Continuous Set Packing and Near-Boolean Functions

Giovanni Rossi

Department of Computer Science and Engineering (DISI), University of Bologna,

Mura Anteo Zamboni 7, 40126, Bologna, Italy

Keywords:

Set Packing, Pseudo-Boolean Function, Polynomial Multilinear Extension, Local Search.

Abstract:

Given a family of feasible subsets of a ground set, the packing problem is to ﬁnd a largest subfamily of pairwise

disjoint family members. Non-approximability renders heuristics attractive viable options, while efﬁcient

methods with worst-case guarantee are a key concern in computational complexity. This work proposes a

novel near-Boolean optimization method relying on a polynomial multilinear form with variables ranging

each in a high-dimensional unit simplex. These variables are the elements of the ground set, and distribute

each a unit membership over those feasible subsets where they are included. The given problem is thus

translated into a continuous version where the objective is to maximize a function taking values on collections

of points in a unit hypercube. Maximizers are shown to always include collections of hypercube disjoint

vertices, i.e. partitions of the ground set, which constitute feasible solutions for the original discrete version of

the problem. A gradient-based local search in the expanded continuous domain is designed. Approximations

with polynomials of bounded degree and near-Boolean coalition formation games are also ﬁnally discussed.

1 INTRODUCTION

Consider a ﬁnite set N = {1, ..., n} of items to be

packed into feasible subsets, where these latter con-

stitute a family F ⊆ 2

= {A : A ⊆ N}. The prob-

lem is to ﬁnd a subfamily F

∗

⊆ F of pairwise dis-

joint feasible subsets with largest size |F

∗

|. In the

weighted version, a function w : F → R

identiﬁes

as optimal those such subfamilies F

∗

with maximum

weight W (F

∗

) =

∑

A∈F

∗

w(A). Maximizing |F

∗

| is

equivalent to setting w(A) = 1 for all A ∈ F . Thus

this work proposes to use the polynomial multilinear

extension, or MLE for short, of set functions (such

as w) in order to evaluate families of fuzzy feasible

subsets. Although unfeasible, such families shall still

drive the search towards locally optimal feasible ones.

Set packing is a key combinatorial optimization

problem (Korte and Vygen, 2002) extensively stud-

ied in computational complexity, where the aim is

to ﬁnd efﬁcient algorithms whose output approxi-

mates optimal solutions within a provable bounded

factor. In that ﬁeld, the focus is placed mostly on non-

approximability results for k-set packing (Trevisan,

2001), where the size of every feasible subset is no

greater than some k  n (and with unit weight for all

as above). Recall that if all feasible subsets have size

k = 2, then the problem is to ﬁnd a maximal match-

ing in a graph with vertex set N, and an efﬁcient (i.e.

with polynomial running time) algorithm capable to

output an exact solution is known to exist (Papadim-

itriou, 1994). In fact, if k > 2, then k-set packing

may be rephrased in terms of vertex clouring in hy-

pergraphs, with special focus on the d-regular and

k-uniform case, where every element of the ground

set is present in precisely d > 1 feasible subsets (i.e.

|{A : i ∈ A ∈ F }| = d for every i ∈ N), each of which,

in turn, has size k (i.e. |A| = k for every A ∈ F )

(Hazan et al., 2006).

Set packing aslo has important applications,

among which combinatorial auctions constitute a

main and lucrative example: the ground set may con-

sist of items to be sold in bundles (or subsets) to-

wards revenue maximization, and once bids are pro-

cessed the issue may be tackled as a maximum-weight

set packing problem, with maximum received bids on

bundles as weights (Sandholm, 2002). Given the ex-

ponentially large size of the search space, revenue

maximization often leads to use heuristics with no

worst-case guarantee or, more simply, to sell each

item independently but simultaneously over a sufﬁ-

ciently long time period (Milgrom, 2004).

The approach to set packing problems proposed

in the sequel replaces standard pseudo-Boolean op-

timization (Boros and Hammer, 2002) with a novel

near-Boolean method. While the former employs

MLE to switch from {0,1} to [0,1] as the domain

of each of the n variables, the proposed near-Boolean

method relies on n variables ranging each in the 2

n−1

Rossi, G.

