Roundings Respecting Hard Constraints

Transcription

1 Roundings Respecting Hard Constraints Benjamin Doerr Mathematisches Seminar II, Christian Albrechts Universität zu Kiel, D Kiel, Germany, bed@numerik.uni-kiel.de Abstract. A problem arising in integer linear programming is to transform a solution of a linear system to an integer one which is close. The customary model to investigate such problems is, given a matrix A and a [0, 1] valued vector x, to find a binary vector y such that A(x y) (the violation of the constraints) is small. Randomized rounding and the algorithm of Beck and Fiala are ways to compute such solutions y, whereas linear discrepancy is a lower bound measure. In many applications one is looking for roundings that, in addition to being close to the original solution, satisfy some constraints without violation. The objective of this paper is to investigate such problems in a unified way. To this aim, we extend the notion of linear discrepancy, the theorem of Beck and Fiala and the method of randomized rounding to this setting. Whereas some of our examples show that additional hard constraints may seriously increase the linear discrepancy, the latter two sets of results demonstrate that a reasonably broad notion of hard constraints may be added to the rounding problem without worsening the obtained solution significantly. Of particular interest might be our results on randomized rounding. We provide a simpler way to randomly round fixed weight vectors (cf. Srinivasan, FOCS 2001). It has the additional advantage that it can be derandomized with standard methods. 1 Introduction and Results 1.1 Rounding Problems, Randomized Rounding and Linear Discrepancy Solving integer linear programs (ILPs) is NP hard, solving linear programs without integrality constraints is easy (in several respects). Therefore a natural and widely used technique is to solve the linear relaxation of the ILP and then transform (typically by rounding) its solution into an integer one. In doing so, one usually has to accept that the constraints are violated to some extent. There are several ways to deal with such violations, including simply accepting them, repairing them and preventing them by solving a linear program with stricter constraints in the first step. We do not want to go into detail here, but note that in any case the central theme is rounding the solution of the relaxation in such a way that the constraints are violated not too much. The underlying theoretical concept is the one of linear discrepancy.

2 Definition 1 (Linear Discrepancy Problem). Given a matrix A R m n and a vector x [0, 1] n, find a y {0, 1} n such that A(x y) is small. We write lindisc(a, x) := lindisc(a) := min A(x y), y {0,1} n max lindisc(a, x). x [0,1] n Thus lindisc(a, x) is the rounding error inflicted by an optimal rounding of x. It is known that this can be quite high. Spencer [Spe87] gives an example of a binary n n matrix A such that lindisc(a) = Ω( n). Whereas linear discrepancies provide bounds on how good roundings can possibly be, there are a number of positive results. A very general approach is the one of randomized rounding introduced in Raghavan and Thompson [RT87,Rag88]. Here the integer vector y is obtained from the solution x of the relaxation by rounding each component j independently with probabilities derived from y j. In particular, if x [0, 1] n, we have Pr(y j = 1) = x j and Pr(y j = 0) = 1 x j for all j. Since the components are rounded independently, the deviation (A(x y)) i in constraint i is a sum of independent random variables. Thus it is highly concentrated around its mean, which by choice of the probabilities is zero. Large deviation bounds like the Chernoff inequality allow to quantify such violations. Derandomizations transform this randomized approach into a deterministic algorithm (see [Rag88,SS96]). Another well-known rounding result is due to Beck and Fiala [BF81]. They give a polynomial time algorithm computing a rounding y such that A(x y) < A 1, where A 1 = max j [n] m i=1 a ij. This result is particularly useful for sparse matrices. A one-sided version was proven by Karp et al. [KLR + 87] and applied to a global routing problem. 1.2 Hard Constraints The notion of linear discrepancy prices all violations of constraints the same. This is feasible if all constraints are of the same kind. There are, however, a number of problems where this is definitely not the case. We sketch a simple one that carries most of the typical structure we are interested in. Raghavan and Thompson [RT87] investigate the following routing problem. Given an undirected graph and several source sink pairs (s i, t i ), we are looking for paths f i from s i to t i such that the maximum edge congestion is minimized. Solving the non-integral relaxation and applying path stripping (cf. [GKR + 99]), we end up with this rounding problem: Round the solution (x P ) P of the linear

