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Operations Research Letters 24 (1999) 105 – www.elsevier.com/locate/orms
On dependent randomized rounding algorithms
Dimitris Bertsimas a;∗, Chungpiaw Teob^ , Rakesh Vohrac
a (^) Sloan School of Management and Operations Research Center, Massachusetts Institute of Technology, Room E53-363, 50 Memorial Drive, Cambridge, MA 02139-1347, USA bDepartment of Decision Sciences, Faculty of Business Administration, National University of Singapore, Singapore cDepartment of Managerial Economics and Management Sciences, Northwestern University, Evanston, Illinois, USA
Received 1 January 1996; received in revised form 1 September 1998
Abstract
In recent years, approximation algorithms based on randomized rounding of fractional optimal solutions have been applied to several classes of discrete optimization problems. In this paper, we describe a class of rounding methods that exploits the structure and geometry of the underlying problem to round fractional solution to 0 –1 solution. This is achieved by introducing dependencies in the rounding process. We show that this technique can be used to establish the integrality of several classical polyhedra (min cut, uncapacitated lot-sizing, Boolean optimization, k-median on cycle) and produces an improved approximation bound for the min-k-sat problem. ©c 1999 Elsevier Science B.V. All rights reserved.
Keywords: Linear programming; Randomized rounding; Approximation algorithm
- Introduction
The idea of using randomized rounding in the study of approximation algorithms was introduced by Ragha- van and Thompson [17]. The generic randomized rounding technique can be described as follows:
- Formulate and solve a continuous relaxation (in polynomial time) for a 0–1 integer programming problem to obtain an optimal (possibly fractional) solution x.
- Devise a randomization scheme to decide whether to round each variable xi to 1 or 0.
The heart of the rounding procedure, given a relaxation, is in the design of the randomization scheme. In a recent survey on combinatorial optimization Grotschel and Lovasz [9] write:
... we can obtain a heuristic primal solution by xing those variables that are integral in the optimum solution of the linear relaxation, and rounding the remaining variables “appropriately”. It seems that this natural and widely used scheme for a heuristic is not suciently analyzed ...
∗ (^) Corresponding author. E-mail address: [email protected] (D. Bertsimas).
0167-6377/99/$ - see front matter ©c 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 6 3 7 7 ( 9 9 ) 0 0 0 1 0 - 3
Raghavan and Thompson [17] derive several approximation bounds for multi-commodity routing problems by rounding independently each variable xi to 1 with probability x (^) i. Goemans and Williamson [7] introduce the idea to round each variable x (^) i independently but with probability f( x (^) i), for some particular nonlinear function f(x). The algorithm (for the maximum-satis ability problem) they obtain matches the best-known guarantee for the problem (originally obtained by Yannakakis [20].) Bertsimas and Vohra [2] use a nonlinear rounding function to obtain a randomized rounding heuristic for the set covering problem. Their method matches the best-known guarantee (originally obtained by Chvatal [4]). Bronnimann and Goodrich [3] show further that the set covering bound can be improved if the Vapnik–Cervonenkis (VC) dimension of the constraint matrix can be suitably bounded. Randomized rounding can also be seen as a generalization of deterministic rounding that exploits structural results of the fractional optimum solution to devise deterministic rounding heuristics. This technique has been used in the analysis of the bin-packing problem (see for instance, [5]), machine scheduling (see [19]) and set covering (see [11]). In all the above applications of randomized rounding, each variable xi was rounded independently. Few structural results of the fractional optimum solution have been used in the design of the rounding heuristics. Goemans and Williamson [8], in their study of the maximum-cut problem, show that the geometry of the fractional solution can be suitably utilized to obtain a rounding heuristic with very strong performance bounds. In their rounding process, the variables x (^) i are rounded in a dependent manner. Our objective in the present paper is to describe a class of randomized rounding techniques that seems to work well on several classes of problems that are variants of the min-cut type. For another application of this technique see Bertsimas et al. [1]. The key advantage of this approach is that by using dependencies in the rounding process, the analysis of the performance of the rounding heuristic becomes extremely simple. In the next section, we use dependent randomized rounding to establish integrality results for several basic combinatorial optimization problems. These include the min s − t cut, boolean optimization, uncapacitated lotsizing and k-median problem on a cycle. In Section 3, we describe several approximation results using the rounding technique. For the feasible-cut problem studied by Yu and Cheriyan [21], our technique obtains a worst case bound of 2, matching that obtained in [21] (and independently by Ravi [18]). For the minimum satis ability problem studied by Kohli et al. [13], our technique gives a 2(1 − 1 = 2 k^ ) bound for the min k-SAT problem. For the min-2-sat problem, this result improves the bound from 2 to 32. Marathe and Ravi [15] have obtained a bound of 2 independently using di erent methods. They have also shown that the minimum satis ability problem is closely related to the node covering problem. In this paper, we restrict the discussion only to new randomized rounding ideas and its applications. We will not discuss, for instance, running time analysis or de-randomization techniques. Furthermore, for ease of exposition, we let ZIP and ZLP denote the optimal integral and optimal fractional solution value, respectively. Z (^) H denotes the value returned by a heuristic H. All cost functions are assumed to be nonnegative. Graphs are assumed to be undirected unless stated otherwise.