Continuous Set Packing and Near-Boolean Functions.

DOI: 10.5220/0005697800840096

In Proceedings of the 5th International Conference on Pattern Recognition Applications and Methods (ICPRAM 2016), pages 84-96

ISBN: 978-989-758-173-1

set of extreme points of a unit simplex, and employs

MLE to include the continuum provided by the whole

simplex. The n variables correspond to the elements

i ∈ N of the ground set, while the extreme points of

each simplex are indexed by those (feasible) subsets

where each element is included. Then, the MLE of the

resulting near-Boolean function evaluates collections

of fuzzy subsets of N or, equivalently, fuzzy subfami-

lies of feasible subsets A ∈ F . The objective function

to be maximized takes thus values over the n-product

of 2

n−1

− 1-dimensional unit simplices, allowing to

design a ﬂexible gradient-based local search.

The following section comprenshively details the

framework for the full-dimensional case F = 2

This seems useful in general and also allows to clearly

see next that by simply introducing the empty set

and all n singletons {i} ∈ 2

into the family F of fea-

sible subsets (with null weights w(

0) = 0 = w({i} if

{i} /∈ F ) the whole class of set packing problems may

be framed within the proposed method. The gradient-

based local search differs when switching from the

full-dimensional case to the lower-dimensional one

F ⊂ 2

, as with the latter a cost function c : F → N

also enters the picture, in line with greedy approaches

to weighted set packing (Chandra and Halldorsson,

2001). The cost c(A) = |{B : B ∈ F ,B ∩A 6=

0}| of in-

cluding a feasible subset in the packing is the number

of members with which it has non-empty intersection

(itself included, hence c(A) ∈ N,A ∈ F ).

Note that maximum-weight set packing may be

tackled through constrained maximization of standard

pseudo-Boolean function v : {0,1}

|F |

→ R

given by v



,. .. ,x

|F |



∑

1≤k≤|F |

w(A

) , s.t.

∩ A

0 ⇒ x

+ x

≤ 1 for all 1 ≤ k < l ≤ |F |,

where x

∈ {0,1} for all A ∈ F = {A

,. .. A

|F |

}. Also,

v can be replaced with xMx ' v, where M is a suit-

able |F | × |F |-matrix and x = (x

,. .. ,x

|F |

) (Ali-

daee et al., 2008). An heuristic then ﬁnds a con-

strained maximizer x

∗

, while the corresponding so-

lution is F

∗

= {A : x

∗

= 1}. This differs from what is

proposed here, in many respects, the most evident of

which being that v has |F | constrained Boolean vari-

ables, while the expanded MLE developed below has

n unconstrained near-Boolean variables.

2 FULL-DIMENSIONAL CASE

The 2

-set {0,1}

of vertices of the n-dimensional

unit hypercube [0,1]

corresponds one-to-one to the

(power) set 2

of subsets A ⊆ N through character-

istic functions χ

: N → {0,1},A ∈ 2

deﬁned by

(i) = 1 if i ∈ A and χ

(i) = 0 if i ∈ N\A = A

while collection {ζ(A,·) : A ∈ 2

} is a linear basis of

the vector space R

of real-valued functions w on 2

where zeta function ζ : 2

×2

→ R is the element of

the incidence algebra (Rota, 1964b; Aigner, 1997) of

Boolean lattice (2

,∩, ∪) deﬁned by ζ(A,B) = 1 if

B ⊇ A and ζ(A,B) = 0 if B 6⊇ A. Linear combination

w(B) =

∑

A∈2

(A)ζ(A,B) =

∑

A⊆B

(A) for B ∈ 2

applies to any w, with M

obius inversion µ

: 2

→ R

uniquely given by (⊂ is strict inclusion) µ

(A) =

∑

B⊆A

(−1)

|A\B|

w(B)



with ζ(B,A) = (−1)

|A\B|



= w(A) −

∑

B⊂A

(B) (recursion, with w(

0) = 0).