3 system Minimize W s. t. x P W, e P e x P = 1, i P P i x P 0, P to an integer one such that the first set of constraints is violated not too much and the second one is satisfied without any violation. The first group of constraints ensures that W is the maximum congestion of an edge. Here a rounding error just enlarges the congestion (our objective value). The second kind of constraints is different. It ensures that each request is satisfied exactly once. Here no violation can be tolerated it would result in demands satisfied more than once or not at all. Further examples of rounding problems with hard constraints include other routing applications ([RT91,Sri01]), many flow problems ([RT87,RT91,GKR + 99]), partial and capacitated covering problems ([GKPS02]), the assignment problem with extra constraints ([AFK02]) and the linear discrepancy problem for hypergraphs in more than two colors ([DS03]). 1.3 Prior Work For linear programs with right hand side of the hard constraints equal to one and hard constraints depending on disjoint sets of variables, Raghavan and Thompson [RT87] presented an easy solution. In the example above, for each i they pick one P P i with probability x P and set y P = 1 and y P = 0 for all P P i \{P }. The general case of the integer splittable flow problem, however, seems to require a more complicated random experiment. In the integer splittable flow problem, each source sink pair has associated an integral demand d i and the task is to find an integer flow f i from s i to t i having value d i. Using the approach sketched in the previous subsection, we would end up with the same rounding problem with the 1 replaced by d i in the second set of constraints. Note that for this rounding problem, the ideas of Raghavan and Thompson (and all promising looking simple extensions) fail. Guruswami et al. [GKR + 99] state on the integral splittable flow problem (ISF) in comparison to the unsplittable flow problem that standard roundings techniques are not as easily applied to ISF. On FOCS 2001, Srinivasan [Sri01] presented a way to compute randomized roundings that respect the constraint that the sum of all variables remains unchanged (cardinality constraint) and fulfill some negative correlation properties (that imply Chernoff bounds). Among other results, this yields a randomized algorithm for the integer splittable flow problem. The deterministic pipage rounding algorithm of Ageev and Sviridenko [AS] allows to round edge weights of in a bipartite graph in such a way that the sum of weights incident with a vertex changes by less than one ( degree preservation ). This yields improved approximation algorithms for maximum coverage problems

4 and max-cut problems with given sizes of parts. Ageev and Sviridenko note that their ideas could be used in a randomized way, but the resulting algorithm will be too sophisticated to admit derandomization. The ideas of [AS] and [Sri01] were combined in Gandhi, Khuller, Parthasarathy and Srinivasan [GKPS02] to obtain randomized roundings of edge weights in bipartite graphs that are degree preserving and fulfill negative correlation properties on sets of edges incident with a common vertex. This again yields improved randomized approximation algorithms for several problems as well as some nice per-user fairness properties. 1.4 Our Contribution As can be seen from the previous subsection, there is now a decent amount of knowledge on rounding problems with hard constraints. However, most of these results focus rather on a particular application than on the common theme of respecting hard constraints. While still having an eye on the application, the main aim of this paper is to investigate rounding problems with hard constraints in a unified way. To this end, we introduce the corresponding linear discrepancy notion and extend previous rounding results to deal with hard constraints. Though we find examples showing that the linear discrepancy can increase unexpectedly by adding hard constraints (Theorem 8), our algorithmic results show that reasonable hard constraints can be added without seriously worsening the optima. We show that for constraints on disjoint sets of variables, a rounding error of 2 A 1 can be achieved, which is twice the bound of Beck and Fiala. For constraints of type By = Bx, where B is an arbitrary totally unimodular m n matrix, we have a bound of (1 + m) A 1. We provide a way to generate randomized roundings that satisfy hard constraints as in [Sri01]. They satisfy the key properties of the ones given there (hence our roundings yield all his results as well), but seem to be conceptually much simpler. This allows to derandomize them with respect to large deviation results. Our approach can be extended to the setting of [GKPS02], but we will not discuss this here. We have to defer detailed descriptions to the remainder of the paper. In simple words though, our results show that many known rounding results (in particular, randomized rounding and its derandomizations) still work when suitable hard constraints are added. For reasons of space, many proofs are omitted in the paper. 2 Definitions and Notation For a number r write [r] := {n N n r}. For a matrix A R m n let A 1 := max j [n] i [m] a ij denote the operator norm induced by the L 1 norm. For matrices A and vectors x we write A I J and x J to denote the restrictions (submatrices or subvectors) on the index sets I J and J respectively.

