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Unsolvable problems At least an NP-complete problem can be solved by some algorithm—the trouble is that this algorithm will be exponential. But it turns out there are perfectly decent computational problems for which no <strong>algorithms</strong> exist at all! One famous problem of this sort is an arithmetical version of SAT. Given a polynomial equation in many variables, perhaps x 3 yz + 2y 4 z 2 − 7xy 5 z = 6, are there integer values of x, y, z that satisfy it? There is no algorithm that solves this problem. No algorithm at all, polynomial, exponential, doubly exponential, or worse! Such problems are called unsolvable. The first unsolvable problem was discovered in 1936 by Alan M. Turing, then a student of mathematics at Cambridge, England. ] When Turing came up with it, there were no computers or programming languages (in fact, it can be argued that these things came about later exactly because this brilliant thought occurred to Turing). But today we can state it in familiar terms. Suppose that you are given a program in your favorite programming language, along with a particular input. Will the program ever terminate, once started on this input? This is a very reasonable question. Many of us would be ecstatic if we had an algorithm, call it terminates(p,x), that took as input a file containing a program p, and a file of data x, and after grinding away, finally told us whether or not p would ever stop if started on x. But how would you go about writing the program terminates? (If you haven’t seen this before, it’s worth thinking about it for a while, to appreciate the difficulty of writing such an “universal infinite-loop detector.”) Well, you can’t. Such an algorithm does not exist! And here is the proof: Suppose we actually had such a program terminates(p,x). Then we could use it as a subroutine of the following evil program: function paradox(z:file) 1: if terminates(z,z) goto 1 Notice what paradox does: it terminates if and only if program z does not terminate when given its own code as input. You should smell trouble. What if we put this program in a file named paradox and we executed paradox(paradox)? Would this execution ever stop? Or not? Neither answer is possible. Since we arrived at this contradiction by assuming that there is an algorithm for telling whether programs terminate, we must conclude that this problem cannot be solved by any algorithm. By the way, all this tells us something important about programming: It will never be automated, it will forever depend on discipline, ingenuity, and hackery. We now know that you can’t tell whether a program has an infinite loop. But can you tell if it has a buffer overrun? Do you see how to use the unsolvability of the “halting problem” to show that this, too, is unsolvable? 262
Exercises 8.1. Optimization versus search. Recall the traveling salesman problem: TSP Input: A matrix of distances; a budget b Output: A tour which passes through all the cities and has length ≤ b, if such a tour exists. The optimization version of this problem asks directly for the shortest tour. TSP-OPT Input: A matrix of distances Output: The shortest tour which passes through all the cities. Show that if TSP can be solved in polynomial time, then so can TSP-OPT. 8.2. Search versus decision. Suppose you have a procedure which runs in polynomial time and tells you whether or not a graph has a Rudrata path. Show that you can use it to develop a polynomialtime algorithm for RUDRATA PATH (which returns the actual path, if it exists). 8.3. STINGY SAT is the following problem: given a set of clauses (each a disjunction of literals) and an integer k, find a satisfying assignment in which at most k variables are true, if such an assignment exists. Prove that STINGY SAT is NP-complete. 8.4. Consider the CLIQUE problem restricted to graphs in which every vertex has degree at most 3. Call this problem CLIQUE-3. (a) Prove that CLIQUE-3 is in NP. (b) What is wrong with the following proof of NP-completeness for CLIQUE-3? We know that the CLIQUE problem in general graphs is NP-complete, so it is enough to present a reduction from CLIQUE-3 to CLIQUE. Given a graph G with vertices of degree ≤ 3, and a parameter g, the reduction leaves the graph and the parameter unchanged: clearly the output of the reduction is a possible input for the CLIQUE problem. Furthermore, the answer to both problems is identical. This proves the correctness of the reduction and, therefore, the NP-completeness of CLIQUE-3. (c) It is true that the VERTEX COVER problem remains NP-complete even when restricted to graphs in which every vertex has degree at most 3. Call this problem VC-3. What is wrong with the following proof of NP-completeness for CLIQUE-3? We present a reduction from VC-3 to CLIQUE-3. Given a graph G = (V, E) with node degrees bounded by 3, and a parameter b, we create an instance of CLIQUE-3 by leaving the graph unchanged and switching the parameter to |V | − b. Now, a subset C ⊆ V is a vertex cover in G if and only if the complementary set V − C is a clique in G. Therefore G has a vertex cover of size ≤ b if and only if it has a clique of size ≥ |V | − b. This proves the correctness of the reduction and, consequently, the NP-completeness of CLIQUE-3. (d) Describe an O(|V | 4 ) algorithm for CLIQUE-3. 8.5. Give a simple reduction from 3D MATCHING to SAT, and another from RUDRATA CYCLE to SAT. (Hint: In the latter case you may use variables x ij whose intuitive meaning is “vertex i is the jth vertex of the Hamilton cycle”; you then need to write clauses that express the constraints of the problem.) 8.6. On page 250 we saw that 3SAT remains NP-complete even when restricted to formulas in which each literal appears at most twice. 263
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Algorithms Copyright c○2006 S. Da
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4 Paths in graphs 109 4.1 Distances
