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## An overview on polynomial approximation of NP-hard problems (2007)

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14035 | Computers and intractability: a guide to the theory of NP-completeness, Freeman, - Garey, Johnson - 1979 |

2799 | Computational Complexity - Papadimitriou - 1994 |

1992 |
Reducibility among combinatorial problems, in
- Karp
- 1972
(Show Context)
Citation Context ... application of the concept of reduction in computer science andscombinatorics arose in the early seventies in the seminal paper by [21], and was soonsfollowed by the equally fund to amental paper by =-=[47]-=-. Actually, both Cook's and Karp'ssreductio olving them. However, thissclassificshow can one compare the approximability of different versions of the sames• rsn dealingsns where conceived to relate to... |

1910 |
Theory of linear and integer programming
- Schrijver
- 1999
(Show Context)
Citation Context ...1 SV ⊆ 2.s*0 \ SVV ⊆ A basic corollary of Theorem 1 is that in order to solve MAX INDEPENDENT SET,sone can first solve its linear relaxation MAX INDEPENDENT SET-R (this can be done inspolynomial time =-=[3, 37, 48, 69]-=-) and storesand then solve the MAX INDEPENDENT SETsin some way in , i.e., the subgraph of Gsinduced by .s1V 1/2[G V ] ] 1/2V Indeed, the solution of the MAX INDEPENDENT SET-R provides sets ,sandsthat ... |

1703 | Combinatorial Optimization: Algorithms and Complexity. - Papadimitriou, Steiglitz - 1982 |

1275 | Approximation algorithms - Vazirani - 2002 |

1046 | The complexity of theorem-proving procedures
- Cook
- 1971
(Show Context)
Citation Context ...y also be used for derivingsnegative (impossible) results.sThe first application of the concept of reduction in computer science andscombinatorics arose in the early seventies in the seminal paper by =-=[21]-=-, and was soonsfollowed by the equally fund to amental paper by [47]. Actually, both Cook's and Karp'ssreductio olving them. However, thissclassificshow can one compare the approximability of differen... |

939 | Introduction to Metamathematics, - Kleene - 1952 |

823 |
Approximation algorithms for combinatorial problems
- Johnson
- 1973
(Show Context)
Citation Context ...d unexpected happens, i.e., until a very highly improbablescomplexity hypothesis (e.g., P = NP) holds.sThe existence of polynomial approximation as a scientific area was started by asseminal paper by =-=[46]-=-. Since then, this research programme is one of the most activesresearch programmes in operational research, combinatorial optimization and theoreticalscomputer science.s2. PRELIMINARIESsThe object of... |

678 |
Approximation Algorithms for NP-Hard Problems
- Hochbaum
- 1997
(Show Context)
Citation Context ...lution , is computable in polynomialstime;s( , )m I S S • .sgoal {min,max}∈ We shall now briefly present some of the basic notions of the polynomialsapproximation theory. More details can be found in =-=[5, 44, 66, 72]-=-.sHaving given an instancesof a combinatorial maximization (resp.,sminimization) problem Π = (I, Sol, m, goal), we denote bysI ( )Iω ,sand opt (A ( , )m I S )Isthe value of the worst solution ofs(in t... |

639 | Graphs and Hypergraphs - Berge - 1973 |

608 | Optimization, approximation, and complexity classes - Papadimitriou, Yannakakis - 1991 |

490 | The ellipsoid method and its consequences in com- binatorial optimization, Combinatorica 1 - Grotschel, Lovsz, et al. - 1981 |

397 | Polynomial time approximation schemes for Euclidian TSP and other geometric problems, in:
- Arora
- 1996
(Show Context)
Citation Context ...imal solution.sV. Th. Paschos / An Overview on Polynomial Approximation of NP-hard Problems 6sThe value of the differential ratio is always in , independently on thesoptimization goal of the problem.s=-=[0,1]-=- For both paradigms, the closer the ratios are to 1, the better the performance ofsthe approximation algorithm A. Let us also note that, as we will see, the results obtainedsby adopting one or the oth... |

