## Exact algorithms for NP-hard problems: A survey

Venue: | Combinatorial Optimization - Eureka, You Shrink!, LNCS |

Citations: | 118 - 3 self |

### BibTeX

@ARTICLE{Woeginger_exactalgorithms,

author = {Gerhard J. Woeginger},

title = {Exact algorithms for NP-hard problems: A survey},

journal = {Combinatorial Optimization - Eureka, You Shrink!, LNCS},

year = {},

pages = {185--207}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. We discuss fast exponential time solutions for NP-complete problems. We survey known results and approaches, we provide pointers to the literature, and we discuss several open problems in this area. The list of discussed NP-complete problems includes the travelling salesman problem, scheduling under precedence constraints, satisfiability, knapsack, graph coloring, independent sets in graphs, bandwidth of a graph, and many more. 1

### Citations

926 | Parameterized Complexity
- Downey, Fellows
- 1999
(Show Context)
Citation Context ...d this has lead to the so-called W-hierarchy, an infinite hierarchy of complexity classes: FPT ⊆ W [1] ⊆ W [2] ⊆ ··· ⊆ W [k] ⊆ ··· ⊆ W [P ].188 G.J. Woeginger We refer the reader to Downey & Fellows =-=[8]-=- for the exact definitions of all these classes. It is commonly believed that all W-classes are pairwise distinct, and that hence all displayed inclusions are strict. Some classes of optimization prob... |

576 |
Optimization, approximation, and complexity classes
- Papadimitriou, Yannakakis
- 1991
(Show Context)
Citation Context ... SUB-EXPonentially solvable problems) if for every fixed ε>0, it can be solved in poly(|x|) · 2ε·m(x) time. The complexity class SNP (the class Strict NP) was introduced by Papadimitriou & Yannakakis =-=[32]-=- for studying the approximability of optimization problems. SNP constitutes a subclass of NP, and it contains all problems that can be formulated in a certain way by a logical formula that starts with... |

178 | The Traveling Salesman Problem
- Applegate, Bixby, et al.
- 2006
(Show Context)
Citation Context ... c○ Springer-Verlag Berlin Heidelberg 2003186 G.J. Woeginger And if the data is nicely structured, then instances with up to 13000 cities can be handled in practice (Applegate, Bixby, Chvátal & Cook =-=[2]-=-). There is a huge gap between the empirical results from testing implementations and the known theoretical results on exact algorithms. – Fast algorithms with exponential running times may actually l... |

154 | Vertex cover: Further observations and further improvements
- Chen, Kanj, et al.
(Show Context)
Citation Context ...orithm with time complexity O ∗ (1.1602 n ) for the restriction of the maximum independent set problem to graphs with maximum degree three! Warning: This is not an easy exercise. See Chen, Kanj & Jia =-=[5]-=- for a solution. (b) Design a sub-exponential time exact algorithm for the restriction of the maximum independent set problem to planar graphs! Hint: Use the planar separator theorem of Lipton & Tarja... |

143 |
A dynamic programming approach to sequencing problems
- Held, Karp
- 1962
(Show Context)
Citation Context ...enoted by d(i, j). The goal is to minimize the total travel length of the salesman. A trivial algorithm for the TSP checks all O(n!) permutations. We now sketch the exact TSP algorithm of Held & Karp =-=[16]-=- that is based on dynamic programming across the subsets. For every non-empty subset S ⊆ {2,... ,n} and for every city i ∈ S, we denote by Opt[S; i] the length of the shortest path that starts in city... |

134 | Which problems have strongly exponential complexity
- Impagliazzo, Paturi, et al.
- 2001
(Show Context)
Citation Context ...al complexity theory depend delicately on the choice of parameters. The right approach seems to be to include an explicit complexity parameter in the problem specification (Impagliazzo, Paturi & Zane =-=[21]-=-). Recall that the decision version of every problem in NP can be formulated in the following way: Given x, decide whether there exists y so that |y| ≤m(x) and R(x, y). Here x is an instance of the pr... |