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List of boxes Bases and logs . . .
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Preface This book evolved over the
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Chapter 0 Prologue Look around you.
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The first question is moot here, as
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for addition, which works on one di
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4. Likewise, any polynomial dominat
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and in general ( Fn ) = F n+1 ( ) n
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Bases and logs Naturally, there is
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Figure 1.1 Multiplication à la Fra
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Figure 1.3 Addition modulo 8. 6 0 0
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Figure 1.4 Modular exponentiation.
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Figure 1.6 A simple extension of Eu
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We say x is the multiplicative inve
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Figure 1.7 An algorithm for testing
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Figure 1.8 An algorithm for testing
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Randomized algorithms: a virtual ch
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Alice’s encryption function is th
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Figure 1.9 RSA. Bob chooses his pub
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There is nothing inherently wrong w
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Exercises 1.1. Show that in any bas
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the depth of the circuit or the lon
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(c) Give an example of positive int
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x = x L x R = 2 n/2 x L + x R y = y
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Figure 2.2 Divide-and-conquer integ
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Binary search The ultimate divide-a
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An n log n lower bound for sorting
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The three sublists S L , S v , and
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to be the best running time possibl
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qp t AB CD EF GH IJ KL MN OP QR Why
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The equivalence of the two polynomi
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Figure 2.6 The complex roots of uni
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A matrix reformulation To get a cle
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The orthogonality property can be s
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FFT n (input: a 0 , . . . , a n−1
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The slow spread of a fast algorithm
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(e) T (n) = 8T (n/2) + n 3 (f) T (n
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2.21. Mean and median. One of the m
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(c) In fact, squaring matrices is n
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Chapter 3 Decompositions of graphs
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Therefore, each edge appears in exa
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Figure 3.3. 1 The previsit and post
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Figure 3.6 (a) A 12-node graph. (b)
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Tree edges are actually part of the
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the same thing. Since a dag is line
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Figure 3.10 The reverse of the grap
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Exercises 3.1. Perform a depth-firs
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(a) Model this as a graph problem:
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For instance, in the graph below (w
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3.30. On page 97, we defined the bi
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Chapter 4 Paths in graphs 4.1 Dista
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Figure 4.4 The result of breadth-fi
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Figure 4.6 Breaking edges into unit
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into account. This viewpoint gives
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Which heap is best? The running tim
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Figure 4.11 (a) A binary heap with
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Figure 4.13 The Bellman-Ford algori
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Figure 4.15 A single-source shortes
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Input: Undirected graph G = (V, E)
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4.16. Section 4.5.2 describes a way
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Let r ∗ be the maximum ratio achi
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Chapter 5 Greedy algorithms A game
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Trees A tree is an undirected graph
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Figure 5.3 The cut property at work
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eturn x As can be expected, makeset
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Figure 5.7 The effect of path compr
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A randomized algorithm for minimum
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Figure 5.9 Top: Prim’s minimum sp
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Figure 5.10 A prefix-free encoding.