374 | P-complete approximation problems
- Sahni, Gonzalez
- 1976
(Show Context)
Citation Context ...NPO roblem osp ′′Π . Then ′′Πsis not approximable w hin bettersthan 1 ( )c r− .sWe conclude this short section ith anoth xample of GAP-reduction settlingsMIN T itsw er e SP.s-appro letesb Example 4. (=-=[35, 68]-=-).Let us prove that (general) MIN TSP is not approximableswithin any constant standard-approximation ratio, unless P = NP. Suppose a contrario,sthat MIN TSP is polynomially approximable within standar... |

371 | Worst-Case Analysis of a New Heuristic for the Travelling Salesman Problem - Christofides - 1976 |

320 | On the ratio of optimal integral and fractional covers - Lovász - 1975 |

317 | Generalized first-order spectra and polynomial-time r cognizable sets - Fagin - 1974 |

309 |
M.: Complexity and Approximation – Combinatorial optimization problems and their approximability properties
- Ausiello, Crescenzi, et al.
- 1999
(Show Context)
Citation Context ...lution , is computable in polynomialstime;s( , )m I S S • .sgoal {min,max}∈ We shall now briefly present some of the basic notions of the polynomialsapproximation theory. More details can be found in =-=[5, 44, 66, 72]-=-.sHaving given an instancesof a combinatorial maximization (resp.,sminimization) problem Π = (I, Sol, m, goal), we denote bysI ( )Iω ,sand opt (A ( , )m I S )Isthe value of the worst solution ofs(in t... |

258 |
On colouring the nodes of a network
- Brooks
- 1941
(Show Context)
Citation Context ...raph Gscan be feasibly colored8swith at most ( )GΔscolors in polynomial time ([53]). This result is, in fact, thesconstructive proof of an existential theorem about such coloring originally stated by =-=[18]-=-.sFix a maximum independent setsof Gsthat containss(by item 1 insTheorem 1 this is always possible). Sincesis the largest of the at mostscolors (independent sets) produced in step 2 of Algorithm IS, i... |

231 |
Fast approximation algorithms for the knapsack and sum of subset problems,
- Ibarra, Kim
- 1975
(Show Context)
Citation Context ...orssand c I aG Gsand of asconstant bsand can be defined in termssof an integer-linear program as follows:smax = a x I c x b ⋅⎧⎨ ⋅ ≤⎩ G G G GsConsider the following algorithm for KNAPSACK presented by =-=[45]-=-:s1. fix an > 0εsand build the instance =1, ,= (( , ) , )i i i nI a c b′ ′ …swithsmax= /( )i ia a n a ε′ ⎣ ⎦ ;s2. output := DYNAMICPROGRAMMING(I )S ′ .sThis dynamic programming algorithm is a classica... |

200 | Proof verification and intractability of approximation problems,” - Arora, Lund, et al. - 1998 |

196 | Zero knowledge and the chromatic number. - Feige, Kilian - 1998 |

173 | A local-ratio theorem for approximating the weighted vertex cover problem - Bar-Yehuda, Even - 1985 |

152 |
Vertex packings: structural properties and algorithms
- Nemhauser, Trotter
- 1975
(Show Context)
Citation Context ...ion of the MAX INDEPENDENT SETsand its linear relaxation, denoted by MAX INDEPENDENT SET-R, where in the given graphs, ( , )G V E Asdenotes its incidence matrix:sThe following seminal theorem, due to =-=[57]-=- gives a very interestingscharacterization for the basic optimal solution of (2).sTheorem 1. ([57]). The basic optimal solution of MAX INDEPENDENT SET-R isssemi-integral, i.e., it assigns values from ... |

149 | Vertex cover might be hard to approximate to within 2− ǫ
- Khot, Regev
(Show Context)
Citation Context ...long effort has only produced ratios of thesform:s• 2 (log log / log )n n−s([10, 54]);s• 2 (2ln ln / ln )n n−s([40]);s• 2 (log log ( )/ log ( ))G G− Δ Δs([40]).sUnfortunately, a more recent result by =-=[50]-=- gives strong evidence that thissimprovement is likely to be impossible.sIn the differential approximation paradigm, MIN VERTEX COVER issequiapproximable with MAX INDEPENDENT SET ([27]). So, it defini... |

