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63
A Threshold of ln n for Approximating Set Cover
 JOURNAL OF THE ACM
, 1998
"... Given a collection F of subsets of S = f1; : : : ; ng, set cover is the problem of selecting as few as possible subsets from F such that their union covers S, and max kcover is the problem of selecting k subsets from F such that their union has maximum cardinality. Both these problems are NPhar ..."
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Cited by 619 (6 self)
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Given a collection F of subsets of S = f1; : : : ; ng, set cover is the problem of selecting as few as possible subsets from F such that their union covers S, and max kcover is the problem of selecting k subsets from F such that their union has maximum cardinality. Both these problems are NPhard. We prove that (1 \Gamma o(1)) ln n is a threshold below which set cover cannot be approximated efficiently, unless NP has slightly superpolynomial time algorithms. This closes the gap (up to low order terms) between the ratio of approximation achievable by the greedy algorithm (which is (1 \Gamma o(1)) ln n), and previous results of Lund and Yannakakis, that showed hardness of approximation within a ratio of (log 2 n)=2 ' 0:72 lnn. For max kcover we show an approximation threshold of (1 \Gamma 1=e) (up to low order terms), under the assumption that P != NP .
Property Testing and its connection to Learning and Approximation
"... We study the question of determining whether an unknown function has a particular property or is fflfar from any function with that property. A property testing algorithm is given a sample of the value of the function on instances drawn according to some distribution, and possibly may query the fun ..."
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Cited by 414 (57 self)
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We study the question of determining whether an unknown function has a particular property or is fflfar from any function with that property. A property testing algorithm is given a sample of the value of the function on instances drawn according to some distribution, and possibly may query the function on instances of its choice. First, we establish some connections between property testing and problems in learning theory. Next, we focus on testing graph properties, and devise algorithms to test whether a graph has properties such as being kcolorable or having a aeclique (clique of density ae w.r.t the vertex set). Our graph property testing algorithms are probabilistic and make assertions which are correct with high probability, utilizing only poly(1=ffl) edgequeries into the graph, where ffl is the distance parameter. Moreover, the property testing algorithms can be used to efficiently (i.e., in time linear in the number of vertices) construct partitions of the graph which corre...
Probabilistic checking of proofs: a new characterization of NP
 Journal of the ACM
, 1998
"... Abstract. We give a new characterization of NP: the class NP contains exactly those languages L for which membership proofs (a proof that an input x is in L) can be verified probabilistically in polynomial time using logarithmic number of random bits and by reading sublogarithmic number of bits from ..."
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Cited by 357 (28 self)
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Abstract. We give a new characterization of NP: the class NP contains exactly those languages L for which membership proofs (a proof that an input x is in L) can be verified probabilistically in polynomial time using logarithmic number of random bits and by reading sublogarithmic number of bits from the proof. We discuss implications of this characterization; specifically, we show that approximating Clique and Independent Set, even in a very weak sense, is NPhard.
A Parallel Repetition Theorem
 SIAM Journal on Computing
, 1998
"... We show that a parallel repetition of any twoprover oneround proof system (MIP(2, 1)) decreases the probability of error at an exponential rate. No constructive bound was previously known. The constant in the exponent (in our analysis) depends only on the original probability of error and on the t ..."
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Cited by 311 (9 self)
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We show that a parallel repetition of any twoprover oneround proof system (MIP(2, 1)) decreases the probability of error at an exponential rate. No constructive bound was previously known. The constant in the exponent (in our analysis) depends only on the original probability of error and on the total number of possible answers of the two provers. The dependency on the total number of possible answers is logarithmic, which was recently proved to be almost the best possible [U. Feige and O. Verbitsky, Proc. 11th Annual IEEE Conference on Computational Complexity, IEEE Computer Society Press, Los Alamitos, CA, 1996, pp. 7076].