5 Throughout the paper let A R m A n, B R m B n and x [0, 1] n such that Bx Z m B. We call the problem to find a y {0, 1} n such that Bx = By and A(x y) is small a rounding problem with hard constraints. Definition 2 (Linear Discrepancy with Hard Constraints). Let A R m A n, B R m B n and x [0, 1] n such that Bx Z m B. Put E(B, x) = {y {0, 1} n Bx = By}. Then lindisc(a, B, x) := lindisc(a, B) := min A(x y), y E(B,x) max lindisc(a, B, x). x [0,1] n Bx Z m B If E(B, x) =, we have lindisc(a, B, x) =. Of course, the interesting case for our problem is that E(B, x) is not empty. Therefore, we will assume that B is totally unimodular. This is justified by the following corollary of the theorems of Hoffman and Kruskal [HK56] and Ghouila-Houri [GH62]. Theorem 1. The following properties are equivalent: (i) B is totally unimodular. (ii) For all x R n there is a y Z n such that x y < 1 and B(x y) < 1. 3 Sparse Matrices In this section, we extend the theorem of Beck and Fiala (cf. Section 1.1) to include hard constraints. Theorem 2. Let B be totally unimodular. Then a) lindisc(a, B) < (1 + m B ) A 1. b) If B 1 = 1, then lindisc(a, B) < 2 A 1 independent of m B. Proof (Theorem 2). Set := A 1. Set y = x. Successively we will round y to a 0, 1 vector. Let δ > 0 to be determined later. We repeat the following rounding process: Put J := {j [n] y j / {0, 1}}, and call these columns floating (the others fixed). Set I A := {i [m A ] j J a ij > δ} and I B := {i [m B ] j J b ij > 0}, and call these rows active (the others ignored). We will ensure that during the rounding process the following conditions are fulfilled (this is clear for the start, because y = x): (i) (A(x y)) IA = 0, (ii) (B(x y)) IB = 0, (iii) y [0, 1] n.

6 If there is no floating column, that is, J =, then our rounding process terminates with y {0, 1} n. Hence assume that there are still floating columns. We consider the system of equations A IA J z J = 0, B IB J z J = 0, z [n]\j = 0. (1) We have J j J i I A a ij = i I A j J a ij > I A δ, hence J > I A δ/. Case 1: I A. The system (1) consists of at most I A + I B + (n J ) equations. We will determine δ later in such a way the system (1) is underdetermined. Then it has a non-trivial solution z. By definition of J and (iii), there is a λ > 0 such that at least one component of y + λz becomes fixed and still y [0, 1] n. Note that y + λz instead of y also fulfills (i) and (ii). Set y := y + λz. Since (i) to (iii) are fulfilled for this new y and also no previously fixed y j becomes floating again (due to (iii)), we can continue this rounding process until all y j {0, 1}. Case 2: I A =. Since B IB J x J is integral and B (and thus B IB J) is totally unimodular, there is a z {0, 1} J such that B IB J z = B IB J x J (cf. e.g. Theorem 1). Define ỹ {0, 1} n by ỹ j = z j for j J and ỹ j = y j else. Note that this implies B(x ỹ) = 0. Since ỹ {0, 1} n we end the rounding process with result ỹ. We show A(x y) < δ for the resulting y. Let i [m A ]. Denote by y (0) and J (0) the values of y and J when the row i first became ignored. We have y (0) j = y j for all j / J (0) and y (0) j y j < 1 for all j J (0). Note that j J a ij δ, since i is ignored. Thus (0) (A(x y)) i = (A(x y (0) )) i + (A(y (0) y)) i = 0 + a ij (y (0) j y j ) < δ. j J (0) It remains to determine δ in such a way that the linear systems regarded are under-determined. Part a) For the general case, put δ = (1 + m B ). Since I A in Case 1, I B m B and J > I A δ/, we have I A + I B + (n J ) < I A + I B + n I A (1 + m B ) n. Part b) Assume now that B 1 = 1, that is, the constraints encoded in B belong to disjoint sets of variables. Then J 2 I B holds throughout the rounding process: If a constraint from B is active, it depends on at least two variables not yet fixed simply because B IB J y J = B IB J x J is integral and B { 1, 0, 1} mb n. Therefore, δ = 2 suffices. We then have I A + I B + (n J ) I A + n 1 2 J < n. The dependence on m B in Part a) is of the right order, as the first example in Section 5 shows. In particular, a bound like lindisc(a, B, x) (1+ B 1 ) A 1 as could be conjectured from a) and b), does not hold. Let us also remark that the rounding algorithms of Karp et al. [KLR + 87] admits similar extensions. We omit the details.

