An Extension of the Lovasz Local Lemma, and its Applications to
Integer Programming
Aravind Srinivasany
Abstract
Though the LLL is powerful, one problem is that
the \dependency" d is high in some cases, precluding
the use of the LLL if p is not small enough. We present
a partial solution to this via an extension of the LLL
(Theorem 3.1), which shows how to essentially reduce
d for a class of events Ei ; this works well when each
Ei denotes a random variable deviating from its mean.
We apply our LLL to improve the known integrality gap
for three classes of NP-hard integer linear programming
problems (ILPs){minimax, packing and covering integer
programs (MIPs/PIPs/CIPs). A key technique, randomized rounding of linear relaxations, was developed
by Raghavan & Thompson [27] to get approximation
algorithms for these ILPs (see also Beck & Spencer [7]).
We use Theorem 3.1 to prove that this technique produces, with non-zero probability, much better feasible
solutions than known before, if the constraint matrices
of the IPs are sparse (having \few" non-zero entries in
all columns). For MIPs, this complements a result of
Karp, Leighton, Rivest, Thompson, Vazirani & Vazirani [18] (see also Beck & Fiala [6]); for PIPs/CIPs, it
improves on recent work [31], which in turn improved on
that of [27]. Such results cannot be got via Lemma 1.1,
as the dependency d, in the sense of Lemma 1.1, can be
as high as (m) for these problems. (Indeed, the striking applications of Lemma 1.1 arise in situations where
d grows as o(m), e.g., if d is O(polylog(m)).)
Theorem 3.1 works well in combination with an
idea that has bloomed in de-randomization and pseudorandomness results, in the last decade or so: (approximately) decomposing a function of several variables into
a sum of terms, each of which depends on only a few of
these variables. Concretely, suppose Z is a sum of random variables Zi . Many tools have been developed to
upper-bound Pr(Z ? E[Z] z) and Pr(jZ ? E[Z]j z)
even if the Zi s are only (almost) k-wise independent for
some \small" k, rather than completely independent.
The idea is to bound the probabilities by considering
E[(Z ? E[Z])k ] or similar expectations, which look at
the Zi k or fewer at a time (via linearity of expectation). The main application of this has been that the
Zi s can then be sampled using \few" random bits, yielding a de-randomization/pseudo-randomness result (e.g.,
The Lovasz Local Lemma (LLL) is a powerful tool in proving
the existence of rare events. We present an extension of this
lemma, which works well when the event to be shown to exist
is a conjunction of individual events, each of which asserts
that a random variable does not deviate much from its mean.
We consider three classes of NP-hard integer programs: minimax, packing, and covering integer programs. A key technique, randomized rounding of linear relaxations, was developed by Raghavan & Thompson to derive good approximation algorithms for such problems. We use our extended LLL
to prove that randomized rounding produces, with non-zero
probability, much better feasible solutions than known before, if the constraint matrices of these integer programs are
sparse (e.g., VLSI routing using short paths, problems on
hypergraphs with small dimension/degree). We also generalize the method of pessimistic estimators due to Raghavan,
to constructivize our packing and covering results.
1 Introduction.
The powerful Lovasz Local Lemma (LLL) [13] is often
used to show the existence of rare combinatorial structures by showing that a random sample from a suitable
sample space produces them with positive probability;
see Chapter 5 of Alon, Spencer & Erd}os [4] for many
such applications. Let e denote the base of natural logarithms as usual. The LLL (symmetric case) shows that
all of a set of \bad" events Ei can be avoided under some
conditions:
Lemma 1.1. ([13]) Let E1; E2; : : :; Em be any
events with Pr(Ei) p 8i. If each Ei is mutually independent of all but at most Vd of the other events Ej and
if ep(d + 1) 1, then Pr( m
i=1 Ei ) > 0.
Work done in parts at the National University of Singapore,
at DIMACS (supported in part by NSF-STC91-19999 and by
support from the N.J. Commission on Science and Technology),
at the Institute for Advanced Study, Princeton, NJ (supported
in part by grant 93-6-6 of the Alfred P. Sloan Foundation),
and while visiting the Max-Planck-Institut fur Informatik, 66123
Saarbrucken, Germany.
y Dept. of Information Systems & Computer Science, National
University of Singapore, Singapore119260, Republic of Singapore.
E-mail: aravind@iscs.nus.sg
1
[3, 22, 8, 23, 24, 29]). Our results show that such ideas
can in fact be used to show that some structures exist! This is one of our main contributions. However,
while many applications of Lemma 1.1 have been constructivized (Beck [5], Alon [1]), our MIP result is only
existential. For PIPs and CIPs, we present a generalization of the powerful method of pessimistic estimators
of Raghavan [26], to constructivize our bounds.
Many proofs/details are omitted here for lack of
space; they will be given in the full version. Our main
contributions are as follows. (a) The LLL extension is of
independent interest: it draws from a de-randomization
tool{decomposing a function of many variables into
a sum of terms, each of which depends on only a
few of these variables. We expect further interaction
between this idea and the rich set of available derandomization tools. (b) This work shows that certain
classes of sparse IPs have better solutions than known
before; sparse problems abound in practice. Two such
results shown later (on routing using short paths and
hypergraph-partitioning), look particularly interesting.
(c) Our generalized method of pessimistic estimators
should prove fruitful in other contexts also. (d) The
central theme of this work is the analysis of various
types of correlations in random processes, which yields
interesting results and nice open questions.
2 Our application domain and approximation
results.
Let Z+ denote the: set of non-negative integers; for
any k 2 Z+ , [k] = f1; : : :; kg. \Random variable" is
abbreviated by \r.v.", and logarithms are to the base 2
unless specied otherwise.
Definition 2.1. An MIP (minimax integer program) has variables W and fP
xi;j : i 2 [n]; j 2 [`i ]g, for
some integers f`i g. Let N = i2[n] `i and let x denote
the N-dimensional vector of the variables xi;j (arranged
in any xed order). An MIP seeks to minimize W, an
unconstrained real, subject to: P
(i) Equality constraints: 8i 2 [n] j 2[` ] xi;j = 1;
~ where
(ii) a system of linear inequalities Ax W,
m
N
~
A 2 [0; 1]
and W is the m-dimensional vector
with the variable W in each component, and
(iii) Integrality constraints: xi;j 2 f0; 1g 8i; j.
