Let us estimate the total error in the approximations, i.e., the Ž.ni ’s summed over all halving steps. The first phase of the algorithms begins
3 The relevant observations can be summarized in the phrases ‘‘an -approximation for a ␦-approximation is an Ž. q ␦ -approximation’’ and ‘‘an -approximation for a disjoint union of two sets of equal cardinality can be obtained as the union of -approximations of equal cardinality for both sets.’’ OPTIMIZATION IN FIXED DIMENSION 585
FIG. 1. Schematic illustration of the algorithm for Theorem 2.3, for d s 1.
with sets of size 2k Ž. for k to be determined . In the first round one performs Ž.d q 1 halving steps with sets of this size, contributing Žd q k 3212 k2 1.Ž2.ŽsOdr kr2yr. . Then the size of the current sets is doubled by the extra merging step, and the second round contributes an error of at Ž.Žkq1.Ž. most d q 1 2 , etc. We may assume ni G 5 for all i, so by 2 the errors of approximation in the successive rounds form a geometrically decreasing sequence, and the total error in the first phase is thus 3 212 k2 k 32 OdŽ r kr2yr. . Choosing a value of k with 2 s Cd r logŽ.rd for large enough absolute constant C, we get that the first phase contributes an error at most 1r2r. 586 CHAZELLE AND MATOUSEKˇ
We now consider the second phase. The terminating condition gives that the error at the last step is at most 1r8r. Each previous step in the second phase halved the size of the current set, and usingŽ. 2 , we get that the errors in the second phase form a geometrically increasing sequence with quotient at least 4r3. Thus, the total error in the second phase is at most Ž.Ž.1r8rr1y3r4s1r2r, so the final set is indeed a Ž. 1rr -approximation for Ž.X, R . Let us analyze the running time. In the first halving step of the k algorithm, we perform the computation as in Lemma 2.1 on <THEOREM 2.4. LetŽ. X, R be a set system with a subsystem oracle of dimension d, for some fixed d, and suppose that the oracle can be imple- Ž. Ž2q␦. mented in NC. Then one can compute a 1rr -aproximation of size Od, ␦ r and a Ž.1 r -net of size O Ž r1q␦.Ž for X, R .in parallel, with at most nr c r Xd, ␦ processors and Ž.log nc parallel time for any fixed ␦ ) 0 Žthe constants c, cX depending on d, ␦ and on the implementation of the oracle.. In particular, the computation can be performed in a polylogarithmic parallel time with Od, rŽ. n processors for e¨ery fixed r. Proof sketch. By inspecting the proof of Theorem 2.3, we find that there is only one nontrivial part in parallelizing the algorithm, namely the application of the halving lemmaŽ at least if we do not care about the specific power of the logarithm in the parallel running time. . Here we can use the results of Berger and Rompelwx BR91Ž or the very similar results of Motwani et al.wx MNN89. . OPTIMIZATION IN FIXED DIMENSION 587
A particular case of their results gives the following: given a set system Ž.A,Sas in Lemma 2.1 and a fixed ␥ ) 0, one can compute a coloring Ä4 Ž. Ž.1r2q␥ :Aªy1, q1 with < S
Remark. The most important special case of the above theorem is for a fixed r. In such a situation, we can produce an NC algorithm with a linear number of processors without the machinery ofwx BR91, MNN89 .
If the size of the sets entering each halving step were only polylogarith- mic in n, we could simply implement the halving lemma sequentially. Note that the version of the algorithm described in the proof of Theorem 2.3 does not have this property: although the size of the sets is bounded by a constant both in the first and last halving steps, it increases geometrically during the first phase and decreases geometrically during the second phase, reaching a small positive power of n in the middle. We can modify the first phase, letting the sizes grow more slowly, but in such a way that the errors of the halving steps still form a convergent series. Namely, we may use the following rule: the extra merging step Ž.increasing the size is inserted at the end of the ith round only if Ž. Ž . Ž. ?5log2 i@ 5 fi)fiy1 , where fis?@5 log 2 i . Thus the sizes grow as 2 f i . The total error during the first phase is then at mostŽŽ. with t given by Ž..Ž.Ž.Ž. Ž.Ž5.Ž5. 1n123qnqnqиии F n 1q 2 n 1q 3 n 1q иии s Ž Ž ..Ž 22.ŽŽ.. On111q1r2q1r3qиии s O n . Thus the total error during the first phase is still proportional to the error in the first halving step, only the constant becomes larger. In this way, the sets entering the halving 588 CHAZELLE AND MATOUSEKˇ step only reach a polylogarithmic size throughout the algorithm, and we get an NC algorithm for any fixed r.