Given this essential combinatorial “analog of the fun-

damental theorem of the calculus” (Rota, 1964b), the

MLE f

: [0,1]

→ R of w takes values w(B) =

= f

(χ

) =

∑

A∈2

∏

i∈A

(i)

(A) =

∑

A⊆B

(A)

on vertices, and f

(q) =

∑

A∈2

∏

i∈A

(A) (1)

on any point q = (q

,. .. ,q

) ∈ [0,1]

. Convention-

ally,

∏

i∈

:= 1 (Boros and Hammer, 2002, p. 157).

Let 2

= {A : i ∈ A ∈ 2

} =

{

,. .. ,A

n−1

}

be the

n−1

-set of subsets containing each i ∈ N. Simplex

∆

(



,. .. ,q

n−1



∈ R

n−1

∑

1≤k≤2

n−1

= 1

)

has dimension 2

n−1

− 1 and generic point q

∈ ∆

Deﬁnition 1. A fuzzy cover q speciﬁes a membership

distribution for each i ∈ N over the 2

n−1

subsets con-

taining it, i.e. q = (q

,. .. ,q

) ∈ ∆

= ×

1≤i≤n

∆

Equivalently, q =



0 6= A ∈ 2

∈ [0,1]



is a 2

− 1-set whose elements q



,. .. ,q



are

n-vectors corresponding to non-empty subsets A ∈ 2

and specifying a membership q

for each i ∈ N, with

∈ [0,1] if i ∈ A while q

= 0 if i ∈ A

. Fuzzy cov-

ers being collections of points in [0,1]

, and the MLE

of w allowing precisely to evaluate such points,

the global worth W (q) of q ∈ ∆

is the sum over all

,A ∈ 2

of f

) as deﬁned by (1). That is,

W (q) =

∑

A∈2

) =

∑

A∈2

∑

B⊆A

∏

i∈B

(B)

or W (q) =

∑

A∈2

∑

B⊇A

∏

i∈A

(A). (2)

Continuous Set Packing and Near-Boolean Functions

Example 2. For N = {1, 2,3}, consider w deﬁned by

w({1}) = w({2}) = w({3}) = 0.2, w({1, 2}) = 0.8,

w({1,3}) = 0.3, w({2,3}) = 0.6, w(N) = 0.7. Mem-

bership distributions of elements i = 1,2, 3 over 2

are q

∈ ∆













, q













, q













If ˆq

= ˆq

= 1, then any membership q

∈ ∆

yields

W ( ˆq

, ˆq

) = w({1,2})



+ q



({3})

= w({1,2}) +w({3}) = 1.

This means that there is a continuum of fuzzy covers

achieving maximum worth, i.e. 1. In order to select

the one

q = ( ˆq

, ˆq

) where ˆq

= 1, attention must

be placed only on exact ones, deﬁned hereafter.

For any two fuzzy covers q = {q

0 6= A ∈ 2

}

and

q = { ˆq

0 6= A ∈ 2

}, deﬁne

q to be a shrinking

of q if there is a subset A, with

∑

i∈A

> 0 and

ˆq



if B 6⊆ A

0 if B = A

for all B ∈ 2

,i ∈ N,

∑

B⊂A

ˆq

= q

∑

B⊂A

for all i ∈ A.

In words, a shrinking reallocates the whole member-

ship mass

∑

i∈A

> 0 from A ∈ 2

to all proper sub-

sets B ⊂ A, involving all and only those elements i ∈ A

with strictly positive membership q

> 0.

Deﬁnition 3. Fuzzy cover q ∈ ∆

is exact as long as

W (q) 6= W(

q) for all shrinkings

q of q.

Proprosition 4. If q is exact, then for all A ∈ 2





i ∈ A : q

> 0





∈ {0,|A|}.

Proof. For

0 ⊂ A

(q) =



i : q

> 0



⊂ A ∈ 2

, with

α = |A

(q)| > 1, notice that

) =

∑

B⊆A

(q)

∏

i∈B

(B). Let shrinking

with ˆq

= q

if B

6∈ 2

(q)

, satisfy conditions

∑

B∈2

∩2

(q)

ˆq

= q

∑

B∈2

∩2

(q)

for all i ∈ A

(q)

and

∏

i∈B

ˆq

∏

i∈B

∏

i∈B

for all B ∈ 2

(q)

,|B| > 1.