We let g denote the maximum column sum in any
column of A, and a be the maximumnumber of non-zero
entries in any column of A.
To see what problems MIPs model, note, from
constraints (i) and (iii) of MIPs, that for all i, any
feasible solution will make the set fxi;j : j 2 [`i ]g have
precisely one 1, with all other elements being 0; MIPs
i
2
thus model many \choice" scenarios. Consider, e.g.,
global routing in VLSI gate arrays [27]. Given are an
undirected graph G = (V; E), a function : V ! V ,
and 8i 2 V , a set Pi of paths in G, each connecting i to
(i); we must connect each i with (i) using exactly
one path from Pi , so that the maximum number of
times that any edge in G is used for, is minimized{
an MIP formulation is obvious, with xi;j being the
indicator variable for picking the jth path in Pi . This
problem, the vector-selection problem of [27], and the
discrepancy-type problems of Section 4, are all modeled
by MIPs; many MIP instances, e.g., global routing, are
NP-hard.
Definition 2.2. Given A 2 [0; 1]mn, b 2 [1; 1)m
and c 2 [0; 1]n with maxj cj = 1, a PIP (resp. CIP)
seeks to maximize (resp. minimize) cT x subject to
x 2 Z+n and Ax b (resp. Ax b). For PIPs, we
also allow constraints of the form 0 xj dj . If
A 2 f0; 1gmn, each entry of b is assumed integral. We
dene B = mini bi , and let a be the maximum number
of non-zero entries in any column of A. A PIP/ CIP is
called unweighted if cj = 1 8j, and weighted otherwise.
Note the parameters g, a and B of denitions 2.1
and 2.2. Though there are usually no restrictions on
the entries of A; b and c in PIPs/CIPs aside of nonnegativity, the above restrictions are without loss of
generality (w.l.o.g.), because of the following. First, we
may assume that 8i; j; Aij is at most bi. If this is not
true for a PIP, then we may as well set xj := 0; if this is
not true for a CIP, we can just reset Aij := bi. Next, by
scaling each row of A such that maxj Ai;j = 1 for each
row i and by scaling c so that maxj cj = 1, we get the
above form for A, b and c. Finally, if A 2 f0; 1gmn,
then for a PIP, we can always reset bi := bbi c for each
i and for a CIP, reset bi := dbi e; hence the assumption
on the integrality of each bi , in this case.
PIPs and CIPs again model many NP-hard problems in combinatorial optimization. Recall that a hypergraph H = (V; E) is a family of subsets E (edges) of
a set V (vertices). A set M E is a matching in H if no
vertex occurs in more than one edge in M; a basic and
well-known NP-hard problem is to nd a maximumcardinality matching in a given nite hypergraph. A
generalization of this is `-matching, for integral ` 1
(Lovasz [21]): each vertex can be in at most ` elements
of M. The `-matching problem is naturally written as
a PIP with 0-1 variables xi , and with B = `. Similarly, CIPs model, e.g., the classical set cover problem{
covering V using the smallest number of edges in E
(such problems have natural weighted versions). The
parameter a for these IPs is the maximum number of
vertices in any edge.
(y (y =m)1=B ) if A 2 f0; 1gmn.
(ii) For covering, an approximation ratio of
We now present a simple lemma, to quantify the approximation results. Part (a) is the Cherno-Hoeding
bound (see, e.g., Appendix A of [4]); (b) is easily derived
from the denition of G (proof omitted).
Lemma 2.1.
Given independent r.v.s X1 ; : : :; Xn 2
P
[0; 1], let X = ni=1 Xi and = E[X].
a. For any > 0, Pr(X (1 + )) G(; ), where
?
G(; ) = e =(1 + )1+ :
p
1 + O(maxfln(mB=y )=B; ln(mB=y )=B g):
Thus, while the work of [27] gives a general approximation bound for MIPs, the above-seen result of [18] gives
good results for sparse MIPs. For PIPs/CIPs, the current best results are those of [31]; however, no better
results were known for sparse PIPs/CIPs.
b. 8 > 0 8p 2 (0; 1); 9 = H(; p) > 0 such that
2.1 Improvements achieved. For MIPs, we use
d e G(; ) p and such that H(; p) is
log(p?1 ) ) if log(p?1 )=2, and is
( log(log(p
?1 )=)
s
?1
( log( + p ) ) otherwise.
Since many MIPs/PIPs/CIPs are NP-hard, we seek
good approximation algorithms for them. One such
approach is to start with their LP relaxation: allowing
the entries of x to be non-negative reals.
Definition 2.3. Given a CIP/PIP/MIP, x and
y denote, resp., an optimal solution to, and the optimum value of, its LP relaxation. (For PIPs, constraints such as \xj 2 f0; 1; : : :; dj g" are relaxed to
\xj 2 [0; dj ]".) If the optimal integral value is C, then
maxfC=y ; y =C g is called the integrality gap of the LP
relaxation.
Given an IP, we can solve its LP relaxation eciently, but need to round fractional entries in x to
integers. The idea of randomized rounding is: given a
real v > 0, round v to bvc + 1 with probability v ? bvc,
and round v to bvc with probability 1 ? v+ bvc. This has
the nice property that the mean outcome is v. Starting with this idea, the analysis of [27] produces an integral solution of value at most y + O(minfy ; mg H(minfy ; mg; 1=m)) for MIPs (though phrased a bit
dierently); this is de-randomized in [26]. But this does
not exploit the sparsity of A; the previously mentioned
result of Karp et al. ([18]) produces an integral solution
of value at most y + g + 1.
