## Algorithms and Resource Requirements for Fundamental Problems (2007)

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Citations: | 10 - 6 self |

### BibTeX

@TECHREPORT{Williams07algorithmsand,

author = {R. Ryan Williams},

title = {Algorithms and Resource Requirements for Fundamental Problems},

institution = {},

year = {2007}

}

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### Abstract

no. DGE-0234630. The views and conclusions contained in this document are those of the author and should not be interpreted as representing the official policies, either expressed or implied, of any sponsoring institution, the U.S. government or any other entity.

### Citations

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Computers and Intractability: A Guide to the Theory of NP-Completeness
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(Show Context)
Citation Context ...nough K. Garey, Johnson, and Stockmeyer [GJS76] proved that Max 2-Sat is NP-complete. For a much larger list of NP-complete problems, the reader is invited to consult Garey and Johnson’s classic text =-=[GJ79]-=-. The first part of this thesis presents time lower bounds on all the above problems (and many other NP complete problems). The second part of the thesis presents novel algorithms for Max Cut and Max ... |

3697 | Artificial Intelligence : A Modern Approach - Russell, Norvig - 1995 |

2345 | Computational Complexity
- Papadimitriou
- 1994
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Citation Context ...quired for our work. Our treatment is neither intended to be completely rigorous, nor rigorously complete. For further background, we invite the reader to try Papadimitriou’s Computational Complexity =-=[Pap94]-=-. 2.1 Asymptotics We start with a quick review of some asymptotic notation. Assume f,g : N → N. • f is O(g) if and only if there are c1,c2 ≥ 0 such that for all n ≥ 1, f(n) ≤ c1g(n)+c2. Thus f is “bou... |

1431 |
Reducibility among Combinatorial Problems
- Karp
- 1972
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Citation Context ...ity problem alone.” [Dew81] 12s• k-Sat, for any constant k > 2: given a Boolean formula F in k-CNF, does the question of the Sat problem hold for it? While k-Sat looks like a restriction of Sat, Karp =-=[Kar72]-=- showed that k-Sat is NP-complete as well, along with the remaining problems on this list. Interestingly, it is known that the 2-Sat problem is in P, and consequently is not known to be NP-complete. •... |

869 | Parameterized Complexity
- Downey, Fellows
- 1999
(Show Context)
Citation Context ...ms Imply Accelerated Sat Algorithms Our first hypothesis concerns the time complexity of k-Dominating Set. In Parameterized Complexity, k-Dominating Set is one of the canonical W[2]-complete problems =-=[DF99]-=-. Given an undirected graph on n nodes and m edges, the task is to find a k-set S of nodes whereby every node of the graph is either in S, or is incident to a node in S. For a long time, the best algo... |

801 |
Matrix multiplications via arithmetic progressions
- Coppersmith, Winograd
- 1990
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Citation Context ...n was shattered with Strassen’s breakthrough algorithm which uses only O(n log 2 7 ) ring operations [Str69]. The current fastest (ring) matrix multiplication algorithm is by Coppersmith and Winograd =-=[CW90]-=- and runs in O(n 2.376 ) ring operations. The main positive result of this work is that a large class of NP-hard problems can be solved significantly faster than exhaustive search, by connecting the s... |

775 | The complexity of theorem-proving procedures
- Cook
- 1971
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Citation Context ...Sat: given a Boolean formula F in conjunctive normal form, is there an assignment to the variables of F that makes the formula true? In a seminal paper laying the groundwork for NP-completeness, Cook =-=[Coo71]-=- showed that Sat is NP-complete. 2 Along these lines, A. K. Dewdney eloquently states a good reason why P �= NP seems so likely: “for all the hundreds of NP-complete problems, the thousands of person-... |