These are 2

− 1 equations with

∑

1≤k≤α





> 2

variables ˆq

,B ⊆ A

(q). Thus there is a continuum

of solutions, each providing precisely a shrinking

where

∑

B∈2

(q)

( ˆq

) = f

) +

∑

B∈2

(q)

This entails that q is not exact.

Partitions (Aigner, 1997) P = {A

,. .. ,A

|P|

} ⊂ 2

of N are families of pairwise disjoint subsets called

blocks, i.e. A

∩ A

0,1 ≤ k < l ≤ |P|, with union

N = ∪

1≤k≤|P|

. Any P corresponds to the collection

{χ

: A ∈ P} of those |P| hypercube vertices identi-

ﬁed by the characteristic functions of its blocks (see

above). Partitions P can thus be seen as p ∈ ∆

where

= 1 for all A ∈ P,i ∈ A, i.e. exact fuzzy covers

where each i ∈ N concentrates its whole membershisp

on a unique A ∈ 2

, thus justifying the following.

Deﬁnition 5. Fuzzy partitions are exact fuzzy covers.

Ojective function W : ∆

→ R includes among

its extremizers (non-fuzzy) partitions. This expands

a result from pseudo-Boolean optimization. Denote

by ex(∆

) the 2

n−1

-set of extreme points of ∆

. For

q ∈ ∆

,i ∈ N, let q = q

−i

, with q

∈ ∆

as well as

−i

∈ ∆

N\i

= ×

j∈N\i

∆

. Then, for any w,

W (q) =

∑

A∈2

) +

∑

∈2

) =

∑

A∈2

∑

B⊆A\i

∏

j∈B



(B ∪ i) + µ

(B)



∑

∈2

∑

⊆A

∏

∈B

)

at all q ∈ ∆

and for all i ∈ N. Now deﬁne

−i

) =

∑

A∈2

∑

B⊆A\i

∏

j∈B

(B ∪ i)

−i

) =

∑

A∈2

∑

B⊆A\i

∏

j∈B

(B)

∑

∈2

∑

⊆A

∏

∈B

)

yielding W (q) = W

−i

) +W

−i

). (3)

Proprosition 6. For all q ∈ ∆

, there are q,q ∈ ∆

such that



(i) W (q) ≤ W (q) ≤ W (q) and,

(ii) q

∈ ex(∆

) for all i ∈ N.

Proof. For i ∈ N, q

−i

∈ ∆

N\i

, deﬁne w

−i

: 2

→ R by

−i

(A) =

∑

B⊆A\i

∏

j∈B

(B ∪ i). (4)

Let A

−i

= argmax w

−i

and A

−

−i

= argmin w

−i

with A

−i

0 6= A

−

−i

at all q

−i

. Most importantly,

) =

∑

A∈2



· w

−i

(A)



= hq

−i

i, (5)

ICPRAM 2016 - International Conference on Pattern Recognition Applications and Methods

where h·, ·i denotes scalar product. Thus for given

membership distributions of all j ∈ N\i, global worth

is affected by i’s membership distribution through a

scalar product. In order to maximize (or minimize)

W by suitably choosing q

for given q

−i

, the whole

of i’s membership mass must be placed over A

−i

(or

−

−i

), anyhow. Hence there are precisely |A

−i

| > 0

(or |A

−

−i

| > 0) available extreme points of ∆

. The

following procedure selects (arbitrarily) one of them.

ROUNDUP(w,q)

Initialize: Set t = 0 and q(0) = q.

Loop: While there is a i ∈ N with q

(t) 6∈ ex(∆

set t = t + 1 and:

(a) select some A

∗

∈ A

−i

(t)

(b) deﬁne, for all j ∈ N,A ∈ 2

(t) =







(t − 1) if j 6= i

1 if j = i and A = A

∗

0 otherwise

Output: Set q = q(t).