For PIPs, the idea of [27] is to solve the LP relaxation, scale down the components of x suitably, and
then perform randomized rounding; for CIPs, we scale
up instead. (See Section 5 for the details.) Starting with
this idea, the work of [27] leads to certain approximation
bounds; similar bounds are achieved through dierent
means by Plotkin, Shmoys & Tardos [25]. Recent work
of this author [31] improved upon these results by observing a \correlation" property of PIPs/CIPs, getting:
(i) For packing integer programs, an integral solution of value (y (y =m)1=(B?1)) in general, and
3
the extended LLL and an idea of E va Tardos that
leads to a bootstrapping of the LLL extension, to
show the existence of an integral solution of value
y + O(minfy ; mg H(minfy ; mg; 1=a)) + O(1); see
Theorem 4.2. Since a m, this is always as good as
the y + O(minfy ; mg H(minfy ; mg; 1=m)) bound of
[27] and is a good improvement, if a m. It also is an
improvement over the additive g factor of [18] in cases
where g is not small compared to y .
Consider, e.g., the global routing problem and its
MIP formulation, sketched above; m here is the number
of edges in G,
and g = a is the maximum length of
S
any path in i Pi. To focus on a specic interesting
case, suppose y , the fractional congestion, is at most
one. Then while the above-cited previous results ([27]
and [18], resp.) give bounds of O(logm= log logm)
and O(a) on an integral solution, we get the improved
bound of O(log a= logloga). Similar improvements
are easily seen for other ranges of y also; e.g., if
y = O(loga), an integral solution of value O(log a)
exists, improving on the previously known bounds of
O(log m= log(2 logm= log a)) and O(a). Thus, routing
along short paths (this is the notion of sparsity for the
global routing problem) is very benecial in keeping the
congestion low. Section 4 presents a scenario where we
get such improvements, for discrepancy-type problems
[30, 4]. In particular, we generalize a hypergraphpartitioning result of Furedi & Kahn [15].
Please refer now to the above-seen bounds of [31]
for PIPs/CIPs; our bounds for PIPs/CIPs depend only
on the set of constraints Ax b (or Ax b), i.e., they
hold for any non-negative objective-function vector c.
Our improvements over [31] get better as y decreases.
For PIPs, we show the existence of an (y =a1=(B?1) )
solution in general, and of an (y =a1=B ) solution if
A 2 f0; 1gmn. This improves on [31] if y n=a,
i.e., for sparse weighted PIPs; in particular, it shows
the existence of an O(1) integrality gap for weighted
hypergraph B-matching, if B = (loga). For CIPs, we
show an integrality gap of
p
1 + O(maxfln(a + 1)=B; ln(a + 1)=B g);
positive integer d, we can dene, for each i 2 [m], a
nite number of r.v.s Ci;1; Ci;2; : : : each taking on only
non-negative values such that:
(i) any Ci;j is mutually independent of all but at most
d of the events Ek , k 6= i, and
(ii) 8I ([m] ? fig),
once again improving on [31] for weighted CIPs. This
CIP bound is better than that of [31] if y mB=a:
this inequality fails for unweighted CIPs and is generally
true for weighted CIPs, since y can get arbitrarily
small in the latter case. In particular, we generalize the
result of Chvatal [10] on weighted set cover. Consider,
e.g., a facility location problem on a directed graph
G = (V; A): given a cost ci 2 [0; 1] for each i 2 V ,
we want a min-cost assignment of facilities to the nodes
such that each node sees at least B facilities in its outneighborhood{multiple facilities at a node are allowed.
If in is the maximum in-degree of G, we show an
integrality gap of
p
1 + O(maxfln(in + 1)=B; ln(B(in + 1))=B g):
This improves on [31] if y jV jB=in; it shows an
O(1) (resp., 1 + o(1)) integrality gap if B grows as fast
as (resp., strictly faster than) log in. Theorem 5.1
presents our packing/covering results.
A key corollary of our results is that for families of
instances of both PIPs and CIPs, we get a good (O(1)
or 1+o(1)) integrality gap, if B grows at least as fast as
loga. For PIPs and CIPs, we use the FKG inequality
in addition to Theorem 3.1; for PIPs, we also need the
powerful Janson's inequality ([16, 9], see also Chapter
8 of [4]). Bounds on the result of a greedy algorithm
for CIPs relative to the optimal integral solution, are
known [11, 12]. Our bound improves that of [11] and
is incomparable with [12]; for any given A, c, and
the unit vector b=jjbjj2, our bound improves on [12] if
B is more than a certain threshold. As it stands,
randomized rounding produces such improved solutions
for several PIPs/CIPs only with a very low, sometimes
exponentially small, probability. Thus, it does not
imply a randomized algorithm, often. To this end, we
generalize Raghavan's method of pessimistic estimators
to constructivize our PIP/CIP results, in x5.2.
Pr(Ei Z(I)) P
X
j
E[Ci;j Z(I)]:
Let pi denote j E[Ci;j ]; clearly, Pr(Ei) pi (set
I = in (ii)). Suppose Vthat for all i 2 [m] we have
epi (d + 1) 1. Then Pr( i Ei) (d=(d + 1))n > 0.
Remark 3.1. Ci;j and Ci;j can \depend" on different subsets of fEk jk 6= ig; the only restriction is that
these subsets be of size at most d. Note that we have essentially reduced the dependency among the Eis, to just
d: epi (d + 1) 1 suces. Another important point is
that the dependency among the r.v.s Ci;j could be much
higher than d: all we count is the number of Ek that
any Ci;j depends on.
Proof of Theorem 3.1. We prove by induction on jI j
suces
that if i 62 I then Pr(Ei Z(I))V epi , which
Q
to prove the theorem since Pr( i Ei) = i2[m] (1 ?