376 |
Some simplified NP-complete graph problems
- Garey, Johnson, et al.
- 1976
(Show Context)
Citation Context ...ables that makes at least K clauses of F true? Notice that the formula F can be a no-instance of 2-Sat, but (F,K) can be a yes-instance of Max 2-Sat for small enough K. Garey, Johnson, and Stockmeyer =-=[GJS76]-=- proved that Max 2-Sat is NP-complete. For a much larger list of NP-complete problems, the reader is invited to consult Garey and Johnson’s classic text [GJ79]. The first part of this thesis presents ... |

375 | A polynomial algorithm in linear programming - Khachiyan - 1979 |

319 |
Introduction to Algorithms, 2 nd edition
- Cormen, Leiserson, et al.
- 2001
(Show Context)
Citation Context ...a literal and there is an edge ℓi → ℓj iff (¬ℓi ∨ ℓj) ∈ F − C1 − C2. First, some preprocessing is performed on G. Compute the transitive closure of G in O(mn+n 2 ) time using standard techniques (cf. =-=[CLRS01]-=-, pp.632-633). More precisely, construct a Boolean matrix M where M[i,j] = 1 ⇐⇒ ℓi → ℓj in this time, for literals ℓi and ℓj. If there is a variable x such that x → ¬x and ¬x → x then return unsatisfi... |

317 |
Relationships between nondeterministic and deterministic tape complexities
- Savitch
- 1970
(Show Context)
Citation Context ... time and O(s) space. The case k = 2 was essentially proved by Kannan [Kan84]. The key idea of Lemma 3.3.2 is to mimic the proof in Chandra, Kozen, and Stockmeyer [CKS81] (following Savitch’s theorem =-=[Sav70]-=-) that DTISP[t,s] ⊆ ATIME[s log t]. In the proof of DTISP[t,s] ⊆ ATIME[s log t], the alternating simulation of a DTISP[t,s] machine M works by repeatedly guessing configurations “in the middle” of the... |

311 |
Linear-time algorithms for testing the satisfiability of propositional Horn formulae
- Dowling, Gallier
- 1984
(Show Context)
Citation Context ...f an improvement over brute force search. 7.3 A Variant on Horn-Sat Can Help Solve Sat Similar to 2-Sat, the Horn-Sat problem is another restriction of Sat that is known to be solvable in linear time =-=[DG84]-=-. An instance of Horn-Sat is a CNF formula with at most one non-negative literal per clause. Horn-Sat is considered to be a more powerful restriction of Sat than 2-Sat, 95ssince Horn-Sat is P-complete... |

226 |
A simple unpredictable pseudo-random number generator
- Blum, Blum, et al.
- 1986
(Show Context)
Citation Context ...ation. • randomadmiss: Takes an integer k and outputs a random k-bit proof annotation, drawn uniformly at random over all such annotations. (Random bits are obtained using the BlumBlum-Shub generator =-=[BBS86]-=-.) To perform the sampling, we adapted a simple method for producing random well-balanced strings, given by Arnold and Sleep [AS80]. • writeLP: Takes an LP instance and filename and writes the LP to t... |

210 |
R (2004) On the complexity of optimal k-anonymity
- Meyerson, Williams
(Show Context)
Citation Context ...of business is often to figure out what’s NP-complete and what’s not. (We have personally experienced this in database research, where a data privacy problem turned out to be unexpectedly NP-complete =-=[MW05]-=-, forcing us to design heuristic approximations for it.) The abundance of complete problems has forced thousands of computer science workers to refine their research agendas in the hopes of working ar... |

197 | A new polynomial time algorithm for linear programming - Karmarkar - 1984 |

184 |
Computational Limitations of Small Depth Circuits
- H˚astad
- 1988
(Show Context)
Citation Context ... Parity is in DTS[n]. We show using a standard construction that if Parity is in ΣkTIME[n 1/k−ε ], then the problem can be solved with depth k+1 circuits of 2 O(n1/k−ε ) size, contradicting H˚astad’s =-=[Has86]-=- celebrated circuit lower bound for Parity. Consider a ΣkTIME[n 1/k−ε ] machine M. Without loss of generality, suppose M guesses n 1/k−ε bits in each alternation. Let M ′ be the “deterministic part” o... |