Every change q

(t − 1) 6= q

(t) = 1 (for any

i ∈ N,A ∈ 2

) induces a non-decreasing variation

W (q(t)) −W(q(t − 1)) ≥ 0. Hence, the sought q is

provided in at most n iterations. Analogously, replac-

ing A

−i

with A

−

−i

yields the sought minimizer q (see

also (Boros and Hammer, 2002, p. 163)).

Remark 7. For i ∈ N, A ∈ 2

, if all j ∈ A\i 6=

0 sat-

isfy q

= 1, then (4) yields w

−i

(A) = w(A) − w(A\i),

while w

−i

({i}) = w({i}) regardless of q

−i

Corollary 8. Some partition P satisﬁes W (p) ≥ W (q)

for all q ∈ ∆

, with W (p) =

∑

A∈P

w(A).

Proof. Follows from propositions 4 and 6, with the

above notation associating p ∈ ∆

to partition P.

Deﬁning global maximizers is clearly immediate.

Deﬁnition 9. Fuzzy partition

q ∈ ∆

is a global max-

imizer if W (

q) ≥ W (q) for all q ∈ ∆

Concerning local maximizers, consider a vector

ω = (ω

,. .. ,ω

) ∈ R

of strictly positive weights,

with ω

∑

j∈N

, and focus on the (Nash) equilib-

rium (Mas-Colell et al., 1995) of the game with ele-

ments i ∈ N as players, each strategically choosing its

membership distribution q

∈ ∆

while being rewarded

with fraction

W (q

,. .. ,q

) of the global worth at-

tained at any strategy proﬁle (q

,. .. ,q

) = q ∈ ∆

Deﬁnition 10. Fuzzy partition

q ∈ ∆

is a local max-

imizer if for all q

∈ ∆

and all i ∈ N inequality

( ˆq

−i

) ≥ W

−i

) holds (see (3)).

This deﬁnition of local maximizer entails that the

neighborhood N (q) ⊂ ∆

of any q ∈ ∆

N (q) =

[

i∈N

q :

q = ˜q

−i

, ˜q

∈ ∆

Deﬁnition 11. The (i,A)-derivative of W at q ∈ ∆

∂W (q)/∂q

= W (q(i, A)) −W (q(i,A)) =

= W



(i,A)|q

−i

(i,A)



−W



(i,A)|q

−i

(i,A)



with q(i,A) =



(i,A), .. ., q

(i,A)



given by

(i,A) =







for all j ∈ N\i, B ∈ 2

1 for j = i, B = A

0 for j = i, B 6= A

and q(i,A) =



(i,A), .. ., q

(i,A)



given by

(i,A) =



for all j ∈ N\i, B ∈ 2

0 for j = i and all B ∈ 2

thus ∇W (q) = {∂W (q)/∂q

: i ∈ N,A ∈ 2

} ∈ R

n−1

is the (full) gradient of W at q. The i-gradient

∇

W (q) ∈ R

n−1

of W at q = q

−i

is set function

∇

W (q) : 2

→ R deﬁned by ∇

W (q)(A) = w

−i

(A)

for all A ∈ 2

, where w

−i

is given by (4).

Remark 12. Membership distribution q

(i,A) is the

null one: its 2

n−1

entries are all 0, hence q

(i,A) 6∈ ∆

The setting obtained thus far allows to conceive

searching for a local maximizer partition p

∗

from

given fuzzy partition q as initial candidate solution,

and while maintaing the whole search within the con-

tinuum of fuzzy partitions. This idea may be speci-

ﬁed in alternative ways yielding different local search

methods. One possibility is the following.

LOCALSEARCH(w,q)

Initialize: Set t = 0 and q(0) = q, with require-

ment |{i : q

> 0}| ∈ {0,|A|} for all A ∈ 2

Loop 1: While 0 <

∑

i∈A

(t) < |A| for a A ∈ 2

set t = t + 1 and

(a) select a A

∗

(t) ∈ 2

such that

∑

i∈A

∗

(t)

−i

(t−1)

∗

(t)) ≥

∑

j∈B

− j

(t−1)

(B)

for all B ∈ 2

such that 0 <

∑

i∈B

(t) < |B|,

(b) for i ∈ A

∗

(t) and A ∈ 2

, deﬁne

(t) =



1 if A = A

∗

(t),

0 if A 6= A

∗

(t),

Continuous Set Packing and Near-Boolean Functions

∗

(t) and A ∈ 2

with A ∩ A

∗

(t) =

deﬁne q

(t) = q

(t − 1)+







w(A)

∑

B∈2

B∩A

∗

(t)6=

(t − 1)













∑

∈2

∩A

∗

(t)=

w(B

)







−1

(d) for j ∈ N\A

∗

(t) and A ∈ 2

with A ∩ A

∗

(t) 6=

deﬁne

(t) = 0.