137 |
A Polynomial Algorithm in Linear Programming", Doklady Akademii Nauk
- Khachian
- 1979
(Show Context)
Citation Context ...1 SV ⊆ 2.s*0 \ SVV ⊆ A basic corollary of Theorem 1 is that in order to solve MAX INDEPENDENT SET,sone can first solve its linear relaxation MAX INDEPENDENT SET-R (this can be done inspolynomial time =-=[3, 37, 48, 69]-=-) and storesand then solve the MAX INDEPENDENT SETsin some way in , i.e., the subgraph of Gsinduced by .s1V 1/2[G V ] ] 1/2V Indeed, the solution of the MAX INDEPENDENT SET-R provides sets ,sandsthat ... |

126 | On Syntactic Versus Computational Views of Approximability.
- Khanna, Motwani, et al.
- 1998
(Show Context)
Citation Context ...issrelated to the fact that it is possible to prove that Max-SNP-complete problems do notsallow polynomial time approximation schemata, unless P = NP ([2]).sThe two approaches have been reconciled by =-=[49]-=-, where the closure ofssyntactically defined classes with respect to an approximation preserving reduction weresproved equal to the more familiar computationally defined classes. As a consequence ofst... |

121 | A tight analysis of the greedy algorithm for set cover - Slavík - 1997 |

113 | Improved approximation algorithms for the vertex cover problem in graphs and hypergraphs. In: - Halperin - 2000 |

91 |
E!cient bounds for the stable set, vertex cover and set packing problems.
- Hochbaum
- 1983
(Show Context)
Citation Context ...independent set per se) and of ansindependent set ofsis an independent set for the whole of G .s0V 1V 1/2V 0V 1/2V 1V 0V 0V 1/2V 1V 1/2V 1V 1/2[G V Let us now consider the following algorithm, due to =-=[43]-=-, denoted by IS:s1. solve MAX INDEPENDENT SET-R in order to determine ,set ;s0V 1V 1/2VsV. Th. Paschos / An Overview on Polynomial Approximation of NP-hard Problems 10s2. colorswith at most 1/2[G V ] ... |

83 | The development of mathematics. - Bell - 1945 |

82 | A still better performance guarantee for approximate graph coloring. - HALLDORSSON - 1993 |

80 | Structure preserving reductions among convex optimization problems - Ausiello, D’Atri, et al. - 1980 |

79 | Completeness in approximation classes
- Crescenzi, Panconesi
- 1991
(Show Context)
Citation Context ...nd MPTAS-reduction.sThe final answer to the problem of finding the suitable kind of reductions(powerful enough to establish completeness results both in NPO and APX) is the APreduction, introduced by =-=[23]-=-.sA large number of other reduct ed throughout the years. Overviews of the world of approximation preservingsreductions and completeness is approximability classes can be found in [8, 9, 22, 23].sOn t... |

71 | Approximations of weighted independent set and hereditary subset problems - Halldórsson |

66 | Numbers and an Approximation Algorithm for the Vertex Cover Problem. - Monien, Speckenmeyer - 1985 |

62 | Clique is Hard to Approximate within n 1−ǫ - Hastad - 1999 |

58 |
A Method for Solving Traveling-Salesman Problems
- Croes
- 1958
(Show Context)
Citation Context .... So, determining the worst solution of MIN TSP is as hard as to determine ansoptimal solution.sLet us consider the following well-known algorithm for MIN TSP, denoted bys2_OPT, originally devised by =-=[25]-=-, wheresdenotes the weight of edge ( , :s( , )d i j )i jv v Figure 6: Algorithm 2_OPTs1. construct some Hamiltonian tour Ts(this can be done, for example, by thesnearest-neighbor heuristic);s2. consid... |