183 |
The equivalence problem for regular expressions with squaring requires exponential space
- Meyer, Stockmeyer
- 1972
(Show Context)
Citation Context ...ise, it is trivial to check that (∃x)F is valid, if F is a DNF. (Recall that the Tautology problem, which is the “coNP version” of Sat, assumes that the formula is given in DNF.) Meyer and Stockmeyer =-=[MS72]-=- showed that Σk-Sat is complete for ΣkP. Letting k > 1, if we define ΣkQL := ΣkTIME[n ·poly(log n)], one can start asking which problems are robustly complete for these new quasilinear time classes. T... |

160 |
A linear-time algorithm for testing the truth of certain quantified boolean formulas
- Aspvall, Plass, et al.
- 1979
(Show Context)
Citation Context ...et in the graph. � 7.2 A Variant of 2-Sat Can Help Solve Sat 2-Sat is the well-studied restriction of Sat to instances with at most two literals per clause, and is known to be solvable in linear time =-=[APT79]-=-. One possible direction for achieving an accelerated algorithm for Sat is to try reducing the problem to 2-Sat in some interesting way. As we do not believe P = NP, this reduction should be exponenti... |

148 |
On the degree of boolean functions as real polynomials
- Nisan, Szegedy
- 1994
(Show Context)
Citation Context ...erated algorithms. It turns out that a degree two polynomial representing a Boolean function depends on at most four variables. In particular, the following can be shown. Theorem 6.5.3 (Nisan-Szegedy =-=[NS94]-=-) Let p be a degree d polynomial representing a Boolean function. Then p depends on at most d2 d−1 of its variables. It follows that Weighted Degree-Two CSP does not include Boolean constraint satisfa... |

134 | Efficient Algorithms for Solving Overdefined Systems of Multivariate Polynomial Equations - Courtois, Klimov, et al. |

128 | A probabilistic algorithm for k-SAT and constraint satisfaction problems
- Schöning
- 1999
(Show Context)
Citation Context ...thms for NP-hard problems in the literature involve either a case analysis of a branch-and-bound strategy (e.g. [GHNR03]), repeated random choice of assignments (e.g. [PPSZ05]), or local search (e.g. =-=[Sch99]-=-). Our design departs from these approaches, and applies a form of dynamic programming akin to earlier algorithms from the 70’s [HS74, SS81]. We call this dynamic programming strategy the split-and-li... |

122 | Which problems have strongly exponential complexity - Impagliazzo, Paturi, et al. - 2001 |

113 | Exact algorithms for NP-hard problems: A survey - Woeginger |

107 |
Multiparty protocols, pseudorandom generators for logspace, and time-space trade-offs
- Babai, Nisan, et al.
- 1992
(Show Context)
Citation Context ...deoff of [Cob66]. To illustrate this kind of proof technique, we briefly describe how to extend Santhanam’s result to multitape Turing machines with multiple heads per tape. Babai, Nisan, and Szegedy =-=[BNS92]-=- prove that the function Gk : {0,1} kn → {0,1} defined by Gk(x1,1,...,x1,n,...,xk,1,... ,xk,n) = ⎛ k� n� ⎝ i=1 j=1 xi,j ⎞ ⎠ mod 2 cannot be computed in Ω(n2−ε ) time and no(1) space on a k-head multit... |

99 |
Alternation
- Chandra, Stockmeyer
- 1981
(Show Context)
Citation Context ...t that a random-access machine using k quantifiers in time t can be simulated by a machine using k quantifiers in time O(t), found in Chandra and Stockmeyer’s original conference paper on alternation =-=[CS76]-=-. Theorem 3.3.1 (“No Complementary Speedup”) For all k ≥ 1 and time constructible t(n) ≥ n, ΣkTIME[t] � ΠkTIME[o(t)]. We call it the “No Complementary Speedup” Theorem, as it intuitively says that not... |