124 |
On cliques in graphs
- Moon, Moser
- 1965
(Show Context)
Citation Context ...nt. For any graph G, there exists a feasible coloring with χ(G) colors in which at least one color class is a maximalExact Algorithms for NP-Hard Problems: A Survey 191 independent set. Moon & Moser =-=[29]-=- have shown that a graph with n vertices contains at most 3 n/3 ≈ 1.4422 n maximal independent sets. By considering a collection of n/3 independent triangles, we see that this bound is best possible. ... |

103 |
On selecting a satisfying truth assignment
- Papadimitriou
(Show Context)
Citation Context ...andom walks. This implies that the time complexity is O ∗ ((4/3) n ) ≈ O ∗ (1.3334 n ). This algorithm and its analysis are due to Schöning [43]. Some of the underlying ideas go back to Papadimitriou =-=[31]-=- who showed that 2-SAT can be solved in polynomial time by a randomized local search procedure. The algorithm easily generalizes to the k-satisfiability problem, and yields a randomized exact algorith... |

96 | The quadratic assignment problem: A survey and recent developments
- Pardalos, Rendl, et al.
- 1994
(Show Context)
Citation Context ...al entries. The objective is to find a permutation π that minimizes the cost function ∑n i=1 is a notoriously hard problem, and no essentially faster algorithms are known (Pardalos, Rendl & Wolkowicz =-=[33]-=-). Prove that (under some reasonable complexity assumptions) the QAP can not be solved in O∗ (cn) time, for any fixed value c. 8 Concluding Remarks Currently, when we are dealing with an optimization ... |

95 |
Algorithms for maximum independent sets
- Robson
- 1986
(Show Context)
Citation Context ..., but performs a smarter (and pretty tedious) structural case analysis of the neighborhood around the high-degree vertex v. The algorithm of Jian [22] has a time complexity of O ∗ (1.2346 n ). Robson =-=[38]-=- further refines the approach. A combinatorial argument about connected regular graphs helps to get the running time down to O ∗ (1.2108 n ). Robson’s algorithm uses exponential space. Beigel [3] pres... |

83 |
Computing partitions with applications to the knapsack problem
- Horowitz, Sahni
- 1974
(Show Context)
Citation Context ...ndition that the total weight does not exceed W . The binary knapsack problem is closely related to the subset sum problem, and it can be solved (trivially) in O∗ (2n) time. In 1974, Horowitz & Sahni =-=[18]-=- used a preprocessing trick to improve the time complexity to O∗ (2n/2). For every I ⊆ {1,... ,⌊n/2⌋} we create a compound item xI with value aI = ∑ i∈I ai and weight wI = ∑ i∈I wi, and we put this it... |

82 |
Finding a maximum independent set
- Tarjan, Trojanowski
- 1977
(Show Context)
Citation Context ... the largest real root of γ 4 = γ 3 +1. This yields the time complexity O ∗ (1.3803 n ). The first published paper that deals with exact algorithms for maximum independent set is Tarjan & Trojanowski =-=[46]-=-. They give an algorithm with running time O ∗ (1.2599 n ). This algorithm follows essentially the above approach, but performs a smarter (and pretty tedious) structural case analysis of the neighborh... |

69 |
Solving satisfiability in less than 2 n steps
- Monien, Speckenmeyer
- 1985
(Show Context)
Citation Context ...olynomial factor of α n where α is the largest real root of α 3 = α 2 +α+1. Since α ≈ 1.8393, this gives a time complexity of O ∗ (1.8393 n ). In a milestone paper in this area, Monien & Speckenmeyer =-=[28]-=- improve the branching step of the above approach. They either detect a clause that can be handled without any branching, or they detect a clause for which the branching only creates formulas that con... |

65 | Satisfiability coding lemma
- Paturi, Pudlák, et al.
- 1999
(Show Context)
Citation Context ...eterministic and/or randomized algorithms for the k-satisfiability problem. More resuls on exact algorithms for k-satisfiability and related problems can be found in the work of Paturi, Pudlak & Zane =-=[34]-=-, Paturi, Pudlak, Saks & Zane [35], Pudlak [37], Rodoˇsek [40], and Williams [48]. 7 How Can We Prove That a Problem Has No Sub-exponential Time Exact Algorithm? All the problems discussed in this pap... |