7 4 Randomized Rounding In this section, we modify the approach of randomized rounding to respect hard constraints. The particular problem is to design a random experiment that at the same time respects the hard constraints and generates independent looking randomized roundings (satisfying Chernoff bounds for example). Our random experiment is different from the one in [Sri01], which enables us to derandomize it. However, it also satisfies the main properties (A1) to (A3) of his approach. To ease reading, we describe our result in its simplest version in the following section and sketch possible extensions in the second one. 4.1 Randomized Construction and Derandomization In this section, we only treat the case that B {0, 1} mb n and B = 1. Hence, we only regard so-called cardinality constraints that contain disjoint sets of variables. Randomized construction: Assume first that all x j are in {0, 1 2, 1}. Since j [n] b ijx j Z for all i [m B ] by assumption, we conclude that all E i := {j [n] x j = 1 2, b ij = 1} have even cardinality. Now partitioning each E i into pairs 1 (j 1, j 2 ) and independently flipping a coin to decide whether (y j1, y j2 ) = (1, 0) or (y j1, y j2 ) = (0, 1) solves the problem in a randomized way (variables x j with j contained in no E i can be rounded independently at random). For x j having finite binary expansion, we iterate this procedure digit by digit: If x has binary length l, write x = x + 2 l+1 x with x {0, 1 2 }n and x [0, 1] n having binary length l 1. Compute y as rounding of x as above. Put x := x + 2 l+1 y. Note that x now has binary length l 1. Repeat this procedure until a binary vector is obtained. For each x having finite binary expansion, this defines a probability distribution D(B, x) on {0, 1} n. Theorem 3. Let y = (y 1,..., y n ) be a sample from D(B, x). Then it holds: (A1) y is a randomized rounding of x: For all j [n], Pr(y j = 1) = x j. (A2) D(B, x) is distributed on E(B, x): Pr(By = Bx) = 1. (A3) For all S [n] and b {0, 1}, Pr( j S : y j = b) j S Pr(y j = b). Proof. (A1): Let j [n]. If x j {0, 1}, the claim is trivial. Let x j therefore have binary length l 1. Let x j be the outcome of the first random experiment (i.e., x j is a random variable having binary length at most l 1). By induction, Pr(y j = 1) = ε { 1,1} = ε { 1,1} Pr( x j = x j + ε2 l ) Pr(y j = 1 x j = x j + ε2 l ) 1 2 (x j + ε2 l ) = x j. 1 As we will see, the particular choice of this partition is completely irrelevant. Assume therefore that we have fixed some deterministic way to choose it (e.g., greedily in the natural order of [n]).