94 |
Algorithms for maximum independent sets
- Robson
- 1986
(Show Context)
Citation Context ...inctly expressing many interesting problems. While Vertex Cover and Independent Set were known to admit accelerated algorithms (the fastest known, to our knowledge, is Robson’s O(1.1889 n ) algorithm =-=[Rob86]-=-), the other four problems were not known to have accelerated algorithms. In the next section, we give an accelerated algorithm for Count 2-CSP, yielding accelerated algorithms for the above problems ... |

91 | Cryptanalysis of the HFE public key cryptosystem by relinearization - Kipnis, Shamir - 1999 |

84 | An improved exponential-time algorithm for k-SAT
- Paturi, Pudlàk, et al.
- 1998
(Show Context)
Citation Context ... Split and List Most exact algorithms for NP-hard problems in the literature involve either a case analysis of a branch-and-bound strategy (e.g. [GHNR03]), repeated random choice of assignments (e.g. =-=[PPSZ05]-=-), or local search (e.g. [Sch99]). Our design departs from these approaches, and applies a form of dynamic programming akin to earlier algorithms from the 70’s [HS74, SS81]. We call this dynamic progr... |

83 |
Cryptanalysis of the Matsumoto and Imai public key scheme of Eurocrypt ’88
- Patarin
- 1995
(Show Context)
Citation Context ...andard (AES) or Rijndael cipher (used by the U.S. Government to encrypt highly sensitive data) [Lan04] and the Hidden Field Equations (HFE) public key cryptosystem depend on the intractability of Mqs =-=[Pat95]-=-. The basic idea behind these systems is to encrypt a string b1 · · · bn of n bits by picking n random quadratic polynomials p1(x1,... ,xn),... ,pn(x1,... ,xn) and send the quadratic equations p1(x1,.... |

83 | Approximation algorithms for constraint satisfaction problems involving at most three variables per constraint - Zwick - 1998 |

81 |
Finding a minimum circuit in a graph
- Itai, Rodeh
- 1978
(Show Context)
Citation Context ...tr(A(G) 3 ) is computable in two matrix multiplications, and it is easy to see that tr(A(G) 3 ) is non-zero if and only if there is a triangle in G. (This observation was first made by Itah and Rodeh =-=[IR78]-=-.) For 3r-cliques when r > 1, build a graph Gr = (Vr,Er) where Vr is the collection of all r-cliques in G, and Er = { {c1,c2} : c1,c2 ∈ Vr, c1 ∪ c2 is a 2r-clique in G}. Observe that each triangle in ... |

79 |
Computing partitions with applications to the knapsack problem
- Horowitz, Sahni
- 1974
(Show Context)
Citation Context ...ze, thus the list of possible solutions to each part will be (exponentially) large. Example. The general paradigm of split-and-list was first used to solve the Subset Sum problem in O ∗ (2 n/2 ) time =-=[HS74]-=-. In this problem, one is given a set S of n integers and a target integer T, and is asked if there is a subset of S whereby � x∈S x = T. A split-and-list algorithm for the problem is obtained by firs... |

70 |
On a class of O(n 2 ) problems in computational geometry
- Gajentaan, Overmars
- 1993
(Show Context)
Citation Context ...m, and we are merely looking for k numbers that sum to zero, each one from three different lists of size n, then the conjecture holds– this is the well-known k-Sum problem from computational geometry =-=[GO95]-=- which can be easily solved in O(b · n ⌈k/2⌉ ) time. If the conjecture is true, then we can use the weighted k-clique algorithm to solve Mqs. 84sTheorem 6.6.1 Conjecture 6.6.1 implies that Mqs has a r... |

69 | Efficient Checking of Polynomials and Proofs and the Hardness of Approximation Problems
- Sudan
- 1996
(Show Context)
Citation Context ...tion, then reduce that problem to edge-weighted k-clique. We use a randomization trick that has been employed in other contexts, such as string matching [Kal02] and probabilistically checkable proofs =-=[Sud92]-=-. Definition 6.6.1 Let K be an extension field of F, and let r = (r1,... ,rm) ∈ K m . Define Pr(x1,... ,xn) := m� ri · pi(x1,... ,xn). i=1 Claim 7 Let K be a (finite) extension field of F, and let a =... |