3. DERANDOMIZING CLARKSON’S ALGORITHM
To simplify our discussion of linear programming, we suppose that Ž.i the vector c determining the objective function is vertical, that 4 is, parallel to the x d-axis, and we look for the lexicographically smallest optimal vertex, Ž.ii we look only for nonnegative solutions Ž to avoid dealing with unbounded solutions and points at infinity. , Ž.iii the problem has an admissible nonnegative solution. If one wishes to relax these assumptions, note thatŽ. i is without loss of generalityŽ we can rotate the coordinate system so that the optimization direction becomes vertical. ; for unbounded solutions, we can formally add ‘‘constraints at infinity,’’ which will play the role of nonnegativity inŽ. ii , and also nonadmissible problems can be handled easily. See alsow Cla88, Sei91x for a similar discussion. We begin by introducing an abstract framework for optimization prob- lems similar to linear programming, due to Sharir and WelzlŽwx SW92 , see alsowx MSW92. . Clarkson’s randomized algorithm for linear programming can be formulated and analyzed in this framework, and with one extra axiom, this will also be the case for the derandomized version. Throughout, we will illustrate the abstract concepts on the specific example of linear programming. In the abstract framework, an optimization problem will be a pair H Ž.H,w, where H is a finite set, and w:2 ªW is a function with values in a linearly ordered set Ž.W , F . The elements of H will be called the constraints, and for a subset G : H, wGŽ.will be called the ¨alue of G. In the linear programming setting, H will be the given set of halfspaces in R d, and W will be the set R d with the lexicographic ordering. For G:H,wGŽ.will be the optimal nonnegative solution for G Ža point in R d .. In general, we define an LP-type problem to be a pair Ž.H, w as above, satisfying the following two axioms: Axiom 1.Ž. Monotonicity For any F : G : H, wF Ž.Ž.FwG. Axiom 2.Ž. Locality For any F : G : H with wF Ž.Ž.swG and any hgH,wGŽ jÄ4h.Ž.)wG implies that wF ŽjÄ4h.Ž.)wF.
4 Considering x d the most significant coordinate and x1 the least significant coordinate. OPTIMIZATION IN FIXED DIMENSION 589
It is easy to check that both axioms hold for linear programming with the above assumptions. Notice that the second axiom follows from unique- ness of the optimal solutionŽ as a point in R d . and does not require any general position assumptions concerning the constraints. For an LP-type problem, a basis B is a set of constraints with wBŽ X . - wBŽ.for all proper subsets BXof B.A basis for a subset G of H is a basis Bwith B : G and wBŽ.swG Ž.. So a basis of G is a minimal subset of G with the same value as G. We say that a constraint h g H ¨iolates a basis B,if wBŽ jÄ4h.Ž.)wB. The maximum cardinality of any basis is called the combinatorial dimen- sion ofŽ. H, w , and it is denoted by dim Ž.H, w . In the sequel, D will stand for the combinatorial dimension of the considered LP-type problem. Note that the combinatorial dimension is a monotone function: if F : G then dimŽ.F, w F dim Ž.G, w . Linear programming with d variables has combi- natorial dimension exactly d Žsince we exclude infeasible linear programs, where a minimum set of constraints witnessing infeasibility may have d q 1 rather than at most d elements. . In order to formulate an algorithm for solving an LP-type problem, we also need some computational assumptions. Clarkson’s randomized algo- rithm requires
Computational assumption 1.Ž. Violation test Given a basis B and a constraint h g H, decide whether h violates B Žand return an error message if the input set B is not a basis. .
