Label Cover Reductions for Unconditional Approximation Hardness of Constraint Satisfaction CENNY WENNER Doctoral Thesis Stockholm, Sweden 2014 Stockholm University, ISBN 978-91-7447-997-3 SE-10691 Stockholm Akademisk avhandling som med tillstånd av Stockholms Universitet framlägges till offentlig granskning för avläggande av Filosfiedoktorsexamen i Datalogi Tors- dagen den 21:e Oktober, 2014, Klockan 13.15 i Föreläsningssal F3, Sing Sing, Kungliga Tekniska Högskolan, Lindstedtsvägen 26, Stockholm. © Cenny Wenner, October, 2014 Tryck: Universitetsservice US AB iii Abstract Combinatorial optimization problems include such common tasks as finding the quickest route to work, scheduling a maximum number of jobs to specialists, and placing bus stops so as to minimize commuter times. We restrict our attention to problems in which one is given a collection of constraints on variables with the objective of finding an assignment satisfy- ing as many constraints as possible, also known as Constraint Satisfaction Problems (CSPs). With only a few exceptions, most CSPs are NP-hard to solve optimally and we instead consider multiplicative-factor approxima- tions. In particular, a solution is a factor-c approximation if the number of constraints satisfied by the solution is at least c times the optimal number. The contributions of this thesis consist of new results demonstrating the approximation limits of CSPs in various settings. The first paper studies the hardness of ordering CSPs. In these prob- lems, one is given constraints on how items should be ordered relative to one another with the objective of ordering the given items from first to last so as to satisfy as many constraints as possible. We give improved approximation hardness results for two classical ordering problems: it is NP-hard to approximate Maximum Acyclic Subgraph with a factor bet- 14 1 ter than 15 and Maximum Betweenness with a factor better than 2 . We show that there are ordering problems which are NP-hard to approximate with any factor that is better than the expected value of random assign- ments, and that there are ordering problems which are arbitrarily hard to approximate. The second paper investigates when collections of simultaneously-sat- isfiable constraints are hard to approximate. It is known that Gaussian elimination can efficiently find exact solutions for satisfiable collections of so-called parity constraints. We show that whenever constraints accept at least one assignment in addition to parities, then the problem is instead NP- hard to approximate noticeably better than picking random assignments. This result is an unconditional analogue of previous work relying on the d- to-1 Conjecture and we largely follow the conditional proofs with general modifications so as to eliminate parameter dependencies and ultimately the need for the conjecture. In contrast to the conditional results however, we are only able to establish the result for constraints acting on at least four variables each. The third paper considers the strong hardness property called use- lessness. This property states that if given a collection where almost all constraints are simultaneously satisfiable and one is furthermore permitted to relax the constraints to accept or reject additional assignments, then it is still NP-hard to find a solution with value significantly better than what is expected from assigning the variables randomly. Uselessness has been characterized assuming the Unique Games Conjecture while the story is less clear without it. We consider the setting where all variables appear unnegated in constraints and provide the first examples of non-trivially useless predicates assuming only P 6= NP. iv Sammanfattning Kombinatoriska optimeringsproblem inkluderar naturliga uppgifter så- som att hitta den snabbaste vägen till sitt arbetet, att schemalägga ett maximalt antal jobb bland specialister, eller att placera nya hållplatser för att minimera resenärers restid. I denna avhandling begränsar vi oss till de problem i vilket man ges en samling vilkor på variabler med målet att hitta en tilldelning av variablerna uppfyllandes så många som möjligt av vilkoren; dessa problem kallas Vilkorsuppfyllningsproblem (eng: Constraint Satisfaction Problems, CSPs). De flesta CSPs är NP- svåra att lösa optimalt och vi beaktar istället approximationer. Specifikt kallas, för 0 ≤ c ≤ 1, en lösning för en faktor-c approximation om an- talet vilkor uppfyllda av lösningen är minst c gånger det största antalet uppfyllda av någon lösning. Denna avhandling består av tre artiklar som presenterar nya resultat begränsande hurpass väl man kan approximera CSPs i diverse situationer. Den första artikeln studerar svårigheten av att approximera diverse ordningsvilkorsproblem. I dessa problem ges man en samling