am˚anga villkor som m¨ojligt, presenteras n˚agra nya effektiva algoritmer f¨or vissa restriktioner. I dessa algoritmer l¨oses det mer generella problemet av minimering av sub- modul¨ara funktioner ¨over vissa ¨andliga latticar. Vi bevisar ocks˚aett resultat som beskriver exakt n¨ar det finns effektiva algoritmer d˚aman endast har tillg˚ang till en typ av villkor. “thesis” — 2009/11/10 — 14:24 — page viii — #8 “thesis” — 2009/11/10 — 14:24 — page ix — #9 Acknowledgements First and foremost I would like to thank my supervisor Peter Jonsson for all the support he has given me during my time as a graduate student. In particular, the fact that Peter always has had time to discuss research questions has been a tremendous resource. I also want to thank: • my co-authors Gustav Nordh, Peter Jonsson, and Andrei Krokhin; • Tommy F¨arnqvistfor reading an earlier version of this manuscript and pointing out lots of errors; • my past and present colleagues at the theoretical computer science laboratory deserve a huge thanks for providing a nice environment to work in; • Anton Blad and Jakob Ros´enfor all the interesting discussions, TRUT, and the geese feeding; • all my friends for making life fun; • my family for always supporting me; • and finally, Linnea for all her love and encouragement and for remind- ing me that there is more to life than research. This research work was funded in part by CUGS (the National Graduate School in Computer Science, Sweden). “thesis” — 2009/11/10 — 14:24 — page x — #10 “thesis” — 2009/11/10 — 14:24 — page xi — #11 List of Papers Parts of this thesis are based on the following publications. • F. Kuivinen. Tight approximability results for the maximum solu- tion equation problem over Zp. In Proceedings of the 30th Interna- tional Symposium on Mathematical Foundations of Computer Science (MFCS ’05), pages 628–639. Springer, 2005. • F. Kuivinen. Approximability of bounded occurrence max ones. In Proceedings of the 31th International Symposium on Mathematical Foundations of Computer Science (MFCS ’06), pages 622–633. Springer, 2006. • P. Jonsson, A. Krokhin, and F. Kuivinen. Hard constraint satisfaction problems have hard gaps at location 1. Theor. Comput. Sci., 410(38– 40):3856–3874, 2009. A conference version of this article first appeared as: – P. Jonsson, A. Krokhin, and F. Kuivinen. Ruling out polynomial- time approximation schemes for hard constraint satisfaction prob- lems. In Proceedings of the Second International Symposium on Computer Science in Russia (CSR ’07), pages 182–193. Springer, 2007. • P. Jonsson, F. Kuivinen, and G. Nordh. Max ones generalized to larger domains. SIAM J. Comput., 38(1):329–365, 2008. A conference version of this article first appeared as: – P. Jonsson, F. Kuivinen, and G. Nordh. Approximability of in- teger programming with generalised constraints. In Proceedings of the 12th International Conference on Principles and Practice of Constraint Programming (CP ’06), pages 256–270. Springer, 2006. • F. Kuivinen. Submodular functions on diamonds. In International Symposium on Combinatorial Optimization (CO ’08), pages 53–54. 2008. “thesis” — 2009/11/10 — 14:24 — page xii — #12 “thesis” — 2009/11/10 — 14:24 — page xiii — #13 Contents I Introduction and Background 1 1 Introduction 3 1.1 How Hard Is It to Colour a Map? . 3 1.2 Constraint Satisfaction Problems . 6 1.3 Why Study CSPs? . 8 1.4 Constraint Language Restrictions . 9 1.5 Maximum Solution . 12 1.6 Maximum CSP . 14 1.7 Main Contributions . 16 1.8 Organisation . 17 2 Technical Preliminaries 19 2.1 Approximability, Reductions, and Completeness . 20 2.2 Co-clones and Polymorphisms . 22 3 Connections to VCSP 25 II Maximum Solution 29 4 Max Sol over Abelian Groups 31 4.1 Introduction . 31 4.2 Inapproximability . 34 4.2.1 Preliminaries . 34 4.2.2 A First Inapproximability Result . 35 4.2.3 Inapproximability of W-Max Sol Eqn . 37 4.2.4 W-Max Sol Eqn(Zp, g) is APX-complete . 39 4.3 Approximability . 41 5 Max Sol on Maximal Constraint Languages 45 5.1 Introduction . 45 5.2 Description of Results . 47 5.3 Algebraic Approach to Max Sol . 49 5.4 Hardness and Membership Results . 50 “thesis” — 2009/11/10 — 14:24 — page xiv — #14 xiv Contents 5.5 Generalised Max-closed Relations . 51 5.6 Classification Results for Four Types of Operations . 56 5.6.1 Constant Operation . 56 5.6.2 Majority Operation . 57 5.6.3 Affine Operation . 57 5.6.4 Binary Commutative Idempotent Operation . 59 5.7 Proof of Theorem 5.3 . 67 6 Bounded Occurrence Max Ones 69 6.1 Introduction . 69 6.2 Preliminaries . 70 6.2.1 Co-clones and Polymorphisms . 72 6.3 Three or More Occurrences . 72 2 6.3.1 The Second Part: IS12 ⊆ hΓi ⊆ IS12 . 74 6.3.2 The Third Part . 76 6.3.3 Putting the Pieces Together . 79 6.4 Two Occurrences . 79 6.4.1 Definitions and Results . 80 6.4.2 Tractability Results for W-Max Ones(Γ)-2 .
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