For linear programming, the violation test as described can be per- formed in OdŽ 3. time: use Gaussian elimination to find the vertex defined by B; then the violation test with h needs OdŽ.additional time. One can, however, organize the algorithm in such a way that the vertex is available together with the basis at no extra cost, then a violation test needs OdŽ. time only. The reader familiar with Clarkson’s algorithm knows that one also needs to solve ‘‘small’’ subproblems directlyŽ ones with fewer than about D2 constraints. . With the violation test available, one can solve a problem with n constraints and combinatorial dimension D simply by a brute force testing of each at most d-element subset of H Ž.a potential basis for ŽŽnn . Ž .. Ž . D optimality. This needs at most n D q иии qs0 n.OnrDq1 viola- tion tests. For specific applications, however, one may use a more sophisti- cated algorithm for these small subproblemsŽ Clarkson suggests using the simplex algorithm for linear programming, or the subexponential random- ized algorithms ofwx MSW92, Gar92¨ can be used for some problems. . For a deterministic counterpart of Clarkson’s algorithm, we will need a stronger assumption. To every LP-type problem Ž.H, w , we associate a set 590 CHAZELLE AND MATOUSEKˇ system Ž.H, R . For every basis B : H, we let VB Ž.be the set of con- straints of H violating B, and we let R s RŽ.H, w be the set of all sets of the form VBŽ.for some basis B. We will need Computational assumption 2. A subsystem oracle for Ž.H, R of dimen- sion D˜ Ž.see Definition 2.2 is available.5 We will postpone the discussion of the subsystem oracle for linear programming to the next section. In general, the dimension D˜ need not be identical to D, the combinato- rial dimension of the considered LP-type problem. Typically it will be equal to D or larger. The following example shows that it can be arbitrar- ily large even for D s 2. EXAMPLE 3.1. There existŽ. quite normal LP-type problems Ž.H, w of combinatorial dimension 2, for which the associated set system Ž.H, R has arbitrarily large VC-dimension. Proof sketch. We let H be a suitable finite collection of convex continuous functions fromwx 0, 1 to real numbers, which are nonconstant at every subinterval ofwx 0, 1 . For G : H, we define wGŽ.s min max gx Ž.. xgwx0,1 ggG By compactness and the nonconstancy assumption, for everyŽ. nonempty G the minimum exists and it is realized in exactly one point ofwx 0, 1 . It is easy to verify Axioms 1 and 2 and see that the combinatorial dimension isŽ at most. 2. We leave it to the reader to check that for a suitably chosen H, the VC-dimension of Ž.H, R can be arbitrarily large. Figure 2 gives a hint as to how to shatter a 3-element set of constraints f123, f , f by R Žthe black points correspond to minima for certain bases; the functions belong- ing to those bases are not shown. . Since the randomized version of Clarkson’s algorithm works with Ax- ioms 1 and 2 and Computational assumption 1 onlywx Wel92 , the assump- tions needed for the randomized version are strictly weaker than we need for the deterministic case. For simplicity, let us assume D s D˜ in the subsequent algorithm and analysis. If this is not the case, then the resulting estimate will hold with maxÄ4D, D˜˜playing the role of D. In applications, we usually have D G D, and the savings in the estimate gained by distinguishing D and D˜ are negligible.
5 Actually, it suffices to have a subsystem oracle of dimension D˜ for any system Ž.H, RX X with R : R ; this is what we will in fact have in applications. OPTIMIZATION IN FIXED DIMENSION 591
FIG. 2. Constructing 3 convex functions which are shattered by R.
Now we can formulate a deterministic version of Clarkson’s algorithmŽ a familiarity with the original Clarkson’s algorithm may be helpful for understanding it. . We formulate it as a recursive procedure DetLp, which takes a single argument G : H, a set of constraints, and outputs a basis for G. The procedure works as followsŽ in the description, we use the notation n s <PROCEDURE. DetLpŽ.G 4 Ž. Step 1. If n F n0 s CD log DCa suitable constant , compute the solution directly, by inspecting all possible at most D-element subsets of G and reporting the smallest one that is not violated by any constraint of H Žor by using any other algorithm that might be available for the specific problem. . Otherwise, move on to the next step. 2 Step 2. Compute S,aŽ. 1rr-net for ŽG, RStep 3. Set W11[ л, G [ G. For i s 1, 2, . . . , perform the next step, until a solution is foundŽ. it will happen for i s D at the latest .
Step 4. Let Biibe a basis for W j S, computed by a recursive call of Ž. Ž. DetLp Wiiij S . Let V s VB be the set of all constraints h g G i violating Bii.If Vsл, then B iis a basis for G and the computation finishes, otherwise set Giq1 s GiiiR V , W q1[ Wiij V , increment i and repeat this step.