Pr(Ei Z([i ? 1]))). For the base case where I = ;,
Pr(Ei Z(I)) = Pr(Ei) pi . For the inductive
step, let Si;j;I =: fk 2 I Ci;j depends on Ek g, and
0 = I ? Si;j;I ; note that jSi;j;I j d. If Si;j;I = ;,
Si;j;I
then E[Ci;j Z(I)] = E[Ci;j ]. Otherwise, letting
Si;j;I = f`1 ; : : :; `r g, we have
Z(S 0 )]
E[C
Ind(Z(S
))
i;j
i;j;I
i;j;I
E[Ci;j Z(I)] =
0 ))
Pr(Z(Si;j;I ) Z(Si;j;I
0
E[Ci;j Z(S
i;j;I )] ;
(3.1)
Pr(Z(S ) Z(S
0
i;j;I
i;j;I ))
3 The extended LLL and an approach to large since Ci;j is non-negative. The numerator of the last
deviations.
term is E[Ci;j ], by assumption. The denominator is
We now present our LLL extension, Theorem 3.1. For
Y
0 )))
(1 ? Pr(E` Z(f`1 ; `2; : : :; `s?1g [ Si;j;I
any event E, dene Ind(E) to be its indicator r.v.: 1
if E holds and 0 otherwise. Suppose we have \bad"
s2[r]
events E1; : : :; Em with a \dependency" d0 (in the sense
of Lemma 1.1) that is \large". Theorem 3.1 shows how which is at least
Y
to essentially replace d0 by a (possibly much) smaller
(1 ? ep` )
d, under some conditions. It generalizes Lemma 1.1
s2[r]
(dene one r.v., Ci;1 = Ind(Ei ), for each i, to get
Lemma 1.1), its proof is very similar to the classical by induction hypothesis, i.e., at least (1 ? 1=(d+1))r proof of Lemma 1.1, and its motivation will be claried (d=(d + 1))d > 1=e.
Hence,PE[Ci;j Z(I)]
eE[Ci;j ]
by the applications.
and thus, Pr(Ei Z(I)) j E[Ci;j Z(I)] epi Theorem 3.1. Given
events E1; : : :; Em and any 1=(d + 1). This concludes the proof.
V
Ei. Suppose that for some
I [m], let Z(I) =:
0
s
s
i2I
4
The crucial point is that the events Ei could have
a large dependency d0 , in the sense of the classical
Lemma 1.1. The main utility of Theorem 3.1 is that
if we can \decompose" each Ei into the r.v.s Ci;j that
satisfy the conditions of the theorem, then there is the
possibility of eectively reducing the dependency by
a lot (d0 can be replaced by the value d). Concrete
instances of this will be studied in later sections.
The tools behind our MIP application are our new
LLL, and a result of [29]. Dene, for z = (z1 ; : : :; zn ) 2
<n, a family of polynomials Sj (z); j = 0; 1; : : :; n, where
S0 (z) 1, and for j 2 [n],
X
Sj (z) =:
zi zi z i :
Proposition 4.1. If 0 < 1 2, then for any
> 0, G(1 ; 2=1) G(2; ).
Theorem 4.1. Given an MIP conforming to Definition 2.1, randomized rounding produces a feasible solution of value at most y + minfy ; mg H(minfy ; mg; 1=(et)), with non-zero probability.
Proof. Conduct randomized rounding: indepen-
dently for each i, randomly round exactly one xi;j to
1, guided by the \probabilities" fxi;j g. We may assume
that fxi;j g is a basic feasible solution to the LP relaxation. Hence, at most m of the fxi;j g will be neither zero
nor one, and only these variables will participate in the
rounding.
Thus, since all the entries of A are in [0; 1],
1 2
we
assume
w.l.o.g. from now on that y m (and that
1i1<i2 <i n
maxi2[n] `i m); this explains the minfy ; mg term in
our stated bounds. If z 2 f0; 1gN denotes the randomly
The relevant theorem of [29] is
i ] = bi by linearity of expecTheoremP
3.2. ([29]) Given r.v.s X1 ; : : :; Xn 2 rounded vector, then E[(Az)
. Dening k = dy H(y ; 1=(et))e
n
tation,
i.e.
,
at
most
y
[0; 1], let X = i=1 Xi and = E[X]. Then,
events E1; E2; : :V:; Em by Ei \(Az)i bi +k", we
(a) For any > 0, any nonempty event Z and any and
now
show that Pr( i2[m] Ei) > 0 using Theorem 3.1.
non-negative integer k (1 + ),
Rewrite the ith constraint of the MIP as
X
X
Pr(X (1 + ) Z) E[Yk Z];
x ;
A
X W; where X =
j
j
where Yk = Sk (X1 ; : : :; Xn )=
?(1+)
k
.
r2[n]
i;r
i;r
s2[`r ]
i;(r;s) r;s
(b) If the Xi s are independent and k = d e, then the notation Ai;(r;s) assumes that the pairs f(r; s) : r 2
[n]; s 2 [`r ]g have been mapped bijectively to [N], in
Pr(X (1 + )) E[Yk ] G(; ), where
G(; ) =
e =(1 + )1+ :
?
Suppose r1 ; r2; : : :rn 2 [0; 1] satisfy
r
q. Then, a simple proof is given in [29],
i
i=1
for the fact that for? any
non-negative integer k q,
Sk (r1 ; r2; : : :; rn) kq . This clearly holds even given
the occurrence of any
nonempty event Z. Thus we get
Pr(X (1 + ) Z) Pr(Yk 1 Z) E[Yk Z],
where the second inequality follows from Markov's inequality. A proof of (b) is given in [29].
Pn Proof.
some xed way. Dening the r.v.
Zi;r =
X
s2[`r ]
Ai;(r;s) zr;s ;
we note that for each i, the r.v.s fZi;r : rP2 [n]g lie in
[0; 1] and are independent. Also, Ei \ r2[n] Zi;r bi + k".
Theorem 3.2 suggests a suitable choice for ?the
crucial r.v.s Ci;j (to apply Theorem 3.1). Let u = nk ;
we now dene the r.v.s fCi;j : i 2 [m]; j 2 [u]g as
follows. Fix any i 2 [m]. Identify each j 2 [u] with
some distinct k-element subset S(j) of [n], and let
Q
4 Approximating Minimax Integer Programs.
Z
:
Ci;j = v?2bS (+jk) i;v :
Suppose we are given an MIP conforming to Denik
tion 2.1. Dene t to be maxi2[m] NZi , where NZi is
the number of rows of A which have a non-zero co- We now need to show that the r.v.s C satisfy the
ecient corresponding to at least one variable among conditions of Theorem 3.1. For any i 2i;j[m], let i =
fxi;j : j 2 [`i ]g. Note that
k=bi . Since bi y , we have, for each i 2 [m],
i
(4.2)
g a t minfm; a maxi2[n] `i g:
G(bi; i ) G(y ; k=y ) by Proposition 4.1
G(y ; H(y ; 1=(et)))
1=(ekt); by the denition of H.
