Computational ComplexityTake-home examHand in via Canvas before March 31 at 17:00Definition 1.We say that a classCof propositional formulas isniceif it has the following property:•There is a polynomial-time algorithm that—given a formulaφ∈ C, a positive integerk∈N, anda partial truth assignmentαto (some of) the variables in Var(φ)—correctly decides if there existsa truth assignmentβ: Var(φ)→ {0,1}that (i) extendsα, (ii) satisfiesφ, and (iii) that sets atleastkvariables among Var(φ) to true.Exercise 1(6pt; a:112pt, b:112pt, c:112pt, d:112pt).(a) Show that if the class of all 2CNF formulas is nice, then P = NP.(b) Specify a classCof propositional formulas that is nice, and such that for every (arbitrary) propo-sitional formulaφ, there exists someψ∈Cthat is logically equivalent toφ. Prove that this is thecase.In the rest of this exercise, we will show that the class of all propositional 3CNF formulas does notpolynomial-size compile into any nice classC, unless the PH collapses.Definition 2.LetC1,C2be two classes of propositional formulas. The classC1polynomial-size compilesintoC2if there exists a polynomialp:N→Nsuch that for everyφ∈ C1there exists af(φ)∈ C2thatis logically equivalent toφand for which holds|f(φ)|≤p(|φ|). Note that there are no requirements onthe running time to computef(φ).Consider the following family{φn}n∈Nof propositional formulas, where eachφncontains variablesin{xi,xi,j|1≤i < j≤n}, and is defined as follows:φn=∧1≤i<j≤n(¬xi,j∨¬xi∨¬xj)(c) LetCbe a class of propositional formulas that is nice. Suppose that there exists a polynomialp:N→Nand a family{ψn}n∈Nof propositional formulas such that for eachn∈N: (i)ψn∈ C,(ii)ψnis logically equivalent toφn, and (iii)|ψn|≤p(n).Show that then there exists a polynomial-time algorithm that—given a graphG= (V,E) withnvertices, an integerk∈N, and the formulaψn—decides ifGhas a clique of sizek.(d) Show that if 3CNF polynomial-size compiles into a classCthat is nice, then PH = Σp2.Hint:use the answer that you gave for(c).1
Exercise 2(3pt; a:1pt, b:2pt).Consider the following problem:Restricted-Positive-1-in-3-SATInput:A propositional formulaφin 3CNF, where each clause contains only positive literals,and where each variablex∈Var(φ) occurs at most 3 times inφ.Question:Does there exist a truth assignmentα: Var(φ)→{0,1}such thatαmakes exactlyone (positive) literal true in each clause ofφ?Do the following:(a) Prove thatRestricted-Positive-1-in-3-SATis solvable in timeO(1.45n)·p(|φ|), for some poly-nomialp:N→N, wherenis the number of variables inφ.(b) Prove that if the ETH is true, thenRestricted-Positive-1-in-3-SATis not solvable in time 2o(n),wherenis the number of variables inφ.Exercise 3(1pt).Consider the following game, played on anm×mboard where each of the positionson the board may be occupied by (i) a tile markedO, (ii) a tile markedX, or (iii) no tile at all. In otherwords, there are tiles marked withOandX, and in each position on the board there is at most one tile.The game is played by a single player, and for each of themcolumns, this player has to play one ofthe following moves: (1) remove all tiles markedOfrom positions in this column, (2) remove all tilesmarkedXfrom positions in this column. The player wins the game if (a) for each of the columns shemade a move, and after these moves (b) for each of themrows, there is at least one tile remaining inthis row.Then consider the following problem, related to the game described above.Game-can-be-wonInstance:Anm×mboard of the game, where each of the positions is occupied by a tilemarked withO, a tile marked withX, or no tile.Question:Can the player win the game starting from this board position?Show that the problemGame-can-be-wonis complete for some complexity class K∈{NP,coNP,Σp2,Πp2,PSPACE}.