8 (A2): By definition of D(B, x), in each rounding step the sum of the values with index in E i is unchanged for all i [m B ]. Hence (By) i = j E i y j = j E i x j = (Bx) i. (A3): Let S [n]. We show the claim for b = 1. Again, if x {0, 1} n, there is nothing to show. Let x therefore have binary length l 1. Let x be the outcome of the first rounding step. This is a random variable, that is uniformly distributed on the set R(x) of possible outcomes (which is determined by x and the way we choose the partition into pairs). Note that for each z R(x), also z := 2x z R(x). Note also that j S z j + j S z j 2 j S x j. Hence by induction Pr( j S : y j = 1) = Pr( x = z) Pr(( j S : y j = 1) x = z) = 1 R(x) z R(x) j S z R(x) z j 1 R(x) ( 1 2 R(x) 2 x j ) = x j = Pr(y i = 1). j S j S j S As shown in [PS97], (A3) implies the usual Chernoff-Hoeffding bounds on large deviations. We build on the following theorem of Raghavan [Rag88], which is a derandomization of the (independent) randomized rounding technique. Theorem 4 (Raghavan (1988)). For any A {0, 1} m n and x [0, 1] n a y {0, 1} n can be computed in O(mn) time such that A(x y) (e 1) s ln(2m), where s = max{ Ax, ln(2m)}. Noting that the pairing trick in a single iteration allows us to write Ay in the form matrix times vector of independent random variables, we prove the following result. Theorem 5. Let A {0, 1} m A n and B {0, 1} m B n such that B 1 = 1. a) Let x [0, 1] n such that Bx Z m B. Then for all l N, a binary vector y such that Bx = By and A(x y) 52 max{ Ax, ln(4m A )} ln(4m A ) + n2 l can be computed in time O(mnl). b) lindisc(a, B) 5 n ln(4m A ). 4.2 Extensions (1) We always assumed that Bx is integral. A trivial reduction (by adding dummy variables) extends our results to arbitrary Bx. We then have: (A2+) For all i [m B ], (By) i is a randomized rounding of (Bx) i. In particular, (By) i { (Bx) i, (Bx) i } with probability one.

9 (2) Raghavan [Rag88] also obtains the bound A(x y) e ln(2m)/ ln(e ln(2m)/ Ax ) for the case that Ax ln(2m). This is strongest for constant Ax, where it yields a bound of O( log m log log m ) instead of our bound of O(log m). Since the typical application of randomized rounding seems to be that Ax is large, we do not try to improve our result in this direction. (3) One subtle aspect in derandomizing Chernoff bounds lies in the computation of the involved pessimistic estimators. There is no problem if one works in a model that allows exact computations of real numbers. In the more realistic RAM model, things are more complicated. Raghavan s derandomization then only works for 0, 1 matrices A. Srivastav and Stangier [SS96] gave a solution that works for matrices having arbitrary entries in [0, 1] Q, though has a higher time complexity of O(mn 2 log(mn)). Here again the simplicity of our approach pays off. Since we only need to derandomize Chernoff type large deviation bounds, we can plug in any algorithmic version of the underlying large deviation inequality. (4) If B { 1, 0, 1}, one can modify the definition of ỹ in the proof above in such a way that Bỹ = 0. An extension to further values, however, is not possible as we might run into the problem that no integral solution exists at all. For example, the single constraint i [3] 4 5 x i = 2 is satisfied by x i = 5 6, but clearly no 0, 1 solution exists. (5) The constant of 52 is not the full truth. Things become much better, if Ax ln(4m A ). In this case, the constant reduces to less than Applications In this subsection, we sketch two applications. Note that and this is one advantage of the results presented above our results in simple words just state that randomized rounding and the corresponding derandomizations work as before even if a few hard constraints are added to the problem. This seems to be particularly useful for real-world application, which usually lack the plainness of problems regarded in theoretical sciences. We start with derandomizing Srinivasan s [Sri01] solution for the integral splittable flow problem (cf. Subsection 1.2 and 1.3). Note that for most of the other randomized results in [Sri01], deterministic algorithms of same quality have already been given earlier by Ageev and Sviridenko [AS]. The integral splittable flow problem extends the unit flow version of Raghavan and Thompson [RT87]. From the problem formulation, it is clear that Theorem 3 and 5 can be applied: The hard constraints depend on disjoint sets of variables, namely the paths obtained from applying the path stripping procedure to the flow satisfying a particular demand. Analogous to the result of Raghavan

