M. Bellare, O. Goldreich, M. Sudan. "Free bits, PCPs and nonapproximability -- towards tight results". SIAM Journal on Computing, 27(3):804--915, 1998.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....from a practical solvability point of view. The same properties apply therefore to QKP as well, which is consequently much more difficult than the classical KP. In particular finding an approximate QKP solution of value not smaller than the optimum divided by n ffl is NP hard for any ffl [5]. In view of these results, one should expect that any upper bound which can be computed efficiently will be extremely bad for some instances. QKP was first studied by Gallo, Hammer and Simeone [13] who proposed exact algorithms where upper bounds are computed by using upper planes, which are ....

M. Bellare, O. Goldreich and M. Sudan (1995), "Free Bits, PCPs and NonApproximability --- Towards Tight Results", Proceedings of the 36th FOCS Conference, 422--431, IEEE Computer Society Press.

....currently is O(n= log 2 n) 8] Moreover, in a sequence of papers [9, 3, 2] it was established that there is some ffl 0 such that it is NP hard to approximate ff(G) within a ratio of n ffl . Subsequent works established hardness results for approximating ff(G) with ratios close to n 1=3 [4]. The question of whether there is some ffl such that a polynomial time algorithm approximates ff(G) within n 1 Gammaffl is open 1 . The conjecture suggested an affirmative answer to this open question. Until recently, there were virtually no techniques for establishing upper bounds on the ....

.... 0 can be made arbitrarily small. The gap location moves from n c to n 1 Gammaffl . The question of distinguishing between graphs of the above types (one with no large independent set, the other with a small chromatic number) is known to be NP hard under randomized reductions for ffi 1=5 [4, 12]. 4.2 Explicit constructions of Ramsey graphs Consider the graphs constructed in [17] with #(G) 3 and (G) n ffi , for some ffi 1. By selecting k log n in Corollary 2.15, we obtain a Ramsey graph with n O(logn) vertices and no clique or independent set of n vertices. The best explicit ....

M. Bellare, O. Goldreich, M. Sudan. "Free bits, PCPs and nonapproximability -- towards tight results". Proc. of 36th IEEE Symp. on Foundations of Computer Science, 422--431, 1995.

....variant of the minimum OBDD cover problem is NP complete. Finally, we investigate in Section 4 the complexity of approximating the minimum OBDD cover. Under the hypothesis NP #= P we can derive from the new theory of probabilistically checkable proof systems (Bellare, Goldreich and Sudan [1]) that the problem has neither approximation schemes nor polynomial time approximation algorithms whose results are only by a constant factor or a slowly increasing function larger # Supported in part by DFG grant We 1066 7 2. than the optimal result. With our results we know that we have to be ....

....bound c(A # ) c opt # (#(n 3 2n 2 n 1) 1) 1 O( 1 n ) # This rather technical result has concrete implications. If MBC (or EBM) can be approximated within a factor s # , then the chromatic number can be approximated within a factor O(n # 3 ) Bellare, Goldreich and Sudan [1] have proved that the chromatic number cannot be approximated within some factor of order n 1 7 , if NP #= P. Hence, we have proved the following result. Theorem 2: If NP #= P, every polynomial time approximation algorithm for EBM or MBC has a worst case performance ratio of # s 1 21 ) ....

M. Bellare, O. Goldreich, M. Sudan: "Free Bits, PCP, and Non-approximability -- Towards Tight Results". Electronic Colloquium on Computational Complexity TR 095 - 024, 1995. Also to appear in the Proc. of 31st Symp. on Foundations of Computer Science, Oct. 1995.

....further refined by Arora et al. 4, 3] one year later. Specifically, they proved that there exists an 0 such that no polynomial time algorithm can approximate the size of the maximum clique within a factor of n , unless P = NP . More recent developments along these lines can be found in [15, 16, 34]. In the light of these negative results, much effort has recently been directed towards devising efficient heuristics for the MCP, for which no formal guarantee of performance may be provided, but are anyway of interest in practical applications [19] An important generalization of the MCP which ....

M. Bellare, S. Goldwasser, and M. Sudan, "Free bits, PCPs and non-approximability---Towards tight results," in Proc. 36th Ann. Symp. Found. Comput. Sci., Milwaukee, WI, 1995, pp. 422-- 431.

....is an altogether inappropriate choice. In contrast to graph isomorphism, in fact, the problem of finding just the cardinality of the maximum clique in a graph is known to be NP complete and, according to recent theoretical results, so is the problem of approximating it within a certain tolerance [1, 6, 18]. 1 The experimental results presented in this paper, however, seem to contradict this claim. By using simple, local optimizing relaxation equations we are able to get results which compare favorably with those obtained using state of the art sophisticated deterministic annealing algorithms, ....

M. Bellare, S. Goldwasser, and M. Sudan, "Free bits, PCPs and non-approximability---Towards tight results," in Proc. 36th Ann. Symp. Found. Comput. Sci., Milwaukee, WI, 1995, pp. 422--431.

....exact equality with a segment of P 1 (x) where the choice of the segment depends on x and y. Compare games arise naturely in MIP(2,1) proof systems. Moreover, the compare structure has been useful in certain applications (hardness of approximating SETCOVER [24] hardness of approximating CLIQUE [7]) 4. Unique: Given x and y, the acceptance predicate V induces a (partial) permutation on P 1 (x) and P 2 (y) That is, for any answer P 1 (x) there is at most one answer P 2 (y) that causes V to accept, and vice versa. Parallel repetition for these games is easier to analyse, but they do not ....

M. Bellare, O. Goldreich, M. Sudan, "Free bits, PCPs and non-approximability -- towards tight results", in Proc. of 36th Annual Symposium on Foundations of Computer Science, 422--431, 1995.

....a practical solvability point of view. The same properties apply therefore to QKP as well, which is consequently much more difficult than the classical KP. In particular finding an approximate QKP solution of value not smaller than the optimum divided by n ffl is NP hard for any ffl 1 4 [5]. In view of these results, one should expect that any upper bound which can be computed efficiently will be extremely bad for some instances. QKP was first studied by Gallo, Hammer and Simeone [13] who proposed exact algorithms where upper bounds are computed by using upper planes, which are ....

M. Bellare, O. Goldreich and M. Sudan (1995), "Free Bits, PCPs and NonApproximability --- Towards Tight Results", Proceedings of the 36th FOCS Conference, 422--431, IEEE Computer Society Press.

.... algorithm if, for every x, T(x) opt(x) r(N ) If there exists a polynomialtime bounded r(N) approximation algorithm for P , then we say that P is approximable within r(N ) There are several optimization problems that have only poor approximation algorithms, such as MIN Graph Coloring [1, 3], MAX Constrained Hamiltonian Circuit [14] and MAX Bounded 0 1 Programming [11] Among others, MAXCLIQUE has received much attention recently because of its novel use of PCP. The latest result [8] says that this problem is not approximable within N 10ffl for any ffl 0 assuming NP6=coRP, ....

M. Bellare, M. Goldreich and M. Sudan, "Free bits PCPs and non-approximability --- towards tight results," FOCS95, pp.422--431, 1995.

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M. Bellare, O. Goldreich, M. Sudan. "Free bits, PCPs and nonapproximability -- towards tight results". SIAM Journal on Computing, 27(3):804--915, 1998.

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