39 | Three short proofs in graph theory. - Lovasz - 1975 |

39 | On approximation preserving reductions: Complete problems and robust measures.
- Orponen, Mannila
- 1987
(Show Context)
Citation Context ...roblems have beensintroduc tructural development analogous to the one that has beenscarried on for the standard paradigm has been elaborated also for the differentialstion problems were introduced by =-=[58]-=-. In particular, this paper presented the socalled strict reduction and provided the first examples of complete problems (MINsVARIABLE-WSAT, MIN 0-1 LINEAR PROGRAMMING and MIN TSP) under approximation... |

30 |
On approximate solutions for combinatorial optimization problems
- Simon
- 1990
(Show Context)
Citation Context ...approximation ratios that are “better than constants”.sAs a last example, let us consider the following classical reduction between MAXsvised eserv WEIGHTED INDEPENDENT SET15 and MAX INDEPENDENT SET (=-=[70]-=-).sLet us consider an instance ( ( , ), )G V E wGsof MAX WEIGHTED INDEPENDENT SETsand sup cespose, in order that the reduction that follows is polynomial, that weights arespolynomial with the order ns... |

27 |
On an approximation measure founded on the links between optimization and polynomial approximation theory
- Paschos
- 1996
(Show Context)
Citation Context ... definitely belongs to Log-APX.sDealing with the differential approximation paradigm, MIN SET COVER issapproximable within differential ratio bounded below by 1.365/Δs([12]) andsinapproximable as is (=-=[27]-=-). So, it belongs to Poly-DAPX.s4.3. Min vertex coversGiven a graph , MIN VERTEX COVER consists of determining asminimum-size set Vsso that, for every ( , ( , )G V E V′ ⊆ )u v E∈ , at least one among ... |

23 | 8/7-approximation algorithm for (1, 2)-TSP,
- Berman, Karpinski
- 2006
(Show Context)
Citation Context ...approximation ratio 3/2 by the celebrated Christofides algorithm ([19]) while the mostsfamous relaxation of MIN METRIC TSP, that is when the edge-weights are either 1 or 2 issapproximable within 8/7 (=-=[17]-=-). Finally, the MIN EUCLIDEAN TSP is in PTAS ([1]).s4.5. Min coloringsGiven a graph , MIN COLORING consists of determining the minimumsnumber of colors (i.e., of independent sets, see also footnote 8)... |

17 | A greedy-heuristic for the set covering problem - Chvátal - 1979 |

15 | A modified greedy heuristic for the set covering problem with improved worst case bound”, - Goldschmidt, Hochbaum, et al. - 1993 |

14 |
A short guide to approximation preserving reductions
- Crescenzi
- 1997
(Show Context)
Citation Context ...y of the existing onessicular conditions on the way to the optimal solutions, or the approximat c. are transformed from one problem to another. For more details, the interestedsreader can referred to =-=[5, 8, 9, 22, 32, 44, 66, 72]-=-.sIn general, given the two NPO problems Π = (I, Sol, m, goal) and Π′ = (I, Sol,sm, goal′), an approximation preserving reduction R from Πsto ′Πs(denoted R ′Π ≤ Π ) issa triple ( , , )f g csof polynom... |

14 | Differential approximation results for the Traveling Salesman and related problems - Monnot - 2002 |

12 | Khachiayan's Linear Programming Algorithm - Aspvall, Stone - 1980 |

11 | Time Slot Scheduling of Compatible Jobs - Paschos - 2001 |

11 |
Maximizing the number of unused colors in the vertex coloring problem
- Hassin, Lahav
- 1994
(Show Context)
Citation Context ...color produces a feasible coloring. Furthermore,sthis coloring cannot be augmented without producing empty colors.sn Consider the following algorithm for MIN COLORING, denoted by COLOR andsdevised by =-=[41]-=- (see also [65]):sV. Th. Paschos / An Overview on Polynomial Approximation of NP-hard Problems 20s1. find an independent setsof size 3 in ; color its vertices with a newscolor and remove it from ;sS G... |