Theorem 4.1 now shows how Theorem 3.1 can help,
for sparse MIPs{those where t m. We will then
bootstrap Theorem 4.1 to get the further improved
Now by Theorem 3.2, we get
Theorem 4.2. We start with the simple
5
Fact 4.1. For all i 2 P
[m] and for all nonempty to 1.) Next, ideas similar to those used in Theorem 4.1
eventsPZ , Pr(Ei Z) j 2[u] E[Ci;j Z]. Also, show that with nonzero probability, we have:
pi =: j 2[u] E[Ci;j ] < G(bi; i ) 1=(ekt).
3 5
1:5 2
Next since any Ci;j involves (a product of) k terms,
each of which \depends" on at most (t ? 1) of the events
fEv : v 2 ([m] ? fig)g by denition of t, we see the
important
Fact 4.2. 8i 2 [m] 8j 2 [u], Ci;j 2 [0; 1] and Ci;j
\depends" on at most d = k(t ? 1) of the set of events
fEv : v 2 ([m] ? fig)g.
From Facts 4.1 and 4.2 and by noting that epi (d + 1) e(kt ? k + 1)=(ekt)
1, we invoke Theorem 3.1, to
V
see that Pr( i2[m] Ei) > 0, concluding the proof of
Theorem 4.1.
Theorem 4.1 gives good results if t m, but can we
improve it further, say by replacing t by a ( t) in it?
As seen from (4.2), the key reason for t a(1) is that
maxi2[n] `i a(1) . If we can essentially \bring down"
maxi2[n] `i by forcing many xi;j to be zero for each i,
then we eectively reduce t (t a maxi `i , see (4.2));
this is so since only those xi;j that are neither zero nor
one take part in the rounding. A way of bootstrapping
Theorem 4.1 to achieve this is shown by:
(i)(Az)i (y ) log t(1+O(1=((y ) log t))) 8i 2 [m];
and
(ii) j
X
j
zi;j ? (y )2 log5 tj O(y log3 t); 8i 2 [n]:
P
A subtle point is that (ii) bounds the lower tail of j zi;j
also: this uses the fact that each of the n events in (ii)
depends on at most t of the other (m + n ? 1) events in
(i) and (ii). Suppose we have a rounding z satisfying
(i) and (ii). PNext, for each i 2 [n] and j 2 [`i ], let
x00i;j :=Pzi;j = u zi;u . From (i) and (ii) we deduce: (a)
1 )) for all i from
Since j zi;j (y )2 log5 t(1 ? O( y log
2t
(ii), we have, 8i 2 [m],
1:5 2
(Ax00 )i y (1 + O(1=((y ) 2log t)))
1 ? O(1=(y log t))
(4.4)
= y (1 + O(1=(y log2 t))):
(b) Importantly, since the z are non-negative integers
i;j
Theorem 4.2. For MIPs, there is an integral summing to at most (y )2 log5 t(1 + O(1=(y log2 t))),
solution of value at most y + O(minfy ; mg at most O((y )2 log5 t) values x00i;j are nonzero, for each
H(minfy ; mg; 1=a)) + O(1).
i 2 [n]. Thus, by losing a little in y (see (4.4)), our
down" method has given
Proof. We present a brief sketch for lack of space. \scaling up{rounding{scaling
00
a much-reduced `i for each
We assume from now on that t is suciently large (oth- a fractional solution 2x with
5 t), essentially. Thus,
i;
`
is
now
O((y
)
log
t has
i
erwise Theorem 4.2 matches Theorem 4.1). Suppose
been reduced to O(a(y )2 log5 t) = O(t1=4+2=7 log5 t),
since (4.3) was assumed false. Repeating this scheme
(4.3)
(y t1=7 ) or (t a4)
O(log logt) times makes t small enough to satisfy (4.3).
here,
holds; Theorem 4.2 will again match Theorem 4.1 then. Case II: t?1=7 < y < 1: The idea is the5 same
So we may assume that (4.3) is false. Also, we shall with the scaling up of xi;j being by (log t)=y . This
show in the full version, as to how the extreme case concludes the proof sketch for Theorem 4.2.
of y < t?5 can be handled: so we also assume that
y t?5 (the common situation). Next if y t?1=7,
We now study our improvements for discrepancyTheorem 4.1 anyway guarantees an integral solution type problems, which are an important class of MIPs
of value O(1), as is promised by Theorem 4.2. Thus, that, among other things, are useful in devising dividesuppose y > t?1=7. The basic idea now is, as sketched and-conquer algorithms. Given is a set-system (X; F),
above, to set many xi;j to zero for each i (without where X = [n] and F = fS1; : : :; SM g 2X . Given a
losing too much on y ), so that maxi `i and hence, t, positive integer `, the problem is to partition X into
will essentially get reduced. Such an approach, whose ` parts, so that each Sj is \split well": we want a
performance will be validated by arguments similar to : X ! [`] which minimizes maxj 2[M ];k2[`] jfi 2
those of Theorem 4.1, is repeatedly applied till (4.3) Sj : (i) = kgj. (The case ` = 2 is the standard
holds, owing to the (continually reduced) t becoming set-discrepancy problem.) To motivate this problem,
small enough to satisfy (4.3). There are two cases:
suppose we have a (di)graph (V; A); we want a partition
Case I: y 51: Solve the LP relaxation, and set of V into V1; : : :; V` such that 8v 2 V , fjfj 2 N(v) \
x0i;j := (y )2(log t)xi;j . Conduct randomized rounding Vk gj : k 2 [`]g are \roughly the same", where N(v) is
on the x0i;j now, rounding each x0i;j independently to the (out-)neighborhood of v. See, e.g., [2, 17] for how
zi;j 2 fbx0i;j c; dx0i;j eg. (Note the key dierence from this aids divide-and-conquer approaches. This problem
Theorem 4.1, where for each i, we round exactly one xi;j is naturally modeled by the above set-system problem.
6
Let be the degree of (X; F), i.e., maxi2[n] jfj :
i 2 Sj gj, and let 0 =: maxS 2F jSj j. Our problem is
naturally written as an MIP with m = M`, `i = ` for
each i, and g = a = , in the notation of Denition 2.1;
y = 0=` here. The analysis of [27] gives an integral
solution of value at most y (1 + O(H(y ; 1=(M`)))),
while [18] presents a solution of value at most y + .
