D. Ranjan and P. Rohatgi. On randomized reductions to sparse sets. In Structure in Complexity Theory Conference, pages 239-242, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....L selective sets, we prove the following theorem. Theorem 9. If GAP is logspace O(1) mc, then GAP 2 L. An important class of languages with low information content is the class of sparse sets. Reducibility to sparse sets has been studied for a long time and the following results are known (see [5, 49] for related and even stronger results) Fact 10 ( 47] If SAT is reducible to a sparse set, then SAT 2 P. Fact 11 ( 61] If CVP is reducible to a sparse set, then CVP 2 L. We make progress on improving the last result by showing the following theorem. Theorem 12. Let C be any of ....

D. Ranjan and P. Rohatgi. On randomized reductions to sparse sets. In Structure in Complexity Theory Conference, pages 239-242, 1992.

....The conclusions of (1.1) and (1.2) guarantee that any polynomial time approximation algorithm fails to output a number within n and n of #(G) for infinitely many graphs G. Is it possible to prove that this approximation failure must occur more often Using the results of Ranjan and Rohatgi [22] and Mahaney [21] about the reducibility of NP to sparse sets, we extend (1.1) and (1.2) to prove the following: RP, then for all # 0, any polynomial time approximation algorithm for MaxClique fails to approximate within n on a nonsparse set of instances. on a nonsparse set of ....

....nonsparse lower bounds on the frequency of approximation failure for polynomial time approximation algorithms. For this we use the following results concerning the reducibility of NP to sparse sets. Theorem 6.5. Mahaney [21] If P NP, then NP m(SPARSE) Theorem 6.6. Ranjan and Rohatgi [22]) If NP RP, then NP co rm (SPARSE) Theorem 6.1 can be extended using these two results as follows. The assumption for the first part has been strengthened from NP ZPP to NP RP but the assumption for the second item remains the same. Theorem 6.7. 1. If NP RP, then for all # 0, ....

D. Ranjan and P. Rohatgi. On randomized reductions to sparse sets. In Proceedings of the Seventh Annual Structure in Complexity Theory Conference, pages 239--242, 1992.

....to the second question, one might intuitively feel that the structure imposed on a set by the fact that it reduces to a sparse set makes it plausible that we can indeed find a simple sparse set that can masquerade as the original sparse set. These two intuitions are in many ways certified by the current literature, and by the results of this paper. The rest of this section summarizes the results of this paper and compares them with previous work. With regard to Question 1: ffl We show that if any NP complete set conjunctively reduces to a sparse set, 2 then P = NP. This result, which has ....

....The rest of this section summarizes the results of this paper and compares them with previous work. With regard to Question 1: ffl We show that if any NP complete set conjunctively reduces to a sparse set, 2 then P = NP. This result, which has been obtained independently by Ranjan and Rohatgi [RR] extends Mahaney s Theorem [Mah82] and is incomparable with the strongest previously known result, which is due to Ogiwara and Watanabe: if any NP complete set bounded truth table reduces to a sparse set then P = NP [OW91] We obtain the above result as a corollary to a more general result we ....

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D. Ranjan and P. Rohatgi. On randomized reductions to sparse sets. In Proceedings of the 7th Structure in Complexity Theory Conference. IEEE Computer Society Press. To appear.

....Watanabe [OW91] when they proved, using a new left set technique, that if NP has sparse hard sets under polynomial time bounded truth table reductions then P = NP. This has been, more recently, followed up by similar results for conjunctive truth table polynomial time reductions [AHH 92, RR92] and in [AHH 92] even for more flexible truth table reductions (e.g. bounded conjunctions on the 1 truth table closure of the conjunctive closure of sparse sets) These results demonstrate the efficacy of the left set technique introduced in [OW91] in fact, the older result of Mahaney has ....

....sets to sparse sets via various types of polynomial time truthtable reductions that yield a collapse of the polynomial hierarchy to P. In Section 4 we consider randomized reductions to sparse sets, and show that if NP R p b (R co rp m (R p c (SPARSE) then RP = NP. D. Ranjan and P. Rohatgi [RR92] have independently shown that if NP R co rp m (SPARSE) then RP = NP. Relatedly, we show in Section 5 that if some solution of the promise problem (1SAT; SAT) is in R p b (R p c (SPARSE) then there is a solution of (1SAT; SAT) in P. We also show that the conclusion RP = NP can be derived ....

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D. Ranjan and P. Rohatgi. Randomized reductions to sparse sets. In Proceedings 7th Structure in Complexity Theory Conference. To appear.

....NP cannot have a sparse hard set under polynomial time many one reductions, and that nearly a decade later, Ogihara and Watanabe [OW91] introduced a new technique to extend Mahaney s result to the case of the so called bounded truth table reductions. Extending their technique, Ranjan and Rohatgi [RR92] showed that NP cannot have a sparse hard set under polynomial time computable co rp reductions (defined below) unless NP = RP. Definition 2.7 (co RP Reduction) A language A is said to be co rp reducible to B if there is a function f ( Delta; Delta) and some fixed e 0 such that for all x, x ....