59 |
On the complexity of the subgraph problem
- Neˇsetˇril, Poljak
- 1985
(Show Context)
Citation Context ...ynomial reduction, and a fast k-clique algorithm on undirected graphs yields the accelerated algorithm. 73s6.3 Fast k-Clique Detecting and Counting We first review an algorithm by Nesetril and Poljak =-=[NP85]-=- for detecting if a graph has a k-clique in less than n k steps. Theorem 6.3.1 ([NP85]) Let r ∈ Z + . Then 3r-clique on undirected graphs is solvable in O(n ωr ) time. Proof. First consider the case k... |

58 |
Rectangular matrix multiplication revisited
- Coppersmith
- 1997
(Show Context)
Citation Context ... Bk = Ak × AT k is an � n � � n � k/2 × k/2 matrix, where Bk[i,j] = 0 iff Si ∪ Sj is a dominating set. Proposition 7.1.2 For k ≥ 7, k-Dominating Set can be solved in n k+o(1) time. Proof. Coppersmith =-=[Cop97]-=- gave an algorithm for multiplying a n × n .294 matrix with a n .294 × n matrix in n2+o(1) ring operations. The product Bk = Ak×AT k is essentially a product of an N ×N2/k matrix with a N2/k × N matri... |

50 | 2 + p-sat: Relation of typical-case complexity to the nature of the phase transition. Random Structures and Algorithms
- Monasson, Zecchina, et al.
- 1999
(Show Context)
Citation Context ...)∧(x5 ∨x6)∧(x1 ∨x2 ∨x3 ∨x4 ∨x5 ∨x6)∧(¬x1 ∨ ¬x2 ∨ ¬x3 ∨ ¬x4 ∨ ¬x5 ∨ ¬x6) is a 2-Sat+2Clauses instance. Such “mixed” instances have been studied in the past, especially in the average-case setting (cf. =-=[MZKBT99]-=-) where one analyzes the solvability of randomly chosen formulas. Let us first give a simple quadratic time algorithm for solving this problem. Theorem 7.2.1 2-Sat+2Clauses is in O(mn+n 2 ) time, wher... |

46 |
Gaussian Elimination is not optimal”. Numerische Mathematik 13
- Strassen
- 1969
(Show Context)
Citation Context ... all possible pairs of rows and columns had to be multiplied separately in Θ(n) time. This intuition was shattered with Strassen’s breakthrough algorithm which uses only O(n log 2 7 ) ring operations =-=[Str69]-=-. The current fastest (ring) matrix multiplication algorithm is by Coppersmith and Winograd [CW90] and runs in O(n 2.376 ) ring operations. The main positive result of this work is that a large class ... |

43 | The minimum equivalent DNF problem and shortest implicants
- Umans
- 1998
(Show Context)
Citation Context ...s functionality. When F is written in DNF, and the question is to determine if there is a DNF F ′ of size K that agrees with F, then Formula Minimization is known to be Σ2P-complete, by work of Umans =-=[Uma01]-=-. Here, our focus is on the collection of problems Σk-Sat for integers k ≥ 1, defined as follows: • For odd k, we are given a CNF formula F on k sets of variables X1,...,Xk and are asked if the first ... |

42 |
The recognition problem for the set of perfect squares
- Cobham
- 1966
(Show Context)
Citation Context ... For example, Santhanam [San01] showed that Sat cannot be solved in n 2−ε time and n o(1) space on multitape Turing machines, by reducing Palindromes to Sat and invoking an old time-space tradeoff of =-=[Cob66]-=-. To illustrate this kind of proof technique, we briefly describe how to extend Santhanam’s result to multitape Turing machines with multiple heads per tape. Babai, Nisan, and Szegedy [BNS92] prove th... |