ordningsvil- kor som anger hur objekt skall ordnas i relation till varandra, och har som mål att ordna objekten för att uppfylla så många av ordningsvilkoren som möjligt. Vi ger förbättrade svårighetsresultat för de två mest välkända ordningsproblem – det är NP-svårt att approximera Maximum Acyklisk 14 Delgraf (eng: Maximum Acyclic Subgraph) till en faktor bättre än 15 och Maximum Mellanliggande (eng: Maximum Betweenness) bätt- 1 re än 2 . Vidare introducerar vi ett ordningsproblem som är NP-svårt att approximera till en faktor bättre än det förväntade värdet av en slump- mässig tilldelning. Slutligen bevisas det att för varje c > 0 så finns det ordningsproblem som är NP-svåra att approximera till faktor c. Artikel Två analyserar appproximerbarheten av konsistenta vilkors- mängder. För konsistenta så kallade paritetsvilkor kan man genom Gausse- limination effektivt hitta fullständiga lösningar. Vi visar att så snart vilko- ren uppfylls av åtminstonde en tilldelning utöver en paritet så är det istället NP-svårt att approximera inom en faktor märkbart bättre än slumpmässiga tilldelningar. Detta resultat var tidigare känt betingat på d-till-1 Hypote- sen och vi följer i stort det betingade beviset med allmäna modifikationer för att eliminera parameterberoende och slutligen behovet av hypotesen. Till skillnad från de betingade resultaten gäller dock vårat bevis blott för vilkor som agerar på minst fyra variabler vardera. I den tredje artikeln betraktar vi en stark svårighetsegenskap av vilkor som kallas oanvändbarhet. Denna egenskap säger att om man ges vilkor där nästan samtliga är gemensamt uppfyllbara och man tillåts förenkla vilkoren genom att gemensamt för vilkoren lägga till eller ta bort uppfyl- lande tilldelningar, så är det fortfarande NP-svårt att hitta en tilldelning med värde något bättre än slumpmässiga tilldelningar. Vi fokuserar på fal- let där variabler endast förekommer onnegerade i vilkor. Oanvändbarhet utan negationer har karakteriserats med hänsyn till Unikaspelshypotesen (eng: Unique Games Conjecture) medans i princip ingenting är känt utan hypotesen. Givet endast P 6= NP ger vi de första exemplen på vilkor som är icke-triviallt oanvändbara utan negationer. Acknowledgements As the saying goes, “You do not know what you have until it is gone” or perhaps more wisely, “Appreciate what you have before it becomes what you had”. The period of doctoral studies might be one of the most luxurious settings you can find yourself in; having lavishly ample time and intellectual freedom to ponder research puzzles of your interests, in the company of a host of like-minded indi- viduals, under the supervision of puzzle-solving sages, experiencing the benefits of both students and employees, with comparatively few responsibilities, and as a cherry on top, you travel the world to find yet more kinspeople and meet the persons behind the names which you merely read about and idealized as an undergraduate. Truly, these have been spoiled years. I count myself particularly lucky as I have had a both brilliant and support- ive advisor. From the inception of and all throughout my studies, Johan Håstad has been emphasizing the role of intuition and understanding in mathematics – non-trivial results are not found by searching blindly in the dark. As noted also by preceding students, a trip to his office always forces you to elucidate your thoughts and leave with new pointed insights. Despite his countless responsibil- ities and results, somehow Johan always makes time for his students. The approximation group has including several excellent short- or long-term researchers during my stay, all of whom have contributed with their unique in- sights and curious discussions: Ola Svensson, Mahdi Cheraghchi, Tobias Mömke, Michael Lampis, Li-Yang Tan; my fellow graduate students Lukáš Poláček and Sangxia Huang; as well as coauthors Rajsekar Manokaran and Per Austrin. Too many for me to dare name, I would also like to extend my gratitude to the members of CSC for upholding the active and delightful working environment, to the numerous friends who have imparted meaning to the fabled outside world, to my defense committee and the opponent Irit Dinur, to the Simon’s Institute for hosting unparalleled congregations in the field, and to the European Union for funding my studies and research. Select noble individuals have offered their time to comment on this work: Antonios Antoniadis, Per Austrin, Henrik Holst, Johan Håstad, Michael Minock, Ljubica Pajevic, Hannah Renner, Björn Terelius, Freddy Wenner, Henry Yuen. Though “it is not much to hang in the xmas tree”, to all the marvelous present and future inhabitants of this world. v Foreword This compilation thesis consists of two logical parts where the first part is a primer to and synopsis of the