Also, since any Sj 2 F intersects at most ( ? 1)0
other elements of F, Lemma 1.1 shows that randomized
rounding produces, with positive probability, a solution
of value at most y (1 + O(H(y ; 1=(e0`)))). This is
the approach taken by [15] for their case of interest:
= 0 , ` = = log.
Theorem 4.2 shows the existence of an integral
solution of value y (1 + O(H(y ; 1=))) + O(1), i.e.,
removes the dependence on 0. This is an improvement
on all the three results above. As a specic interesting
case, suppose ` grows at most as fast as 0 = log. Then
we see that good integral solutions{those that grow at
the rate of O(y ) or better{exist, and this was not
known before. (The above approach of [15] shows such
a result for ` = O(0= log(maxf; 0g)). Our bound of
O(0= log) is always better than this, and especially
so if 0 .)
that Ei \Ai X > i (1 + i )", where i = E[Ai X]
and i = (bi ? Ai s)=i ? 1. Also, Em+1 \cT X <
m+1 (1 ? m+1 )", where m+1 = E[cT X] and m+1 =
1 ? (y =(2) ? cT s)=m+1 .
For CIPs, the idea is to solve the LP relaxation,
and for an > 1 to be xed later, to set x0j = xj ,
for each j 2 [n]. We then construct a random integral
solution z by setting, independently for each j 2 [n],
zj = bx0j c + 1 with probability x0j ? bx0j c, and zj = bx0j c
with probability 1 ? (x0j ? bx0j c). Let Ai , si , and
X1 ; X2 ; : : :; Xn be as dened above for PIPs. Here
again, we want an () approximation. The bad events
now are Ei \Ai X < i (1 ? i )" 8i 2 [m]; and
Em+1 \cT X > m+1 (1 + m+1 )", where i =
E[Ai X] and i = 1 ? (bi ? Ai s)=i for i 2 [m],
m+1 = E[cT X], and m+1 = (y ? cT s)=m+1 ? 1.
Let us also dene
Definition 5.1. For a matrix D 2 [0; 1]mn,
ppack (D; ; ) : =: (e=)+1 , if D 2 f0; 1gmn;
ppack (D; ; ) = (e=) otherwise.
We also need a simple lemma from [31], which combines
the Cherno-Hoeding bound (Lemma 2.1(a)) with
some simple algebraic observations about PIPs/CIPs:
Lemma 5.1. (i) For a PIP, maxi2[m] Pr(Ei) <
ppack (A; ; B); (ii) for a CIP, maxi2[m] Pr(Ei) < pcov =:
e?B(?1)2 =(2) .
Similarly, we can show that the \worst case" (in
terms of having a large value for the integrality gap
) for our analysis of PIPs/CIPs, occurs (as usual)
when si = 0 8i above; in this case, m+1 = y = for
PIPs and m+1 = y for CIPs. Thus, our assumptions
henceforth are that
(5.5) m+1 = y = and m+1 = 1=2 for PIPs;
(5.6) m+1 = y and m+1 = ? 1 for CIPs:
j
5 Approximating packing and covering integer
programs.
For PIPs and CIPs, we extend some of the ideas of
Theorem 3.1, and also use some \correlation" results
of [31]. A crucial way in which we will need to extend
Theorem 3.1 for PIPs is in allowing one Ei , Em+1 , to
have Ci;j s that can become negative: we use Janson's
inequality [16, 9] to aid in this. For PIPs, we solve the
LP relaxation and set x0i := xi = for some > 1 to
be xed later; this scaling down is done to boost the
chance that the constraints in the PIP are all satised.
Dene a random z 2 Z+m , the outcome of randomized
rounding, as follows. Independently for each i, set zi to
be bx0i c + 1 with probability x0i ? bx0i c, and bx0i c with
probability 1 ? (x0i ? bx0i c). We need to show that all
constraints are satised and that cT z is not \much
below" y , with positive probability; we also need to
choose . Note that E[(Az)i ] = (Ax0 )i bi = and
E[cT z] = y =. For some > 1 to be xed later,
dene events E1; : : :; Em by Ei \(Az)i > bi ", and
let Em+1 \cT z < y =()".
Now, z is an (){
Vm+1
approximate solution to PIP if i=1 Ei holds. We now
focusV on achieving a \small" value for (), such that
Pr( mi=1+1 Ei ) > 0 holds; in fact, we x = 2 for PIPs.
For every j 2 [n], let sj = bx0j c, and pj =
0
xj ? sj 2 [0; 1). Let Ai denote the ith row of A.
Let X1 ; X2 ; : : :; Xn 2 f0; 1g be independent r.v.s with
Pr(Xj = 1) = pj 8j 2 [n]. For i 2 [m], it is clear
5.1 Correlation properties of PIPs and CIPs.
Our focusV from now on is to nd \small" ; > 1 so
that Pr( mi=1+1 Ei ) > 0 (for both PIPs and CIPs); some
preliminary lemmas are in order.
Definition 5.2. (a) Events T1 ; T2; : : :; Tm are positively
correlatedQ if for every nonempty S [m],
V
Pr( i2S Ti ) i2S Pr(Ti). (b) An event E is negatively correlated with a set of events VfT1 ; T2; : : :; Tm g if
for every nonempty S [m], Pr(E i2S Ti ) Pr(E).
The \intuitively clear" Lemma 5.2 is a crucial part
and an immediate consequence of, the results of [31].
Its proof follows from the FKG inequality [14, 28]; the
reader is referred to [31] for the proof.
Lemma 5.2. Let r.v.s X1 ; X2 ; : : :; Xn and events
E1 ; E2; : : :; Em+1 be as above, for PIPs/CIPs.
(a) The events fE1; E2; : : :; Em g are positively corre7
lated for PIPs; similarly for CIPs.