A SubConstant ErrorProbability LowDegree Test, and a SubConstant ErrorProbability PCP Characterization of NP
 IN PROC. 29TH ACM SYMP. ON THEORY OF COMPUTING, 475484. EL PASO
, 1997
"... We introduce a new lowdegreetest, one that uses the restriction of lowdegree polynomials to planes (i.e., affine subspaces of dimension 2), rather than the restriction to lines (i.e., affine subspaces of dimension 1). We prove the new test to be of a very small errorprobability (in particular, ..."
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Cited by 274 (19 self)
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We introduce a new lowdegreetest, one that uses the restriction of lowdegree polynomials to planes (i.e., affine subspaces of dimension 2), rather than the restriction to lines (i.e., affine subspaces of dimension 1). We prove the new test to be of a very small errorprobability (in particular, much smaller than constant). The new test enables us to prove a lowerror characterization of NP in terms of PCP. Specifically, our theorem states that, for any given ffl ? 0, membership in any NP language can be verified with O(1) accesses, each reading logarithmic number of bits, and such that the errorprobability is 2 \Gamma log 1\Gammaffl n . Our results are in fact stronger, as stated below. One application of the new characterization of NP is that approximating SETCOVER to within a logarithmic factors is NPhard. Previous analysis for lowdegreetests, as well as previous characterizations of NP in terms of PCP, have managed to achieve, with constant number of accesses, error...
Truth revelation in approximately efficient combinatorial auctions
 Journal of the ACM
, 2002
"... Abstract. Some important classical mechanisms considered in Microeconomics and Game Theory require the solution of a difficult optimization problem. This is true of mechanisms for combinatorial auctions, which have in recent years assumed practical importance, and in particular of the gold standard ..."
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Cited by 182 (1 self)
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Abstract. Some important classical mechanisms considered in Microeconomics and Game Theory require the solution of a difficult optimization problem. This is true of mechanisms for combinatorial auctions, which have in recent years assumed practical importance, and in particular of the gold standard for combinatorial auctions, the Generalized Vickrey Auction (GVA). Traditional analysis of these mechanisms—in particular, their truth revelation properties—assumes that the optimization problems are solved precisely. In reality, these optimization problems can usually be solved only in an approximate fashion. We investigate the impact on such mechanisms of replacing exact solutions by approximate ones. Specifically, we look at a particular greedy optimization method. We show that the GVA payment scheme does not provide for a truth revealing mechanism. We introduce another scheme that does guarantee truthfulness for a restricted class of players. We demonstrate the latter property by identifying natural properties for combinatorial auctions and showing that, for our restricted class of players, they imply that truthful strategies are dominant. Those properties have applicability beyond the specific auction studied.
Approximate graph coloring by semidefinite programming
 Proc. 35 th IEEE FOCS, IEEE
, 1994
"... a coloring is called the chromatic number of�, and is usually denoted by��.Determining the chromatic number of a graph is known to be NPhard (cf. [19]). Besides its theoretical significance as a canonical NPhard problem, graph coloring arises naturally in a variety of applications such as register ..."
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Cited by 178 (6 self)
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a coloring is called the chromatic number of�, and is usually denoted by��.Determining the chromatic number of a graph is known to be NPhard (cf. [19]). Besides its theoretical significance as a canonical NPhard problem, graph coloring arises naturally in a variety of applications such as register allocation [11, 12, 13] is the maximum degree of any vertex. Beand timetable/examination scheduling [8, 40]. In many We consider the problem of coloring�colorable graphs with the fewest possible colors. We give a randomized polynomial time algorithm which colors a 3colorable graph on vertices with� � ���� colors where sides giving the best known approximation ratio in terms of, this marks the first nontrivial approximation result as a function of the maximum degree. This result can be generalized to�colorable graphs to obtain a coloring using�� � ��� � � � �colors. Our results are inspired by the recent work of Goemans and Williamson who used an algorithm for semidefinite optimization problems, which generalize linear programs, to obtain improved approximations for the MAX CUT and MAX 2SAT problems. An intriguing outcome of our work is a duality relationship established between the value of the optimum solution to our semidefinite program and the Lovász�function. We show lower bounds on the gap between the optimum solution of our semidefinite program and the actual chromatic number; by duality this also demonstrates interesting new facts about the�function. 1
Zero Knowledge and the Chromatic Number
 Journal of Computer and System Sciences
, 1996
"... We present a new technique, inspired by zeroknowledge proof systems, for proving lower bounds on approximating the chromatic number of a graph. To illustrate this technique we present simple reductions from max3coloring and max3sat, showing that it is hard to approximate the chromatic number wi ..."