Let us first prove correctnessŽ following Clarksonwx Cla88. . For every i we have Giij W s G, so the returned basis indeed defines an optimum for G. The crucial observation is that if Viq1 / л, then, for every basis B for G, Viq1 must contain at least one constraint of B R Wi. Indeed, if it is Ž Ä4.Ž. not the case, we have wGR h swG for every h g Viq1, and by a Ž.Ž. repeated application of Axiom 2 we find that wGRViq1 swG. Also, Ž. Ž since none of the constraints of G R Viq1 violates Bii, wB swGR .Ž. Viq1 swG, thus Bi defines an optimum of G. Since any basis of G has at most D elements, the algorithm must terminate after at most D repetitions of Step 4. Let us analyze the running time. The fact that S is chosen as aŽ. 1rr -net for Ž.G, R implies that any basis violated by more than nrr constraints of G is also violated by a constraint of S. Since no constraint of S is violated << << . by Bii, it follows that V q1F nrr, hence WiF inrr - nr4D. Let TnŽ.denote the worst-case running time of the algorithm for n constraints. For Step 1, we get that the brute force search for solution Ž.D Ž.3Dq4Ž.Dq1 requires at most On00rDq1 n sOD log D violation tests. We charge every violation test a unit cost; a more refined analysis counting violation tests separately is straightforward. Step 2 consumes no more than wODŽ.7DDlog Dnxtime, by Theorem 2.3. Then, in each of the at most D recursive calls, we have a subproblem with Ž 3 . at most <
3Dq4 Dq1 TnŽ.FOD Ž . Žlog D . for n F n0,
7D D TnŽ.FOD Ž .log DnqDT Ž nr2 D . for n ) n0.
7DD This recurrence yields TnŽ.FwODŽ.log Dnx. OPTIMIZATION IN FIXED DIMENSION 593
We have thus proved the following:
THEOREM 3.2. LetŽ. H, wbeanLP-type problem with n constraints of combinatorial dimension at most D, satisfying Computational assumptions 1 and 2, with D˜˜F DŽ else, replace D by maxÄ4D, D below.. Then the optimum ofŽ. H, w can be found deterministically in time at most C Ž. D n, where 7DD CDŽ.sOD Ž.log D. Remark. A reader familiar with Clarkson’s work may know that Clark- son has proposed another variant of his algorithm, one with smaller size and ‘‘reweighting’’ of the constaints. We could also derandomize this variant, but it seems that this brings no improvement, since the main overhead comes from the deterministic computation of the sample, and this remains roughly the same for both methods.
4. SAMPLE OF APPLICATIONS
In this section we give few examples of application of Theorem 3.2 to specific geometric optimization problems. First we finish the discussion of linear programming.
Linear Programming It remains to construct the subsystem oracle. A constraint h violates a basis B iff its bounding hyperplane lies above the vertex x defined by B, that is, if the bounding hyperplane intersects the open vertical semiline emanating from x upwards.6 Thus, any set of R corresponds to a set of bounding hyperplanes intersecting a certain vertical semiline. For a set A of constraints, let A be the bounding hyperplanes, and consider the arrangement of A. Then the semilines with endpoint within the same cellŽ. and lower-dimensional faces bounding that cell from below give rise to the same subsets. At this point it is convenient to use the simplifying assumptionsŽ. i ᎐ Žiii . from Section 3. Since we only look for the lowest nonnegati¨e admissible vertex, it suffices to consider the portion of the arrangement of A in the nonnegative orthant. Then each cell has at least one bottommost vertex, which is defined by some k hyperplanes of A and d y k of the coordinate hyperplanes bounding the nonnegative or- thant. All such vertices and the sets of hyperplanes lying above them can Ž3 .ŽŽmm . Ž .. Ž . be inspected, in at most Od qmd d q иии q 0 time m s <6 Constraints with vertical bounding hyperplanes formally require a special treatment; this is easy to handle. 594 CHAZELLE AND MATOUSEKˇ vertex. Hence, a subsystem oracle of dimension d is available and we conclude that: d THEOREM 4.1. The linear programming problem with n constraints in R 7d oŽd. can be sol¨ed in d q n deterministic time.