63 |
A note on the complexity of the chromatic number problem
- Lawler
- 1976
(Show Context)
Citation Context ... this bound is best possible. Paull & Unger [36] designed a procedure that generates all maximal independent sets in a graph in O(n 2 ) time per generated set. Based on the ideas introduced by Lawler =-=[26]-=-, we present a dynamic program across the subsets with a time complexity of O ∗ (2.4422 n ). For a subset S ⊆ V of the vertices, we denote by G[S] the subgraph of G that is induced by the vertices in ... |

55 | Finding maximum independent sets in sparse and general graphs
- Beigel
- 1999
(Show Context)
Citation Context ...son [38] further refines the approach. A combinatorial argument about connected regular graphs helps to get the running time down to O ∗ (1.2108 n ). Robson’s algorithm uses exponential space. Beigel =-=[3]-=- presents another algorithm with a weaker time complexity of O ∗ (1.2227 n ), but polynomial space complexity. Robson [39] is currently working on a new algorithm which is supposed to run in time O ∗ ... |

46 | On the Complexity of k-SAT
- Impagliazzo, Paturi
(Show Context)
Citation Context ... ≥ 3. TheExact Algorithms for NP-Hard Problems: A Survey 203 exponential time hypothesis conjectures sk > 0 for all k ≥ 3, and that the numbers sk converge to some limit s∞ > 0. Impagliazzo & Paturi =-=[20]-=- prove that under ETH, sk ≤ (1 − α/k) · s∞ holds, where α is some small positive constant. Consequently, under ETH we can never have sk = s∞ and the time complexities for k-satisfiability must increas... |

45 | Improved algorithms for 3-coloring, 3-edge-coloring, and constraint satisfaction
- Eppstein
- 2001
(Show Context)
Citation Context ...ime of O ∗ (1.3446 n )byapplying the technique of pruning the search tree; see Section 4 of this survey. The current champion algorithm has a time complexity of O ∗ (1.3289 n ) and is due to Eppstein =-=[10]-=-. This algorithm combines pruning of the search tree with several tricks based on network flows and matching. Exercise 34 (Nielsen [30]) Find an O ∗ (1.7851 n ) exact algorithm that decides for a grap... |

44 | Small maximal independent sets and faster exact graph coloring
- Eppstein
(Show Context)
Citation Context ... time complexity of n∑ k=0 ( ) n k k 2 3 k/3 ≤ n 2 n∑ k=0 ( ) n 3 k k/3 = n 2 (1+3 1/3 ) n . Since 1 + 3 1/3 ≈ 2.4422, this yields the claimed time complexity O ∗ (2.4422 n ). Very recently, Eppstein =-=[11]-=- managed to improve this time complexity to O ∗ (2.4150 n ) where 2.4150 ≈ 4/3+3 4/3 /4. His improvement is based on carefully counting the small maximal independent sets in a graph. Finally, we turn ... |

41 |
Fixed-parameter tractability and completeness IV: On completeness for W [P] and PSPACE analogues. Annals of pure and applied logic
- Abrahamson, Downey, et al.
- 1995
(Show Context)
Citation Context ... our case the complexity parameter m(x). A whole theory has evolved around such parameterizations, and this has lead to the so-called W-hierarchy, an infinite hierarchy of complexity classes: FPT ⊆ W =-=[1]-=- ⊆ W [2] ⊆ ··· ⊆ W [k] ⊆ ··· ⊆ W [P ].188 G.J. Woeginger We refer the reader to Downey & Fellows [8] for the exact definitions of all these classes. It is commonly believed that all W-classes are pai... |

38 |
e D.S. Johnson [1979] “Computers and Intractability: A Guide to theory of NPCompleteness”, Freeman and
- Garey
(Show Context)
Citation Context ...rriers for the corresponding classes of optimization problems. More technical remarks. All optimization problems considered in this survey are known to be NP-complete. We refer the reader to the book =-=[14]-=- by Garey & Johnson for (references to) the NP-completeness proofs. We denote the base two logarithm of a real number z by log(z). 3 Technique: Dynamic Programming across the Subsets A standard approa... |