10 | Polynomial approximation and graph-coloring. - Paschos - 2003 |

9 |
Free bits and non-approximability — towards tight results
- Bellare, Goldreich, et al.
- 1998
(Show Context)
Citation Context ...oximation of NP-hard Problems 22sOn the other hand, it is inapproximable within better than:s• 1n ε−sfor any constant > 0ε , unlesss([34]);sNP coRP⊆ • (1/5)n ε−sfor any > 0ε , assuming that NP coRP≠s(=-=[15]-=-);s• (1/7)n ε−sfor any > 0εsunless P ≠ NP ([15]).sMore about the complexity classes mentioned in the results above can be foundsin [59].s4.6. Min weighted bipartite coloringsConsider a vertex-weighted... |

7 | Paschos. Completeness in differential approximation classes
- Ausiello, Bazgan, et al.
(Show Context)
Citation Context ... computedsduring instance's revealing divided by the value of the optimal solution of the wholesthat is much younger than the former since it has been defined at the beginningsof the 90's by [27]. In =-=[4]-=- and in [11] the approximability classes DAPX and DPTAS aresintroduced, suitable approximation preserving reductions are defined and compl s in NPO, 0-DAPX, DAPX, DPTAS and Poly-DAPX, under such kind ... |

6 | Th.: Improved approximations for weighted and unweighted graph problems. Theory of Computing Systems - Demange, Paschos |

5 |
Lattice-theoretical ordering properties for NPcomplete optimization problems
- Ausiello, D’Atri, et al.
- 1981
(Show Context)
Citation Context ...n the same way?sIs there some stronger kind of reducibility than the simple polynomial reducibility thatswill explain s as we define them? ([46]).sThe first answer to the above questions was given by =-=[6, 7]-=- where the notion ofsV. Th. Paschos / An Overview on Polynomial Approximation of NP-hard Problems 35sMAX VARIABLE-WSAT (a weighted version of MAX SAT) in the class of NPO problems issproved. Still it ... |

5 | Reductions, completeness and the hardness of approximability - Paschos - 2006 |

5 | A survey about how optimal solutions to some covering and packing problems can be approximated - Paschos - 1997 |

4 | On the differential approximation of min set cover. Theoret - Paschos, Serrière |

4 | Th Paschos. Differential approximation for optimal satisfiability and related problems - Bazgan, V |

4 |
A sub-constant error probability low-degree test and a sub-constant error probability PCP characterization of NP
- Raz, Safra
- 1997
(Show Context)
Citation Context ...2 3{ , , }S S S 1= 3 2k−Δ × 1( 1)/3 = (ln(3 2 ))kk O −+ × More recently, another analysis of Algorithm GREEDYSC has been presentedsby [71] providing a tight ratio of . On the other hand, as stated in =-=[67]-=- (sees[31] for an informal proof), it is impossible to approximate MIN SET COVER within bettersthan , unless a highly unlikely complexity classess(log | |)O C (log | |) (log log | |)O C O C−sV. Th. Pa... |

4 |
Time slot scheduling of compatible jobs
- Werra, Demange, et al.
(Show Context)
Citation Context ... for ,sis defined as in (25). = 1, ,i k… ( )iw S MIN WEIGHTED COLORING is obviously NP-hard in general graphs since settings, , it becomes the classical MIN COLORING problem. However, it is proved ins=-=[26]-=- that it is NP-hard, even in bipartite graphs (MIN WEIGHTED BIPARTITE COLORING).sLet us note that MIN COLORING is polynomial in these graphs since they are 2-colorable.s= 1iw iv V∈ In what follows in ... |

3 | Improved approximations for maximum independent set via approximation chains - Paschos - 1997 |

3 |
Approximation algorithms for the traveling salesman problem
- Paschos, Toulouse
(Show Context)
Citation Context ...e;s4. output Tsthe finally produced tour.sAlgorithm 2_OPT is polynomial when, for instance,sthe maximum edgeweight is bounded above by a polynomial of . For other cases where 2_OPT isspolynomial, see =-=[56]-=-.smaxd n Theorem 5. ([56]) The algorithm 2_OPT achieves a differential-approximationsratio bounded below by 1/2.sProof. Assume that Tsis represented as the set of its edges, i.e.:s( ) ( ) ( ){ }1 2 1 ... |