(b) For PIPs, 81 i < j n, the event \Xi = 1"
and the event \Xi = Xj = 1", are each negatively
correlated with the set of events fE1; : : :; EWmg. Also
for any S [m], the event \(Xi = 1) ^ ( j 2S Ei)"
is negatively correlated with the set of events fEj :
j 2 [m] ? S g, for PIPs.
via the FKG inequality, Lemma 5.2(a). By arguments
similar to those of Lemma 5.1(i), we can show that for
any j 2 [m],
Pr(((Az)j ? Aji Xi ) > (B ? 1)) < ppack (A; ; B ? 1):
Using this, (5.8), and the fact that jQij a (from (P1)),
We now present an important lemma, to get cor- with (5.7), we conclude the proof of part (i).
relation inequalities which \point" in the \direction" (ii) For part(ii), we use some ideas from the proof of
opposite to FKG (i.e., instead of etc.); it will be a Lemma 1.1 and Theorem 3.1. Let Qi , Q0i , Z1 and Z2
key ingredient in proving Theorem 5.1.
be as in part(i); (P1) and (P2) hold now also.
Now,
Z2 )
Lemma 5.3. (i) In a PIP, 8i 2 [n],
Pr
(((
X
=1)
^
Z
)
1
. The
Pr(Xi = 1 (Z1 ^ Z2 )) =
m
Pr(Z1 Z2 )
^
Pr(Xi = 1 Ej ) pi (1 ? appack (A; ; B ? 1)):
((X =1)
Z2 ) , which equals Pr(X =1) ,
r.h.s. is at most PrPr
j =1
Z
(Z1 2 )
Pr(Z1 Z2)
Pr
by (P2). This fraction, in turn, is at most Pr(X(Z=1)
by
(ii) In a CIP, 8i 2 [n],
1)
Lemma 5.2(a), i.e., at most Pr(Xi = 1)=(1 ? pcov )a .
m
^
a
This concludes the proof of part(ii).
Ej ) pi =(1 ? pcov ) :
Pr(Xi = 1
i
i
i
i
j =1
Next, a simple conditional version of Markov's and
Proof. (i) Fix i. Let Qi [m] be the indices of
Chebyshev's
inequalities (proof omitted).
the rows of A, which have a non-zero entry in the ith
P
0
Proposition 5.1. Let Y = `i=1 Yi be a sum of
column; let Qi = [m] ? Qi. Note, importantly, that
P1: \jQij a" (see the denition of a in Defn. 2.2), independent r.v.s Yi , each of which lies in [0; 1]; let
E[Y ] =P. Then,
and
(i) 2 1i<j ` E[Yi Yj ] < 2 .
P2: \Xi is mutually independent of the set of events
(ii) For any u > 0 and any positive probability event
fEj : j 2 Q0iV
g"
V
2, where T denotes
Z , Pr(jY P
? j u Z)
T=uP
hold. Let Z1 ( j 2Q Ej ) and Z2 ( j 2Q Ej ); thus,
Vm
`
E[Y
Y
the sum i=1 E[Yi Z]
+
2
i
j
(Z1 ^ Z2 ) ( j =1 Ej ). Now,
1
i<j
`
P`
2
Z] + ? 2 i=1 E[Yi Z].
Pr(Xi = 1 (Z1 ^ Z2 )) Pr(((Xi = 1) ^ Z1 ) Z2 );
(iii) For any u > 0 andany positive probability
event Z ,
and the right-hand-side
equals Pr(Xi = 1 Z2 ) ?
P`
Pr(Y u Z) i=1 E[Yi Z] =u.
Pr(((X = 1) ^ Z ) Z ); this dierence, in turn, is
0
i
1
i
i
2
at least Pr(Xi = 1) ? Pr((Xi = 1) ^ Z1), from (P2) and
Lemma 5.2(b). Thus we get
(5.7) Pr(Xi = 1 (Z1 ^ Z2 )) Pr((Xi = 1) ^ Z1 ):
The above approach was inspired by the simple
proof of Janson's inequality due to [9]. To simplify (5.7),
we need to lower-bound Pr((Xi = 1) ^ Z1 ) (an upper
bound is immediate, via FKG). To do this, note that
((Xi = 1) ^ V ) implies ((Xi = 1) ^ Z1 );
V
where V j 2Q (((Az)j ? Aji Xi ) (B ? 1)). Thus,
Pr((Xi = 1) ^ Z1 ) Pr((Xi = 1) ^ V )
= Pr(Xi = 1)Pr(V );
since \Xi = 1" is independent of V . So the quantity of
interest, Pr((Xi = 1) ^ Z1 ), is at least
Y
(5.8) pi
Pr(((Az)j ? AjiXi ) (B ? 1));
Lemmas 5.2 and 5.3 and Proposition 5.1 now lead
to our main theorem for PIPs/CIPs:
Theorem 5.1. (i) For PIPs, there is a xed c1 >
0 such that if = c1a1=(B?1) V(or c1 a1=B if A 2
f0; 1gmn) and = 2, then Pr( mi=1+1 Ei) > 0 holds,
i.e., there exists an integral solution of value at least
y =().
(ii) For CIPs, there is a xed c2 > 0 such that if ; > 1
are chosen as:
= c2 ln(a + 1)=B and = 2; if ln(a + 1) B; and
p
= = 1 + c2 ln(a + 1)=B; if ln(a + 1) < B;
i
j 2Qi
then, there exists a feasible solution of value at most
y . Thus, the p
integrality gap is at most 1 +
O(maxfln(a + 1)=B; ln(a + 1)=B g).
V
Proof. For PIPs, let Z i2[m] Ei ; thus,
8
V
Pr( mi=1+1 Ei) = Pr(Em+1
Z) Pr(Z).
By
Q
implicitly dened L f0; 1gn. Theorem n5.2 shows an
approach to eciently nd some v 2 f0; 1g ?L. For any
function f : [n] ! f0; 1g, let f[i 7! j] be the function g
such that g(k) = f(k) if k V6= i, with g(i) = j; for any
S [n], let Z(S; f) denote i2S (Xi = f(i)).
Lemma 5.2(a), Pr(Z) i2[m] Pr(Ei) (1 ? ppack )m .
Also, ppack < 1 since V
> 1. Hence, Pr(Z) > 0 and
m+1 E ) > 0, it suces to show
thus, to show that
Pr(
i=1 i
V
that Pr(Em+1 i2[m] Ei) < 1; similarly for CIPs.