41 | Upper bounds for Max Sat: Further Improved - Bansal, Raman - 1999 |

40 |
On determinism versus nondeterminism and related problems
- Paul, Pippenger, et al.
- 1983
(Show Context)
Citation Context ...Kan84, MS87, WL92, For97, LV99, FvM00, Tou01] have followed this alternation-trading scheme, often in an implicit manner. For example, the celebrated result of Paul, Pippenger, Szemeredi, and Trotter =-=[PPST83]-=- that NTIME[n] �= DTIME[n] for multitape machines can be said to follow the alternation-trading scheme: 1. Assume NTIME[n] = DTIME[n]. 2. Paul-Pippenger-Szemeredi-Trotter prove DTIME[t] ⊆ Σ4TIME[t/log... |

38 | Complexity of k-sat
- Impagliazzo, Paturi
- 1999
(Show Context)
Citation Context ...algorithm. On the other hand, while many researchers are skeptical that an accelerated algorithm for Sat exists, we have not found much evidence for this skepticism. Results of Impagliazzo and Paturi =-=[IP01]-=- imply that if a O∗ (2δn ) algorithm for Sat exists, then a O∗ 1 δ(1− (2 e·k) ) algorithm for k-Sat exists. But this in itself is not truly evidence; for example, if δ = .99, the implied k-Sat algorit... |

37 | Determinism versus Nondeterminism for Linear Time RAMs with Memory Restrictions
- Ajtai
(Show Context)
Citation Context ... Applying an efficient reduction from problems in NTIME[n] to Sat, Fortnow proved that either Sat is not solvable in nondeterministic logspace, or Sat is not solvable in n 1+o(1) time. In 1999, Ajtai =-=[Ajt02]-=- studied the element distinctness problem: given a list of O(log n)-bit strings, determine if all strings are different. He proved the following time-space tradeoff bound for any random access machine... |

36 | Improved rounding techniques for the max 2-sat and max di-cut problems - Livnat, Zwick - 2002 |

36 | New upper bounds for maximum satisfiability - Niedermeier, Rossmanith - 2003 |

35 | Satisfiability is Quasilinear Complete in NQL
- Schnorr
- 1978
(Show Context)
Citation Context ...near time reduction to problem P with the property that each bit of the reduction can be computed in polylogarithmic time and logarithmic space. Building on work of Gurevich-Shelah [GS89] and Schnorr =-=[Sch78]-=-, Fortnow et al. proved that Sat is robustly complete. Theorem 2.5.1 (Fortnow-Lipton-Van Melkebeek-Viglas [FLvMV05]) Sat for formulas in conjunctive normal form is robustly complete for NQL. The robus... |

31 | Faster exact algorithms for hard problems: a parameterized point of view, Discrete Math - Alber, Gramm, et al. - 2001 |

28 |
Linear FPT reductions and computational lower bounds
- Chen, Huang, et al.
- 2004
(Show Context)
Citation Context ...ed algorithm of the form poly(m)· 2 δn . A weaker connection between k-Dominating Set and Sat has been recently established in the literature. We give its formal statement: Theorem 7.1.2 (Chen et al. =-=[CHKX04]-=-, Theorem 5.3) Unless FPT = W[1], k-Dominating Set is not in f(k)n o(k) time for any function f. It is not necessary to know the meaning of the classes FPT and W[1]; to define them would take us too f... |

28 | Worst-case upper bounds for MAX-2-SAT with an application to
- Gramm, Hirsch, et al.
(Show Context)
Citation Context ... all possible instances. 6.2.1 Outline of our approach: Split and List Most exact algorithms for NP-hard problems in the literature involve either a case analysis of a branch-and-bound strategy (e.g. =-=[GHNR03]-=-), repeated random choice of assignments (e.g. [PPSZ05]), or local search (e.g. [Sch99]). Our design departs from these approaches, and applies a form of dynamic programming akin to earlier algorithms... |