D. Ranjan and P. Rohatgi. On randomized reductions to sparse sets. In Proc. 7th Annual IEEE Conference on Structure in Complexity Theory, pages 239--242, 1992.

.... the polynomial hierarchy collapses to Sigma p 2 [8] The more recent result of Ogiwara and Watanabe [14] that the existence of sparse hard sets for NP under polynomial time bounded truth table reductions implies P = NP, has been followed up by analogous results for more general reductions in [1, 15, 2]. Recently, Book and Lutz [7] have considered reductions to sets whose characteristic sequences are of very high space bounded Kolmogorov complexity (they call the class of such sets HIGH) Book and Lutz obtained the surprising result that every set in ESPACE that is (polynomial time) bounded ....

D. Ranjan and P. Rohatgi. Randomized reductions to sparse sets. Proceedings of the 7th Structure in Complexity Theory Conference, IEEE Computer Society Press, 239-242, 1992.

....later, Ogiwara and Watanabe [OW91] succeeded in extending Mahaney s theorem to sparse sets complete under bounded truthtable reductions. Since then, there has been a large body of research to extend the OgiwaraWatanabe theorem for various weaker reducibilities and for other complexity classes [RR92, OL91, AKM92, HL91] (see [HOW92, You92a, You92b] for a survey) In contrast, there is another equally prominent conjecture by Hartmanis about sparse sets on which, until recently, not much progress had been made. In 1978, while studying the isomorphism problem for P complete and NL complete problems, Hartmanis ....

....(with one sided error) Combining this result with the Valiant Vazirani reduction [VV85] from satisfiability to unique satisfiability, we show that if NP has a sparse hard set under randomized reductions computable in polynomial time, then NP = RP. This answers an open question raised in [RR92]. Results about randomized reductions are described in Section 4. To obtain sharp space bounds for the truth table reductions, and for the case of truthtable reductions to quasipolynomially sparse sets (that is, sets with density 2 (log n) O(1) we need to compute in parallel the rank of an m ....

[Article contains additional citation context not shown here]

D. Ranjan and P. Rohatgi. On randomized reductions to sparse sets. In Proc. 7th Structures, pages 239--242, 1992.

....on the existence of sparse hard sets for NP under randomized many one reductions with two sided error: There is no sparse hard set for NP under polynomial time randomized two sided error many one reductions with confidence at least inversely polynomial unless NP RP. Earlier, Ranjan and Rohatgi [22] had shown this theorem for reductions with one sided error on the non membership side. Arvind, Kobler and Mundhenk [2] improved upon that, but the question whether the statement was true of reductions with one sided error on the membership side, and of reductions with two sided error remained ....

D. Ranjan and P. Rohatgi. On randomized reductions to sparse sets. In Proceedings of the 7th IEEE Structure in Complexity Theory Conference, pages 239--242. IEEE, New York, 1992.

....parallel for each fixed k truth table on any given level of the assignment tree. The theorem of Ogihara and Watanabe is proved. 4. 1 Extensions The breakthrough by Ogihara and Watanabe has provoked a flurry of results about sparse hard complete set problems [HL94, AHH 93, AKM92b, AKM92a, AA95, RR92, OL93] see [HOW92] for a survey) Below we state the best known results on polynomial time sparse hard sets for NP. Theorem 4.2 1. AKM92b] NP is included in the P btt reducibility closure of the languages that are P ctt reducible to sparse sets if and only if P = NP. 2. AKM92a] ....

D. Ranjan and P. Rohatgi. On randomized reductions to sparse sets. In Proc. 7th Conf. on Structure in Complex. Theory, pages 239--242. IEEE, June 1992. 1. Sparse Sets versus Complexity Classes 30

....parallel for each fixed k truth table on any given level of the assignment tree. The theorem of Ogihara and Watanabe is proved. 4. 1 Extensions The breakthrough by Ogihara and Watanabe has provoked a flurry of results about sparse hard complete set problems [HL94, AHH 93, AKM92b, AKM92a, AA95, RR92, OL93] see [HOW92] for a survey) Below we state the best known results on polynomial time sparse hard sets for NP. Theorem 4.2 1. AKM92b] NP is included in the P btt reducibility closure of the languages that are P ctt reducible to sparse sets if and only if P = NP. 2. AKM92a] ....

D. Ranjan and P. Rohatgi. On randomized reductions to sparse sets. In Proc. 7th Conf. on Structure in Complex. Theory, pages 239--242. IEEE, June 1992.

....the notion of left sets, which are NP sets with a special self reducibility structure. The left set method turned out to be a well suited tool to prove collapse results concerning sparse sets. Using this method similar results were obtained for polynomial time conjunctive reductions [AHH 93, RR92] Also the proof in [AHH 93] showing that no bounded Turing hard set for NP conjunctively reduces to a sparse set unless P = NP uses the left set technique. Furthermore, it makes use of the fact that the sets in R p bT (R p c (SPARSE) are monotonously reducible to a sparse set. The reason ....