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149
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5.3 Horn formulas In order to displ
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Figure 5.11 (a) Eleven towns. (b) T
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Exercises 5.1. Consider the followi
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(a) Code: {0, 10, 11} (b) Code: {0,
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Input: Undirected graph G = (V, E);
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Chapter 6 Dynamic programming In th
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Figure 6.2 The dag of increasing su
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Programming? The origin of the term
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Figure 6.4 (a) The table of subprob
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Common subproblems Finding the righ
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6.4 Knapsack During a robbery, a bu
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Memoization In dynamic programming,
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Figure 6.7 (a) ((A × B) × C) × D
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ecause it leads right back to the O
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d ij . We must pick the best such i
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Figure 6.11 I(u) is the size of the
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6.8. Given two strings x = x 1 x 2
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Figure 6.12 Two binary search trees
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6.26. Sequence alignment. When a ne
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Chapter 7 Linear programming and re
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It is a general rule of linear prog
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the feasible region. The point of f
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Figure 7.3 A communications network
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Reductions Sometimes a computationa
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Matrix-vector notation A linear fun
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Figure 7.5 An illustration of the m
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L e R s t e ′ We claim that size(
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Figure 7.6 Continued Current Flow R
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from the algorithm, which in such c
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Figure 7.10 A generic primal LP in
Page 211 and 212: Now suppose the two of them play th
Page 213 and 214: In LP form, this is min w −3y 1 +
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Page 217 and 218: Figure 7.13 Simplex in action. Init
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Page 221 and 222: Gaussian elimination Under our alge
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Page 225 and 226: Exercises 7.1. Consider the followi
Page 227 and 228: 7.12. For the linear program max x
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Page 231 and 232: 7.28. A linear program for shortest
Page 233 and 234: Chapter 8 NP-complete problems 8.1
Page 235 and 236: Satisfiability SATISFIABILITY, or S
Page 237 and 238: cost as the budget. Fine, but how d
Page 239 and 240: Figure 8.3 What is the smallest cut
Page 241 and 242: Figure 8.4 A more elaborate matchma
Page 243 and 244: 8.2 NP-complete problems Hard probl
Page 245 and 246: I. These two translation procedures
Page 247 and 248: Figure 8.7 Reductions between searc
Page 249 and 250: Figure 8.8 The graph corresponding
Page 251 and 252: Figure 8.9 S is a vertex cover if a
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Page 255 and 256: Figure 8.10 Rudrata cycle with pair
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Page 265 and 266: Input: An undirected graph G = (V,
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Page 269 and 270: Chapter 9 Coping with NP-completene
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Page 275 and 276: 9.2 Approximation algorithms In an
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Page 279 and 280: 9.2.3 TSP The triangle inequality p
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Page 285 and 286: Figure 9.8 Local search. Figure 9.7
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Page 289 and 290: Figure 9.10 The search space for a
Page 291 and 292: Exercises 9.1. In the backtracking
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Page 295 and 296: Chapter 10 Quantum algorithms This
Page 297 and 298: Figure 10.2 Measurement of a superp
Page 299 and 300: Figure 10.3 A quantum algorithm tak
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Page 303 and 304: The Fourier transform of a periodic
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Page 307 and 308: 10.6 Factoring as periodicity We ha
Page 309 and 310: Figure 10.6 Quantum factoring. 0 QF
Page 311 and 312: Exercises 10.1. ∣ ψ 〉 = 1 √2
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Historical notes and further readin
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Index O(·), 13 Ω(·), 14 Θ(·),
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interior-point method, 217 interpol
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