3 | Complexité et approximation polynomiale - Paschos - 2004 |

3 |
A tight analysis of the greedy algorithm for set cover
- Slavk
- 1997
(Show Context)
Citation Context ...red, . Consequently, the approximation ratio achieved here is .s1 2 3{ , , }S S S 1= 3 2k−Δ × 1( 1)/3 = (ln(3 2 ))kk O −+ × More recently, another analysis of Algorithm GREEDYSC has been presentedsby =-=[71]-=- providing a tight ratio of . On the other hand, as stated in [67] (sees[31] for an informal proof), it is impossible to approximate MIN SET COVER within bettersthan , unless a highly unlikely complex... |

2 | Reductions that preserve approximability - Paschos - 2007 |

2 |
l Complexity
- Res
- 1979
(Show Context)
Citation Context ...rd Problems 12sThis algorithm has been independently analyzed by [46, 52]. Its version forsWEIGHTED MIN SET COVER where we ask for determining a set cover of minimum totalsweight has been analyzed by =-=[20]-=-. It is easy to see that Algorithm GREEDYSC runs inspolynomial time.sTheorem 3. The standard-approximation ratio of Algorithm GREEDYSC issbounded above by 1 l , where n+ Δ = {|max S S S∈ |}Δ .sProof. ... |

1 | Paschos. Completeness in approximation classes beyond apx. Theoret - Escoffier, Th |

1 |
A survey on the structure of approximation classes
- Escoffier, Paschos, et al.
- 2007
(Show Context)
Citation Context ...y of the existing onessicular conditions on the way to the optimal solutions, or the approximat c. are transformed from one problem to another. For more details, the interestedsreader can referred to =-=[5, 8, 9, 22, 32, 44, 66, 72]-=-.sIn general, given the two NPO problems Π = (I, Sol, m, goal) and Π′ = (I, Sol,sm, goal′), an approximation preserving reduction R from Πsto ′Πs(denoted R ′Π ≤ Π ) issa triple ( , , )f g csof polynom... |

1 | To χαµóγǫλo τoυ Turing (Turing’s smile). N´ǫα Σ´υνoρα - Papadimitriou - 2000 |

1 | A note on the improvement of the maximum independent set’s approximation ratio - Paschos - 1995 |

1 |
the traveling salesman problem
- Hypergraphs, Amsterdam
- 1973
(Show Context)
Citation Context ...oblems discussed in detail in this paper will be defined.sFor the other ones, the interested reader can referred to [35]. Also, for the notions andsdefinitions of the graph theory, one can refered to =-=[16]-=-.s3. APPROXIMATION CLASSESsAccording to the best approximation ratios known for them, NP-hard problemssare classified into approximability classes. These classes create a kind of hierarchy in thesclas... |

1 |
Reductions that preserve approximability” in
- Sci
- 2006
(Show Context)
Citation Context ...S S S 1= 3 2k−Δ × 1( 1)/3 = (ln(3 2 ))kk O −+ × More recently, another analysis of Algorithm GREEDYSC has been presentedsby [71] providing a tight ratio of . On the other hand, as stated in [67] (sees=-=[31]-=- for an informal proof), it is impossible to approximate MIN SET COVER within bettersthan , unless a highly unlikely complexity classess(log | |)O C (log | |) (log log | |)O C O C−sV. Th. Paschos / An... |

1 | On the differential approximation of min set cover”, Theoret - Bazgan, Monnot, et al. - 1992 |

1 |
Worst-case analysis of a new heuristic for
- Christofides
- 1976
(Show Context)
Citation Context ...proximation paradigm, the MIN TSP is in Exp-APXs(see Section 6). On the other hand, the MIN METRIC TSP is approximable within standardapproximation ratio 3/2 by the celebrated Christofides algorithm (=-=[19]-=-) while the mostsfamous relaxation of MIN METRIC TSP, that is when the edge-weights are either 1 or 2 issapproximable within 8/7 ([17]). Finally, the MIN EUCLIDEAN TSP is in PTAS ([1]).s4.5. Min color... |

