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From: harold@sanjuan.UVic.CA (Todd Wareham)
Subject: Q: Computational Complexity of Generalized Mover's Problem
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I have several questions concerning the Generalized Mover's Problem 
(GMP), as formulated and shown PSPACE-hard by Reif (1987), and solved 
by the algorithms of Schwartz and Sharir (1983) and Canny (1987).

1) Given that for an instance of GMP, r is the number of degrees of
    freedom, n is the number of constraint polynomials describing the
    FP space of the system, and d and w are the maximum degree and
    coefficient magnitudes of the constraint polynomials, then Canny's
    algorithm (Canny (1987), p. 7) runs in time

	n^r log n (2d)^{O(r)} log^3 w + (2dr)^{O(r^2)} log^2 w

    The algorithm of Schwartz and Sharir (1983) runs in time 
    O(n^{2^{O(r)}}) (Reif (1987), p. 272). Though Canny's algorithm
    is singly-exponential in r w.r.t. n in comparison with that of
    Schwartz and Sharir, it is claimed that " ... unfortunately the
    constant of proportionality, which depends of the maximum degree of
    the ... constraints, is prohibitively large" (Sharir (1989), p. 13).
    However, the algorithm of Schwartz and Sharir uses the decomposition
    method of Collins (1975), which has running time 

	(2d)^{2^{2r' + 8}} n^{2^{r' + 6}}(log w)^3 a

    where r' is the number of variables in the polynomial (which, if I 
    remember correctly, is a lower bound on the degrees of freedom, 
    i.e., r' <= r) and a is the number of atomic formula (?) (Collins 
    (1975), p. 135).
   It seems to me that both Canny's and Schwartz and Sharir's algorithm
    have leading constants of the form r exponential in d. Is this
    correct, or have I misinterpreted something?

2) What is the magnitude of d, the maximum degree of the constraint
    polynomials, in instances of GMP encountered in practice?

3) What are the magnitudes of d in the instances of GMP constructed in
    the PSPACE-hardness reduction of Reif (1987) and the version of this
    reduction given in Reif (1979)?

Any help on these questions would be greatly appreciated, as well as
any references on subsequent algorithms/complexity-hardness for GMP. 
Thanks in advance,

- Todd

			REFERENCES

Canny, J. F. (1987) THE COMPLEXITY OF ROBOT MOTION PLANNING. The MIT
     Press; Cambridge, MA.

Collins, G. E. (1975) Quantifier elimination for real closed fields
     by cylindrical algebraic decomposition. In G. Goos and and
     J. Hartmanis (eds.) 2nd GI CONFERENCE ON AUTOMATA THEORY AND
     FORMAL LANGUAGES. Lecture Notes in Computer Science no. 33.
     Springer-Verlag; Berlin. 134-183.

Reif, J. H. (1987) Complexity of the Generalized Mover's Problem. In
     J. T. Schwartz, M. Sharir, and J. Hopcroft (eds.) PLANNING, 
     GEOMETRY, AND COMPLEXITY OF ROBOT MOTION. Ablex Publishing 
     Corporation; Norwood, NJ. 267-281. An earlier version appeared in 
     1979 in FOCS 20, 421-427.

Schwartz, J. T., Sharir, M. and Hopcroft, J. (eds.) (1987) PLANNING, 
     GEOMETRY, AND COMPLEXITY OF ROBOT MOTION. Ablex Publishing 
     Corporation; Norwood, NJ.

Schwartz, J. T. and Sharir, M. (1983) On the Piano Mover's Problem II:
     Computing Topological Properties of Real Algebraic Manifolds.
     ADVANCES IN APPLIED MATHEMATICS, 4, 298-351.

Sharir, M. (1989) Algorithmic Motion Planning in Robotics. COMPUTER,
     March, 9-20.

Todd Wareham	harold@csr.uvic.ca	|"Success in science depends not
Department of Computer Science		| only on rational argument but
University of Victoria, P. O. Box 3055	| on a mixture of subterfuge,
Victoria, BC, Canada	V8W 3P6		| rhetoric, and propaganda"
					|    - Feyerabend