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Cited by 175 (8 self)
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We present a new technique, inspired by zeroknowledge proof systems, for proving lower bounds on approximating the chromatic number of a graph. To illustrate this technique we present simple reductions from max3coloring and max3sat, showing that it is hard to approximate the chromatic number within \Omega\Gamma N ffi ), for some ffi ? 0. We then apply our technique in conjunction with the probabilistically checkable proofs of Hastad, and show that it is hard to approximate the chromatic number to within\Omega\Gamma N 1\Gammaffl ) for any ffl ? 0, assuming NP 6` ZPP. Here, ZPP denotes the class of languages decidable by a random expected polynomialtime algorithm that makes no errors. Our result matches (up to low order terms) the known gap for approximating the size of the largest independent set. Previous O(N ffi ) gaps for approximating the chromatic number (such as those by Lund and Yannakakis, and by Furer) did not match the gap for independent set, and do not extend...
Polynomial Time Approximation Schemes for Dense Instances of NPHard Problems
, 1995
"... We present a unified framework for designing polynomial time approximation schemes (PTASs) for "dense" instances of many NPhard optimization problems, including maximum cut, graph bisection, graph separation, minimum kway cut with and without specified terminals, and maximum 3satisfiability. By d ..."
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Cited by 172 (27 self)
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We present a unified framework for designing polynomial time approximation schemes (PTASs) for "dense" instances of many NPhard optimization problems, including maximum cut, graph bisection, graph separation, minimum kway cut with and without specified terminals, and maximum 3satisfiability. By dense graphs we mean graphs with minimum degree Ω(n), although our algorithms solve most of these problems so long as the average degree is Ω(n). Denseness for nongraph problems is defined similarly. The unified framework begins with the idea of exhaustive sampling: picking a small random set of vertices, guessing where they go on the optimum solution, and then using their placement to determine the placement of everything else. The approach then develops into a PTAS for approximating certain smooth integer programs where the objective function and the constraints are "dense" polynomials of constant degree.
Which Problems Have Strongly Exponential Complexity?
 Journal of Computer and System Sciences
, 1998
"... For several NPcomplete problems, there have been a progression of better but still exponential algorithms. In this paper, we address the relative likelihood of subexponential algorithms for these problems. We introduce a generalized reduction which we call SubExponential Reduction Family (SERF) t ..."
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Cited by 120 (5 self)
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For several NPcomplete problems, there have been a progression of better but still exponential algorithms. In this paper, we address the relative likelihood of subexponential algorithms for these problems. We introduce a generalized reduction which we call SubExponential Reduction Family (SERF) that preserves subexponential complexity. We show that CircuitSAT is SERFcomplete for all NPsearch problems, and that for any fixed k, kSAT, kColorability, kSet Cover, Independent Set, Clique, Vertex Cover, are SERFcomplete for the class SNP of search problems expressible by second order existential formulas whose first order part is universal. In particular, subexponential complexity for any one of the above problems implies the same for all others. We also look at the issue of proving strongly exponential lower bounds for AC 0 ; that is, bounds of the form 2 \Omega\Gamma n) . This problem is even open for depth3 circuits. In fact, such a bound for depth3 circuits with even l...