36 |
3-coloring in time O(1.3446 n ): a no-MIS algorithm
- Beigel, Eppstein
- 1995
(Show Context)
Citation Context ... − S] is bipartite. Schiermeyer [42] describes a rather complicated modification of this idea that improves the time complexity to O ∗ (1.415 n ). The first major progress is due to Beigel & Eppstein =-=[4]-=- who get a running time of O ∗ (1.3446 n )byapplying the technique of pruning the search tree; see Section 4 of this survey. The current champion algorithm has a time complexity of O ∗ (1.3289 n ) and... |

33 |
Minimizing the number of states in incompletely specified sequential switching functions
- Paull, Unger
- 1959
(Show Context)
Citation Context ...at a graph with n vertices contains at most 3 n/3 ≈ 1.4422 n maximal independent sets. By considering a collection of n/3 independent triangles, we see that this bound is best possible. Paull & Unger =-=[36]-=- designed a procedure that generates all maximal independent sets in a graph in O(n 2 ) time per generated set. Based on the ideas introduced by Lawler [26], we present a dynamic program across the su... |

29 | A probabilistic 3-SAT algorithm further improved
- Hofmeister, Schöning, et al.
- 2002
(Show Context)
Citation Context ...nd yields a randomized exact algorithm with time complexity O ∗ ((2(k − 1)/k) n ). The fastest known randomized exact algorithm for 3-satisfiability is due to Hofmeister, Schöning, Schuler & Watanabe =-=[17]-=-, and has a running time of O ∗ (1.3302 n ). It is based on a refinement of the above random walk algorithm. Open problem 62 Design better deterministic and/or randomized algorithms for the k-satisfia... |

29 |
An O(2 0.304n ) algorithm for solving the maximum independent set problem
- Jian
- 1986
(Show Context)
Citation Context ... This algorithm follows essentially the above approach, but performs a smarter (and pretty tedious) structural case analysis of the neighborhood around the high-degree vertex v. The algorithm of Jian =-=[22]-=- has a time complexity of O ∗ (1.2346 n ). Robson [38] further refines the approach. A combinatorial argument about connected regular graphs helps to get the running time down to O ∗ (1.2108 n ). Robs... |

25 | On limited versus polynomial nondeterminism
- Feige, Kilian
- 1997
(Show Context)
Citation Context ...ating classical NP-hardness proofs from the 1970s into SERF-reductions. The main technical problem is to keep the complexity parameters m(x) under control. In another line of research, Feige & Kilian =-=[12]-=- show that if in graphs with n vertices independent sets of size O(log n) can be found in polynomial time, then the 3-satisfiability problem can be solved in sub-exponential time. This result probably... |

21 | Worst-case analysis, 3-SAT decision and lower bounds: Approaches for improved SAT algorithms
- Kullmann
- 1997
(Show Context)
Citation Context ...es these ideas for 3-satisfiability even further, and performs a quantitative analysis of the number of 2-clauses in the resulting subtrees. This yields a time complexity of O ∗ (1.5783 n ). Kullmann =-=[24, 25]-=- writes half a book on the analysis of this approach, and gets time complexities of O ∗ (1.5045 n ) and O ∗ (1.4963 n ) for 3-satisfiability. The current champion algorithms for satisfiability are, ho... |

14 |
The generalized searching over separators strategy to solve some NP-Hard problems in subexponential time
- Hwang, Chang, et al.
- 1993
(Show Context)
Citation Context ...h such a time complexity O∗ (cn) with c<2 for the closely related, but slightly simpler Hamiltonian cycle problem (given a graph G on n vertices, does it contain a spanning cycle). Hwang, Chang & Lee =-=[19]-=- describe a sub-exponential time O(c √ n log n ) exact algorithm with some constant c > 1 for the Euclidean TSP. The Euclidean TSP is a special case of the TSP where the cities are points in the Eucli... |

10 |
Satisfiability – algorithms and logic
- Pudlák
- 1998
(Show Context)
Citation Context ...he k-satisfiability problem. More resuls on exact algorithms for k-satisfiability and related problems can be found in the work of Paturi, Pudlak & Zane [34], Paturi, Pudlak, Saks & Zane [35], Pudlak =-=[37]-=-, Rodoˇsek [40], and Williams [48]. 7 How Can We Prove That a Problem Has No Sub-exponential Time Exact Algorithm? All the problems discussed in this paper are NP-complete, and almost all of the devel... |