- Dependent rounding and integrality proofs
In this section, we study the connection of dependent randomized rounding and some basic combinatorial optimization problems. In particular, with the right randomization scheme, we show that the rounding argument leads to direct integrality proofs of several well-known polyhedra.
2.1. s − t cut
In this section we give a direct probabilistic proof that the polyhedron de ned by the min s − t cut problem de ned on the graph G = (V; E) is integral (originally established in [6]). The s − t mincut problem can be
2.2. Boolean optimization
The quadratic optimization problem is to
minimize
i; j
Qij x (^) i x (^) j +
i
ci x (^) i
subject to xi ∈ { 0 ; 1 }:
When the Q (^) ij are arbitrary, the problem is NP-hard. Several researchers have thus focused on identifying properties of Qij that allow the quadratic optimization problem to be solved in polynomial time. Sign-balanced graph: Construct a graph G that has an edge between i and j if and only if Qij 6 =
- The edges which correspond to positive (resp. negative) Q (^) ij are called positive edges (resp. negative edges). G is called a sign-balance graph if it does not contain any cycle with an odd number of positive edges. The notion of sign-balancedness essentially ensures that the graph G can be decomposed into G 1 ∪ G 2 , where G 1 ∩ G 2 = ∅; G 1 ⊂ G; G 2 ⊂ G, and (G 1 ; G 2 ) contains the set of positive edges. Hansen and Sime- one [10] show that the sign-balanced graph problem (i.e., a restricted version of the quadratic optimiza- tion problem where the coecients Q (^) ij give rise to a sign-balance graph) is solvable in polynomial time. Note that this problem contains the maximum independent set problem on bipartite graphs as a special case. Consider the following LP formulation for the problem:
minimize
i; j
Qij z (^) ij +
i
ci x (^) i
subject to zij 6 x (^) i ; if Qij ¡ 0 ;
z (^) ij 6 x (^) j ; if Qij ¡ 0 ;
z (^) ij ¿x (^) i + xj − 1 ; if Qij ¿ 0 ;
z (^) ij ; xi¿ 0 ; ∀ i; j;
z (^) ij ; xi 61 ; ∀ i; j:
We show next that the integrality result of the above LP relaxation can be obtained in a direct manner. We round the fractional solution as follows:
- Generate a single random number U uniformly in [0; 1].
- Starting from an optimal solution of the LP relaxation x; z, round xi to 1 if (i) i ∈ G 1 and x (^) i¿U , or (ii) i ∈ G 2 and x (^) i¿ 1 − U.
Theorem 2. ZLP = ZIP if G is sign-balanced.
Proof. Since 1 − U is also uniformly distributed in [0; 1], P(xi = 1) = x (^) i. For i; j both in G 1 or both in G 2 , P(x (^) i xj = 1) = min{ x (^) i ; x (^) j }: For i ∈ G 1 and j ∈ G 2 ,
P(x (^) i x (^) j = 1) = P(U 6 x (^) i ; 1 − U 6 x (^) j ) = max(0; x (^) i + x (^) j − 1):
At optimality, ∑ z (^) ij = min( x (^) i ; x (^) j ) if Qij ¡ 0, and z (^) ij = max(0; x (^) i + x (^) j − 1) if Qij ¿ 0. Then ZLP 6 ZIP 6 E(ZH ) =
i; j Q(i; j)E[xi^ x^ j^ ] +^
i c^ i^ x^ i^ =^
i; j Q(i; j) z^ ij^ +^
i ci^ x^ i^ =^ ZLP^.