(i) We handle PIPs rst. Using (5.5) and setting
=: m+1 = y =, we get, by Proposition 5.1(ii), that
Pr(Em+1 Z) is at most
(5.9)
T = (=2)2 ;
Pn
Z) +
where
T
denotes
the
sum
c
Pr(X
=
1
i
i
i
=1
P
Z) + 2 ?
2 P1i<j n ci cj Pr(Xi = Xj = 1
n
2 i=1 ci Pr(Xi = 1 Z).
We wish to pick suitably, to make (5.9) smaller
than 1. Note, now, that 81 i < j n,
(5.10)
Pr(Xi = 1 Z) Pr(Xi = 1) and
(5.11)Pr(Xi = Xj = 1 Z) Pr(Xi = Xj = 1);
by Lemma 5.2(b). More
importantly, we need to lower
bound Pr(Xi = 1 Z), to handle terms such as
?2ciPr(Xi = 1 Z); this is done by Lemma 5.3(i).
Using Lemma 5.3(i) along with (5.10,5.11) and Proposition 5.1(i), we see that (5.9) is at most 42 ( + 22 ?
22(1 ? appack (A; ; B ? 1))), i.e., at most
4 (1 + 2ap (A; ; B ? 1)):
(5.12)
pack
Now, we may assume that 1, since there is always
a feasible solution of value at least 1 for PIPs. Simple
algebra then shows that choosing > 1 as in part(i)
of the statement of the theorem, makes (5.12) strictly
smaller than 1, concluding the
proof of part (i).
V
(ii) For CIPs, setting Z i2[m] Ei and =: m+1 =
y , we see, from Proposition
5.1(iii), that
Pr(Em+1 Pn
Z) is at most R = ( i=1 ci Pr(Xi = 1 Z))=(). We
again seek to make R < 1. Lemma 5.3(ii) shows that
R 1=((1 ? pcov )a ). It is once again simple algebra
to show that choosing ; > 1 as in part (ii) of the
statement of the theorem, ensures that R < 1.
Theorem 5.2. Suppose there is an eciently computable function U that takes some S [n] and some
f : [n] ! f0; 1g as inputs, produces a non-negative real
output, and satises the following:
(a) U(; f) < 1 for any f : [n] ! f0; 1g;
(b) 8S [n] 8f : [n] ! f0; 1g,
(b1) U(S; f) Pr((X1; : : :; Xn ) 2 L Z(S; f));
(b2) If U(S; f) < 1 but 8i 2 ([n] ? S) 8j 2 f0; 1g,
U(S [ fig; f[i 7! j]) 1 holds, then there is an eciently computable v 2 f0; 1gn ? L.
We now present such a function for PIPs; the CIP
construction will be shown in the full version. Let
Xi , Ei , pi, etc. be as before for PIPs, and ; , and
= y = be as in Theorem 5.1. We want a function U
as above for the Xi , with L being the subset of f0; 1gn
corresponding to points that are either infeasible for
the PIP or with objective function value smaller than
y =(), or both. For i 2 [n], let Qi [m] be, as
before, the indices of the rows of A, which have a nonzero entry in the ith column. For j 2 Qi, let V (i; j)
denote the event \((Az)j ? Aji Xi ) (B ? 1)". The
proof of Theorem
Q 5.1 suggests a natural choice for U:
U(S; f) =: F i2[m] (1 ? Pr(Ei Z(S; f))), where F is
shorthand for
X
1 ? 42 ((1 ? 2) ciE[Xi Z(S; f)] +
i2[n]
X
2
1i<j n
X
ci
X
i2[n] j 2Qi
(Pr(Xi Z(S; f))Pr(V (i; j) Z(S; f)))):
To be able to compute Pr(Ei Z(S; f)),
Pr(V (i; j) Z(S; f)) etc. eciently, we rst perturb
the nonzero entries of the matrix A suitably, so that
they all become rationals with the same denominator
that is a suciently large polynomial in n and m; this
can be done in a way that aects the quality of our approximation negligibly. Pr(Ei Z(S; f)) etc. can now
be computed using polynomial interpolation, in polynomial time. The approach of Theorem 5.1 can be used to
show that our function U satises requirements (a) and
(b1) of Theorem 5.2; that it satises (b2) will be shown
in the full paper. Similarly, the CIP constructivization
combines the approach of Theorem 5.1 with some technical results.
5.2 Constructivization. It can be shown that for
many problems, randomized rounding produces the
PIP/CIP solutions shown to exist by Theorem 5.1, with
very low probability. Thus we need to constructivize
Theorem 5.1. The powerful method of pessimistic
estimators [26] does not seem to work for this. So, we
present an extension of this method in Theorem 5.2 and
show how it applies for us, thus getting a deterministic
polynomial-time version of Theorem 5.1. Our discussion
will be very sketchy for lack of space.
Let X1 ; : : :; Xn 2 f0; 1g be r.v.s with Pr(Xi = 1) =
pi. Suppose Pr((X1 ; X2; : : :; Xn) 2 L) < 1, for some
ci cj E[XiXj Z(S; f)] + 2 +
9
Acknowledgements. This work started while visiting
subgraph in a random graph, In Random Graphs '87,
Karonski et al., eds., John Wiley & Sons, Chichester,
the Sandia National Laboratories in the summer of
1990, pp. 73{87.
1994; I thank Leslie Goldberg and Z Sweedyk who were
involved in the early stages of this work. I would like to [17] H. J. Karlo and D. B. Shmoys, Ecient parallel algorithms for edge coloring problems, Journal of Algothank E va Tardos for suggesting the idea that helped
rithms,
8 (1987), pp. 39{52.
bootstrap Theorem 4.1 to get Theorem 4.2. I also [18] R. M. Karp,
F. T. Leighton, R. L. Rivest, C. D.
thank Noga Alon, Prabhakar Raghavan and the referees
Thompson, U. V. Vazirani, and V. V. Vazirani, Global
for their helpful comments and suggestions, and thank
wire routing in two-dimensional arrays, Algorithmica,
Alessandro Panconesi for his help with [7].
2 (1987), pp. 113{129.
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