8 |
Fixed parameter intractability
- Downey, Fellows
- 1992
(Show Context)
Citation Context ...As far as this survey is concerned, all we need to know about SNP is that it is a fairly broad complexity class that contains many of the natural combinatorial optimization problems. Downey & Fellows =-=[7]-=- introduced parameterized complexity theory for investigating the complexity of problems that involve a parameter. This parameter may for instance be the treewidth or the genus of an underlying graph,... |

8 |
What are the least tractable instances of max independent set
- Johnson, Szegedy
- 1999
(Show Context)
Citation Context ... Hamiltonian cycle problem and the independent set problem (both with the number of vertices as complexity parameter) can not be solved in sub-exponential time, unless SNP ⊆ SUBEXP. Johnson & Szegedy =-=[23]-=- strengthen the result on the independent set problem by showing that the independent set problem in arbitrary graphs is equally difficult as in graphs with maximum degree three: Either both of these ... |

8 | Algorithms for quantified Boolean formulas
- Williams
- 2002
(Show Context)
Citation Context ...resuls on exact algorithms for k-satisfiability and related problems can be found in the work of Paturi, Pudlak & Zane [34], Paturi, Pudlak, Saks & Zane [35], Pudlak [37], Rodoˇsek [40], and Williams =-=[48]-=-. 7 How Can We Prove That a Problem Has No Sub-exponential Time Exact Algorithm? All the problems discussed in this paper are NP-complete, and almost all of the developped algorithms use exponential t... |

7 |
Faster exact solutions for some NP-hard problems
- Drori, Peleg
(Show Context)
Citation Context ...ether there exists a subset Y ⊆ X such that |Y ∩ T | =1for all T ∈S. Use the technique of preprocessing the data to get an exact algorithm with time complexity O∗ (2n/2) ≈ O∗ (1.4145n). Drori & Peleg =-=[9]-=- use the technique of pruning the search tree to get a time complexity of O∗ (1.2494n) for the Exact-Hitting-Set problem. Exercise 52 (Van Vliet [47]) In the Three-Partition problem, the input consist... |

3 |
A new approach on solving 3-satisfiability
- Rodoˇsek
- 1996
(Show Context)
Citation Context ...lity problem. More resuls on exact algorithms for k-satisfiability and related problems can be found in the work of Paturi, Pudlak & Zane [34], Paturi, Pudlak, Saks & Zane [35], Pudlak [37], Rodoˇsek =-=[40]-=-, and Williams [48]. 7 How Can We Prove That a Problem Has No Sub-exponential Time Exact Algorithm? All the problems discussed in this paper are NP-complete, and almost all of the developped algorithm... |

2 |
Exponential time algorithms for computing the bandwidth of a graph
- Feige, Kilian
(Show Context)
Citation Context ... in O ∗ (n!) time. We will sketch an exact O ∗ (20 n ) algorithm for the bandwidth problem that is based on the technique of pruning the search tree. This beautiful algorithm is due to Feige & Kilian =-=[13]-=-. The algorithm checks for every integer b with 1 ≤ b ≤ n in O ∗ (20 n ) time whether the bandwidth of the input graph G is less or equal to b. To simplify the presentation, we assume that both n and ... |

2 |
1998]. An improved exponential time algorithm for k-SAT
- Paturi, Pudlak, et al.
(Show Context)
Citation Context ...orithms for the k-satisfiability problem. More resuls on exact algorithms for k-satisfiability and related problems can be found in the work of Paturi, Pudlak & Zane [34], Paturi, Pudlak, Saks & Zane =-=[35]-=-, Pudlak [37], Rodoˇsek [40], and Williams [48]. 7 How Can We Prove That a Problem Has No Sub-exponential Time Exact Algorithm? All the problems discussed in this paper are NP-complete, and almost all... |

1 |
Schöning [2001]. A deterministic (2− 2 k+1 )n algorithm for k-SAT based on local search
- Dantsin, Goerdt, et al.
(Show Context)
Citation Context ...igible value e−100 . In fact, the whole algorithm can be derandomized without substantially increasing the running time. Dantsin, Goerdt, Hirsch, Kannan, Kleinberg, Papadimitriou, Raghavan & Schöning =-=[6]-=- do not choose the centers of the balls at random, but they take all centers from a so-called covering code so that the resulting balls cover the whole solution space. They show that such covering cod... |