2.3. Uncapacitated lot-sizing
Given a time horizon T , setup costs di (i = 1; : : : ; T ) and production-inventory costs cij (indicating the cost of producing a unit in period i to satisfy a unit of demand in period j), the goal of the uncapacitated lot-sizing problem is to nd a production schedule to minimize the total setup and production-inventory cost, and to satisfy the demand (denoted by fi, i = 1; 2 ; : : : ; T ) at all time periods. We assume further that back-ordering is not allowed in the model, and that production lead time is zero. Let y (^) i be a 0 − 1 decision variable that indicates whether we produce during period i. Let wi; j be the fraction of the demand fj in period j that is met from production in period i 6 j. One formulation of the uncapacitated lot-sizing problem (see [16]) is as follows:
minimize
i; j
fj cij w (^) i; j +
i
di yi
subject to
∑^ j
k=
wk; j = 1; ∀j;
w (^) ij 6 y (^) i ; ∀i; j;
y (^) i ∈ { 0 ; 1 }; ∀i:
It is well known that the resulting LP is integral (see [16]) when ci; j ¿cl; j for all i 6 l ¡ j. This condition is satis ed, for instance, when the unit production cost is constant throughout all time periods. Here we prove this result using randomized rounding. For ease of exposition, we prove the result only for the case c (^) i; j ¿ cl; l for all i ¡ l ¡ j. The argument can be adapted to prove the result for the general case. In the optimal LP solution ( w; y), we must have
w (^) i; j 6 w (^) i; k ; i 6 k ¡ j:
Otherwise, from the constraints
∑j l=1 w^ l; j^ = 1 and^
∑k l=1 w^ l; k^ = 1, there exists some time period^ i ′ (^) such that
w (^) i′; j ¡ w (^) i′; k whereas w (^) i; j ¿ w (^) i; k ; i; i′ 6 j ¡ k:
If i′^ ¿ i, then transfering an ( ¿ 0) amount of ow from w (^) i; j to w (^) i′; j leads to feasible solution with smaller cost (due to savings in inventory holding). Similarly, if i′^ ¡ i, then transfering an amount of ow from w (^) i′; k to w (^) i; k leads to a feasible solution with smaller cost. Hence, without loss of generality, we can augment the LP relaxations with inequalities of the type
w (^) i; j 6 wi; k if j ¿ k:
Let ZLP denote the value of this augmented LP relaxation.
Theorem 3. ZLP = ZIP.
Proof. Let ( w; y) be an optimal LP solution. Consider the following rounding method:
- Set r = 1:
- Set y (^) r = 1. Generate a random number Ur uniformly in [0; y (^) r ]. Let i be the index such that w (^) r; i¿U ¿ w (^) r; i+1. Set wr; l to 1, for all l = r; : : : ; i.
- Repeat step 2 with r ← i + 1 until r ¿ T.
Note that the optimal solution x can be computed from y:
x (^) i; ij =
min
y (^) i (^) j ; 1 −
l:l¡j
y (^) i (^) l
if
l:l¡j
y (^) il ¡ 1 ;
0 otherwise:
On the other hand,
P(x (^) i; ij = 1) = P(S ∩ Ii (^) j 6 = ∅; S ∩ Ii (^) l = ∅ ∀l ¡ j)
min
y (^) i (^) j ; 1 −
l:l¡j
y (^) i (^) l
if
l:l¡j
y (^) i (^) l ¡ 1 ;
0 otherwise:
Hence E(x (^) i; ij ) = x (^) i; ij.
- Dependent rounding and approximation algorithms
In this section, we use the technique proposed in the previous section to obtain approximation results for two classes of problems: the feasible cut and the min-k-sat problem. Our analysis shows that the LP relaxations are within 2 times of the optimum for both problems. Hochbaum [12] has recently obtained an re nement of these results by showing that the LP relaxations are half-integral, and hence the 2-approximation results follow immediately.