....trade offs between the density of the set B, the number k(n) of queries in the Hausdorff reduction, and the randomized time complexity of A. As a special case we obtain that no bounded Turing hard set for NP co rp many one reduces to a sparse set unless RP = NP. This extends the result in [RR92] that an NP complete set is not co rp m reducible to a sparse set unless RP = NP. In Section 6 we consider the problem of reducing some solution of the promise problem (1SAT; SAT) to sparse sets. In particular, we show that the conclusion RP = NP can be derived from the apparently weaker ....

[Article contains additional citation context not shown here]

D. Ranjan and P. Rohatgi. Randomized reductions to sparse sets. Proceedings of the 7th Structure in Complexity Theory Conference, IEEE Computer Society Press, 239-242, 1992.

....to the second question, one might intuitively feel that the structure imposed on a set by the fact that it reduces to a sparse set makes it plausible that we can indeed find a simple sparse set that can masquerade as the original sparse set. These two intuitions are in many ways certified by the current literature, and by the results of this paper. The rest of this section summarizes the results of this paper and compares them with previous work. With regard to Question 1: ffl We show that if any NP complete set conjunctively reduces to a sparse set, 2 then P = NP. This result, which has ....

....The rest of this section summarizes the results of this paper and compares them with previous work. With regard to Question 1: ffl We show that if any NP complete set conjunctively reduces to a sparse set, 2 then P = NP. This result, which has been obtained independently by Ranjan and Rohatgi [RR] extends Mahaney s Theorem [Mah82] and is incomparable with the strongest previously known result, which is due to Ogiwara and Watanabe: if any NP complete set bounded truth table reduces to a sparse set, then P = NP [OW91] In fact, a more general result holds regarding the impossibility (if P ....

[Article contains additional citation context not shown here]

D. Ranjan and P. Rohatgi. On randomized reductions to sparse sets. In Proceedings of the 7th Structure in Complexity Theory Conference. IEEE Computer Society Press. To appear.

....means that there is no satisfying assignment either through node n nor to the left of n; so again node n can be safely eliminated. 18 Added June, 1992: This extension of Yap s 83 result has been obtained independently in several very recent papers on reductions to sparse sets, AHHKLMOSST 92] [RR 92]) In either case, if the the elements queried by the truth table at node n are a subset of the elements queried to the left of node n; then node n can be safely eliminated. Thus we may assume that as we do the left to right search, we accumulate at least one new query y i at each node that is ....

D. Ranjan and P. Rohatgi, "Randomized reductions to sparse sets," Proc Structure in Complexity Theory Conference 7 (1992), 239-42.

....on the existence of sparse hard sets for NP under randomized many one reductions with two sided error: There is no sparse hard set for NP under polynomial time randomized two sided error many one reductions with con dence at least inversely polynomial unless NP RP. Earlier, Ranjan and Rohatgi [29] had shown this theorem for reductions with one sided error on the non membership side. Arvind, K#bler and Mundhenk [2] improved upon that, but the question whether the statement was true of reductions with one sided error on the membership side, and of reductions with two sided error remained ....

D. Ranjan and P. Rohatgi. On randomized reductions to sparse sets. In Proceedings of the 7th IEEE Structure in Complexity Theory Conference, pages 239242, New York, 1992. IEEE.

....later, Ogiwara and Watanabe [OW91] succeeded in extending Mahaney s theorem to sparse sets complete under bounded truth table reductions. Since then, there has been a large body of research to extend the OgiwaraWatanabe theorem for various weaker reducibilities and for other complexity classes [RR92, OL91, AKM92] (see [HOW92, You92a, You92b] for a survey) Department of Computer Science, State University of New York at Buffalo, Buffalo, NY 14260. Research supported in part by NSF grants CCR 9057486 and CCR9319093, and an Alfred P. Sloan Fellowship. Email: cai cs.buffalo.edu Computer Science ....

....unique satisfiability, we show that if NP has a sparse hard set under randomized reductions computable in polynomial time, then NP = RP. These are the first known results about the existence of sparse hard sets under randomized reductions with two sided error and answer open questions raised in [RR92, AKM92]. Results about randomized reductions are described in Section 4. To obtain sharp space bounds for the truth table reductions, and for the case of truth table reductions to quasipolynomially sparse sets (that is, sets with density 2 (log n) O(1) we need to compute in parallel the rank of an ....

[Article contains additional citation context not shown here]

D. Ranjan and P. Rohatgi. On randomized reductions to sparse sets. In Proc. 7th Structures, pages 239--242, 1992.

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D. Ranjan and P. Rohatgi, On randomized reductions to sparse sets, in Proceedings of the 7th Structure in Complexity Theory Conference, IEEE Computer Society Press, June 1992, 239-242.

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