1 |
A Guide to the Theory of
- Intractability
(Show Context)
Citation Context ...ew on thescompleteness result in the approximation classes.sTo be brief, only the problems discussed in detail in this paper will be defined.sFor the other ones, the interested reader can referred to =-=[35]-=-. Also, for the notions andsdefinitions of the graph theory, one can refered to [16].s3. APPROXIMATION CLASSESsAccording to the best approximation ratios known for them, NP-hard problemssare classifie... |

1 |
Generalized first-order spectra and polynomial-time reco Karp, (ed), Complexity of Computations, American Mathematics Society
- in
- 1974
(Show Context)
Citation Context ...VER. A lot of unsuccessful effort has been putsinto such improvement until now. All this long effort has only produced ratios of thesform:s• 2 (log log / log )n n−s([10, 54]);s• 2 (2ln ln / ln )n n−s(=-=[40]-=-);s• 2 (log log ( )/ log ( ))G G− Δ Δs([40]).sUnfortunately, a more recent result by [50] gives strong evidence that thissimprovement is likely to be impossible.sIn the differential approximation para... |

1 |
Clique is hard to approximate within ε−1n
- Håstad
- 1999
(Show Context)
Citation Context ...ET is approximable within ratios([39]) andswithins([29]);s2( /logO n n) ) (log / ( ) loglog )O n G nΔ MAX INDEPENDENT SET is inapproximable within better thansforsanys1(O nε − > 0ε , unless P = NP (=-=[42]-=-).s4.2. Min set coversGiven a ground setsand a collection 1sofssubsets of , MIN SET COVER consists of determining a minimum-size sub-collectionsthat covers C , i.e., such that .s1= { , , }nC c c… = { ... |

1 |
On the ratio of optimal integral and fractional covers”, Discrete
- Nostrand, NJ
- 1975
(Show Context)
Citation Context ...ade use of geometrical arguments in order tossolve algebraic problems.sIn recent times, a particular type of transformation, called reduction has beensintroduced by logicians in computability theory (=-=[51]-=-). In this case, a reduction from asproblem Πsto a problem ′Πssolution of the latter but, possibly more important in such context, it allowssshowing that if problem Πsis unsolvable (i.e., no algorithm... |

1 |
Το χαμόγελο του Turing (Turing's Smile). Νέα Σύνορα
- riou, H
- 2000
(Show Context)
Citation Context ...ems 31shave tosshortly explain the former of these techniques, that is the olderstechniqu ould contradict a commonly accepted complexitysthe sh first apprehend the complete spectrum of the possible” (=-=[60]-=-, translation fromsGreek by the author).sThere are three fundamental techniques to prove inapproximability results: thesGAP-reductions, the PCP theorem and the approximability preserving reductions. I... |

1 |
approximation and complexity
- “Optimization
(Show Context)
Citation Context ...earch, during the same years, the study of logicalsproperties of optimization problems has led to the syntactic characterization of ansimportant class of approximable problems, the class Max-SNP (see =-=[62]-=-) strongly basedsupon characterizatiosof N n terms of L-reductions and natural complete problems (e.g., MAX 3SAT, MAXs2SAT, MIN VERTEX COVER etc.) have been found. The relevance of such an approach is... |

1 |
A survey about how optimal solutions to some cov problems can be approximated
- Paschos, Th
- 1997
(Show Context)
Citation Context ... note that the strongest approximation resultssknown for this problem are the following:s MAX INDEPENDENT SET is asymptotically approximable (i.e., for largeswithin ratio , for every fixed constants(=-=[28, 64]-=-);s( )GΔ / ( )k GΔ k MAX INDEPENDENT SET is approximable within ratios([39]) andswithins([29]);s2( /logO n n) ) (log / ( ) loglog )O n G nΔ MAX INDEPENDENT SET is inapproximable within better than... |