1 |
2000]. Faster exact solutions for
- Gramm, Niedermeier
(Show Context)
Citation Context ... ) for some c<2. (b) Design an exact algorithm for the restriction of the Max-Cut problem to graphs with maximum degree three that has a time complexity O ∗ (c n ) for some c<1.5. Gramm & Niedermeier =-=[15]-=- state an algorithm with time complexity O ∗ (1.5160 n ). The bandwidth problem. Given a graph G =(V,E) with n vertices, a linear arrangement is a bijective numbering f : V →{1,... ,n} of the vertices... |

1 |
1999]. New methods for 3-SAT decision and worst case analysis
- Kullmann
(Show Context)
Citation Context ...es these ideas for 3-satisfiability even further, and performs a quantitative analysis of the number of 2-clauses in the resulting subtrees. This yields a time complexity of O ∗ (1.5783 n ). Kullmann =-=[24, 25]-=- writes half a book on the analysis of this approach, and gets time complexities of O ∗ (1.5045 n ) and O ∗ (1.4963 n ) for 3-satisfiability. The current champion algorithms for satisfiability are, ho... |

1 |
Tarjan [1979]. A separator theorem for planar graphs
- Lipton, E
(Show Context)
Citation Context ...th formulas F in CNF for which the graph GF is planar. Design a sub-exponential time exact algorithm for the planar 3-satisfiability problem! Hint: Use the planar separator theorem of Lipton & Tarjan =-=[27]-=- to break the formula F into two smaller, independent pieces. Running times of roughly O ∗ (c √ n ) are possible. The independent set problem. Given a graph G =(V,E) with n vertices, the goal is to fi... |

1 |
1992]. Solving 3-satisfiability in less than O(1.579 n ) steps. Selected papers from
- Schiermeyer
(Show Context)
Citation Context ...ul analysis yields a time complexity of O ∗ (β n ) for k-satisfiability, where β is the largest real root of β =2− 1/β k−1 . For 3-satisfiability, this time complexity is O ∗ (1.6181 n ). Schiermeyer =-=[41]-=- refines these ideas for 3-satisfiability even further, and performs a quantitative analysis of the number of 2-clauses in the resulting subtrees. This yields a time complexity of O ∗ (1.5783 n ). Kul... |

1 |
Deciding 3-colorability in less than O(1.415 n ) steps
- Schiermeyer
(Show Context)
Citation Context ...f deciding whether χ(G) = 3. Lawler [26] gives a simple O ∗ (1.4422 n ) algorithm: Generate all maximal independent sets S, and check whether their complement graph G[V − S] is bipartite. Schiermeyer =-=[42]-=- describes a rather complicated modification of this idea that improves the time complexity to O ∗ (1.415 n ). The first major progress is due to Beigel & Eppstein [4] who get a running time of O ∗ (1... |

1 |
Schöning [1999]. A probabilistic algorithm for k-SAT and constraint satisfaction problems
- unknown authors
(Show Context)
Citation Context ...f the algorithm is proportional to the number of performed random walks. This implies that the time complexity is O ∗ ((4/3) n ) ≈ O ∗ (1.3334 n ). This algorithm and its analysis are due to Schöning =-=[43]-=-. Some of the underlying ideas go back to Papadimitriou [31] who showed that 2-SAT can be solved in polynomial time by a randomized local search procedure. The algorithm easily generalizes to the k-sa... |

1 |
Schöning [2001]. New algorithms for k-SAT based on the local search principle
- unknown authors
(Show Context)
Citation Context ...∗ ( √ 3 n ) ≈ O ∗ (1.7321 n ) for 3-satisfiability. It is debatable whether this algorithm should be classified under pruning the search tree or under local search. In any case, it is due to Schöning =-=[44]-=-. Second local search approach to 3-satisfiability. In the first approach, we essentially covered the whole solution space by two balls of radius d = n/2 centered at 0n and 1n . The second approach wo... |