10 and Thompson for unit flows and derandomizing Srinivasan [Sri01] (with larger constants), we obtain the following. Theorem 6. A solution of the relaxation with objective value W ln(4 E ) can efficiently be transformed into an integer solution with objective value W + 52 W ln(4m A ). As a second example, let us consider the packing problem max c t x such that Ax k, x {0, 1} n. We may view this as a scheduling problem. We want to select a set of jobs maximizing our profit in such a way that all m machines are busy for at most k time units. Using an additional scaling trick, Raghavan [Rag88] showed that for k = Ω(log m A ), approximations with additive error exist. In a real world scenario, additional constraints often are present (or show up while a first solution is analyzed). Here, one may assume that different parties have a particular interest in some jobs to be scheduled. In this case, we have disjoint sets F 1,..., F l of jobs favored by party i [l], and a fairness condition might impose that from each set F i, at least a given number of r jobs has to be scheduled. Note that r can (and usually will) be small compared to k. Hence large deviation bounds will not be applicable. However, the following easily solves the problem: (i) Solve the relaxation with additional constraints j F i x j r, i [l]. Denote the solution by ˆx. (ii) Apply randomized rounding or its derandomization on ˆx with the additional hard constraints that j F i y j is a randomized rounding of j F i ˆx j for all i [l] (cf. the extensions subsection for a remark on these dependencies). We thus obtain an integer solution of similar quality as Raghavan s that also satisfies our fairness requirements. 4.4 Comparison to the Approach of Srinivasan In Srinivasan [Sri01], randomized roundings satisfying hard constraints as in Theorem 3 were generated. His approach is to repeat regarding two variables only, and fixing one to an integer value and propagating the other with an updated probability distribution. This sequential rounding approach seems to be much harder to work with. We currently do not see how this algorithm can be derandomized. Also, we feel that proving the properties (A1) to (A3) must be quite complicated (proofs are omitted in [Sri01]). Note that the complexity of both approaches is very similar. Working with real numbers in [Sri01] hides part of complexity that is present in the bit-wise model used in this paper. 5 Examples and Lower Bounds The following simple example shows that hard constraints may increase the rounding error significantly. It also shows that the dependence on m B in part a) of Theorem 2 is of the right order.

11 Example 1: Let n be a multiple of 4. Let A = ( ) R 1 n, m B = n 1 and B {0, 1} m B n such that b ij = 1 if and only if j {i, i + 1}. Let x = n. Then lindisc(a, x) = 0, lindisc(a, x ) 1 2 for all x [0, 1] n, lindisc(a, B, x) = 1 4 n (= 1 4 (1 + m B) A 1 ). Example 2: The linear discrepancy problem for hypergraphs is to compute for a given mixed coloring (each vertex receives a weighted mixture of colors) a pure coloring in such a way that each hyperedges in total contains (roughly) the same amount of each color with respect to both colorings. Definition 3 (Linear Discrepancy Problem for Hypergraphs). Let c N 2. Let H = (V, E) be a hypergraph. A mapping p : V [0, 1] c such that d [c] p(v) d = 1 for all v V is called mixed coloring of H. It is called pure coloring, if for all v V there is a (unique) d [c] such that p(v) d = 1. In this case, we say that v has color d and write ˆp(v) = d. The discrepancy of two mixed colorings p, q is disc(h, p, q) = max d [c] max E E v E p(v) d v E q(v) d. The objective in the linear discrepancy problem for hypergraphs is to find for given hypergraph H and mixed coloring p a pure one q such that disc(h, p, q) is small. Put lindisc(h, c) := max p min q disc(h, p, q). A hypergraph is called totally unimodular, if its incidence matrix is totally unimodular. It is well known that totally unimodular hypergraph behave nicely in linear discrepancy problems. Theorem 7. Let H = (V, E) be a totally unimodular hypergraph. a) De Werra [dw71]: For all numbers c of colors, the combinatorial discrepancy lindisc(h, 1 c 1 V ) is less than 1. b) Hoffman, Kruskal [HK56]: The linear discrepancy lindisc(h, 2) of H in 2 colors is less than 1. The constant in b) was recently [Doe01] improved to the sharp bound of V /( V + 1). Contrary to what one might expect, a combination of a) and b) is not true: Theorem 8. For all c 3 there is a totally unimodular hypergraph H such that lindisc(h, c) ln(c + 1) 1. In consequence, the bound lindisc(h, c) < 1 for totally unimodular hypergraphs holds only in the case c = 2. References [AFK02] S. Arora, A. Frieze, and H. Kaplan. A new rounding procedure for the assignment problem with applications to dense graph arrangement problems. Math. Program., 92:1 36, 2002.

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