Extremal Ellipsoids The smallest enclosing ellipsoid problem is the following: Given an n-point set P in R d, find the smallest volume ellipsoid containing P Žalso called minimum spanning ellipsoid, Lowner¨ ᎐John ellipsoid. . We have cho- sen this example because there is an extensive literature concerning it, and it has been considered in several recent papers related to our theme wxPos84, Wel91, SW92, Dye92 . Here the points of P play the role of the constraints, and the function w is the volume of the smallest ellipsoid enclosing a given subset. Axiom 1 is satisfied obviously, Axiom 2 follows easily from the well-known uniqueness of the Lowner¨ ᎐John ellipsoid wxDLL57 . The combinatorial dimension is D s Ž.Žd q 3 dr2 this is the number of degrees of freedom of an ellipsoid, seew DLL57, Juh90, Wel91, SW92x. . The violation test in this case is somewhat more problematic. Specifi- Ä4 cally, it means the following: Given a set B s b12, b ,...,brof r F D points in R d and an extra point h, decide whether h is contained in the unique minimal ellipsoid containing B. The minimum enclosing ellipsoid is determined by a system of nonlinear inequalities. Postwx Pos84 mentions that explicit formulas for solution can be given for d s 2. However, for a general dimension, no better algorithm is known to us than to apply general methods for solving systems of polynomial inequalities. The set of points x of an ellipsoid E in R d can be described by the inequality T Ž.Ž.x y cQxyc F1, Ž. 3 d where c g R is the center of the ellipsoid and Q is a symmetric positive definite d = d matrix. Juhnkewx Juh90 formulates necessary and sufficient conditions for such an ellipsoid E to be the smallest enclosing ellipsoid of B: this is iff Q is symmetric positive definite and there exist real numbers
01G 0, ) 0,...,r)0 such that T Ž.Ž.bjjycQbyc s1 js1,...,r rr bc ÝÝjjsž/ j js1 js1 r T Ž.0det QEsÝjjQbŽ.Ž.ycb jyc. js1 OPTIMIZATION IN FIXED DIMENSION 595
Ž.Edenotes the unit matrix. Then the membership of another point h in this ellipsoid is expressed usingŽ. 3 . Positive definiteness can be enforced by requiring all the upper left submatrices of Q to be positive. Hence, the violation test reduces to deciding the solvability of a system of 2r q D q 2 q d polynomial equalities and inequalities in D q r variablesŽ the unknowns are the entries of Q,ofc, and of the j’s. of maximum degree dq1. Renegarwx Ren92 shows that the solvability of a system of m inequalities of maximum degree d in D variables can be decided in Ž.md OŽD.time.7 Hence, a violation test can be performed in DOŽD.time. It would be interesting to develop some more efficient methods for determin- ing the minimum enclosing ellipsoid for F D pointsŽ this is also a bottleneck for the randomized algorithms. . It remains to discuss the subsystem oracle. Here perhaps the easiest way is to use a ‘‘lifting transform’’Ž pioneered by Yao and Yaowx YY85 for problems of this flavor.Ž. . The left-hand side of the inequality 3 describing an ellipsoid is quadratic in the xi’s, but we can ‘‘linearize’’ it by mapping the problem into a higher dimensional space. Given a point x s Ž.d Ž. D x1,..., xd gR , we map it to a point x in R , given by
2 2 2 Ž.x s Ž.x1,..., xd, x11213, xx,xx,..., xx 1d,x223,xx,..., xd.
Then the condition x g E orŽ. 3 is equivalent to Žx .g h, where h s hQŽ.,c is a certain halfspace in R D, determined by Q and c. For a given A : P, we want to find all sets definable by ellipsoids determined by bases of P. Being somewhat generous, we include subsets of A definable by all ellipsoids. In fact, we map A into R D by , and we list the preimages of all subsets of Ž.A definable by halfspaces in R D. The number of such subsets is still OAŽ<<.D, and we can list them in OAŽ<<.Dq1timeŽ using essentially the method discussed above for linear programming. . Hence, a subsystem oracle of dimension D is available and we can apply the general result. Let us remark that, strictly speaking, the algorithm computes only the minimal subset determining the ellipsoid; the ellipsoid itself is given implicitly as a solution of the above system of inequalities and equalities. A problem of a similar flavor is finding the maximum volume ellipsoid inscribed into the intersection of n given halfspaces in R d. This ellipsoid is again uniquewx DLL57 , and the combinatorial dimension is again D. Both violation test and subsystem oracle can be handled in a similar way as for the previous problem. We get:
THEOREM 4.2. The minimum ¨olume enclosing ellipsoid for a set of n d points in R or the maximum ¨olume ellipsoid inscribed into the intersection
7 This assumes the real RAM model of computation. 596 CHAZELLE AND MATOUSEKˇ of n halfspaces in R d can be computed deterministically in DOŽ D.n time, 7DoŽD. DsddŽ.q3r2. The computation needs Dq n arithmetic oeprations 3 D oŽ D. plus D q n ¨iolation tests. We believe that the above examples sufficiently illustrate the technique of applying the general result to specific geometric optimization problems. In general, problems similar to convex programming involving bounded degree polynomials should be amenable to such treatment. It would be interesting to find also nongeometric applications for the Sharir᎐Welzl framework and the algorithms.
ACKNOWLEDGMENTS
We thank the referees for helpful suggestions to improve the presentation of this paper.
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