3.1. Feasible cut
The feasible cut problem on a graph G = (V; E) was introduced in Yu and Cheriyan [21]. Let M be a set of pairs of nodes in G. The problem asks for a cut of minimum weight, which contains a designated vertex s, but not any node pair (u; v) ∈ M. Yu and Cheriyan showed that the node covering problem can be reduced to this problem. Furthermore, the reduction preserves the approximation bound. Hence, any 2- approximation algorithm for the feasible cut problem would imply the same improvement for the node covering problem. Yu and Cheriyan (and independently Ravi [18]) proposed a 2-approximation algorithm for this problem. We show next how to obtain a similar bound using the rounding idea of Section 2.1. Consider the following formulation of the feasible cut problem.
minimize
(u;v)∈E
c(u; v)x(u; v)
subject to x(u; v)¿y(u) − y(v); (u; v) ∈ E; x(u; v)¿y(v) − y(u); (u; v) ∈ E; y(u) + y(v) 61 ; (u; v) ∈ M; y(s) = 1; y(u); x(u; v) ∈ { 0 ; 1 } :
The randomized rounding algorithm is as follows:
- Starting with an optimal solution of the LP relaxation (x; y), position the nodes in [0; 1] according to the value of y(u).
- Generate a single random variable U uniformly distributed in [ 12 ; 1]. Round all nodes u with y(u) ¡ U to y(u) = 0, and all nodes u with y(u) ¿ U to y(u) = 1.
Since for (u; v) ∈ M at least one from y(u) and y(v) is larger than 12 , the rounding process produces a feasible cut.
Theorem 5. ZIP 6 E(ZH ) 62 ZLP.
Proof. If max(y(u); y(v)) 6 12 , then
E(x(u; v)) = 0:
If min(y(u); y(v)) 6 12 6 max(y(u); y(v)), then
E(x(u; v)) = P(U ∈ [ 12 ; max(y(u); y(v))])
= 2(max(y(u); y(v) − 12 ) 62 |y(u) − y(v)|:
If 12 6 min(y(u); y(v)), then
E(x(u; v)) = 2|y(u) − y(v)|:
In all cases E(x(u; v)) 62 |y(u) − y(v)| 62 x(u; v).
As before, the bound of 2 holds even in the presence of precedence constraints y(u) 6 y(v).
3.2. Minimum satis ability
Kohli et al. [13] introduced the minimum satis ability problem as an analog of the maximum satis ability problem. They proved that this version of the satis ability problem remains NP-hard, even when each clause contains at most two literals (min-2-sat). Given a set of literals and clauses, let xi be a literal and Cj the jth clause. Let I (^) j+ be the set of unnegated
literals in clause C (^) j and I (^) j− the set of negated literals in Cj. Each literal is assigned to be “true” or “false”. The clause C (^) j is a satis ed clause only if one of the literals in I (^) j+ is assigned to be “true” or if one of the
literals in I (^) j− is assigned to be false. The min-sat problem is to nd an assignment of the literals to minimize a weighted sum of satis ed clauses. In the rest of this section, we provide an improvement of their result in the case when the number of literals in each clause is bounded. Let k denote an upper bound on the number of literals in each clause. The problem can be formulated as follows:
minimize
j
w (^) j zj
subject to zj ¿x (^) i ; ∀i ∈ I (^) j+ ; z (^) j ¿ 1 − x (^) i ; ∀i ∈ I (^) j− ; x (^) i ; zj ∈ { 0 ; 1 } :
minimize
S
z (^) S
subject to zS ¿xi ; ∀i ∈ S; ∀S; z (^) S ¿ 1 − xi ; ∀i 6 ∈ S; ∀S; x (^) i ; zS ∈ { 0 ; 1 } : The optimal solution of the linear programming relaxation has xi = 12 and therefore, ZLP = 2k−^1. We next nd the optimal solution to the integer programming problem. Consider an arbitrary integer solution. Let T be such that x (^) i = 1; i ∈ T and xi = 0; i 6 ∈ T. Then, all variables zS (S 6 = T ) are forced to equal 1 except variable z (^) T. Since this is true for every T , we conclude that ZIP = 2k^ − 1. Therefore, for this example,
ZIP = 2
2 k
ZLP:
Acknowledgements
We would like to thank the associate editor and referee for making suggestions that improved the presenta- tion of the current paper. We would like to thank the referee especially for pointing out several discrepencies in an earlier version of the paper.
References
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