Loop 2: While q

(t) = 1,|A| > 1 for a i ∈ N and

w(A) < w({i}) + w(A\i), set t = t + 1 and deﬁne:

(t) =



1 if |

A| = 1

0 otherwise

for all

A ∈ 2

(t) =



1 if B = A\i

0 otherwise

for all j ∈ A\i, B ∈ 2

(t) = q

(t − 1) for all j

∈ A

B ∈ 2

Output: Set q

∗

= q(t).

Both ROUNDUP above and LOCALSEARCH yield

a sequence q(0), .. ., q(t

∗

) = q

∗

where q

∗

∈ ex(∆

) for

all i ∈ N. In the former at the end of each iteration t

the novel q(t) ∈ N (q(t − 1)) is in the neighborhood

of its predecessor. In the latter q(t) 6∈ N (q(t − 1))

in general, as in |P| ≤ n iterations of Loop 1 a parti-

tion {A

∗

(1),. .. ,A

∗

(|P|)} = P is generated. Selected

subsets A

∗

(t) ∈ 2

,t = 1,. .. ,|P| are any of those

where the sum over members i ∈ A

∗

(t) of (i, A

∗

(t))-

derivatives ∂W (q(t − 1))/∂q

∗

(t)

(t − 1) is maximal.

Once a block A

∗

(t) is selected, then lines (c) and (d)

make all elements j ∈ N\A

∗

(t) redistribute the entire

membership mass currently placed on subsets A

∈ 2

with non-empty intersection A

∩A

∗

(t) 6=

0 over those

remaining A ∈ 2

such that, conversely, A∩A

∗

(t) =

The redistribution is such that each of these latter gets

a fraction w(A)/

∑

B∈2

:B∩A

∗

(t)=

w(B) of the newly

freed membership mass

∑

∈2

∩A

∗

(t)6=

(t − 1).

The subsequent Loop 2 checks whether the partition

generated by Loop 1 may be improved by exctract-

ing some elements from existing blocks and putting

them in singleton blocks of the ﬁnal output. In the

limit, set function w may be such that for some el-

ement i ∈ N global worth decreases when the ele-

ment joins any subset A ∈ 2

,|A| > 1, that is to say

w(A)−w(A\i)−w({i}) =

∑

B∈2

A\i

:|B|>1

(B) < 0.

Proprosition 13. LOCALSEARCH(W,q) outputs a

local maximizer q

∗

Proof. It is plain that the output is a partition P or,

with the notation of corollary 8 above, q

∗

= p. Ac-

cordingly, any element i ∈ N is either in a single-

ton block {i} ∈ P or else in a block A ∈ P,i ∈ A

such that |A| > 1. In the former case, any mem-

bership reallocation deviating from p

{i}

= 1, given

memberships p

, j ∈ N\i, yields a cover (fuzzy or

not) where global worth is the same as at p, be-

cause

∏

j∈B\i

= 0 for all B ∈ 2

\{i} (see exam-

ple 2 above). In the latter case, any membership re-

allocation q

deviating from p

= 1 (given member-

hips p

, j ∈ N\i) yields a cover which is best seen by

distinguishing between 2

\A and A. Also recall that

w(A) − w(A\i) =

∑

B∈2

A\i

(B). Again, all mem-

bership mass

∑

B∈2

> 0 simply collapses on sin-

gleton {i} because

∏

j∈B\i

= 0 for all B ∈ 2

\A.

Hence W (p) −W (q

−i

) = w(A) − w({i})+

−





∑

B∈2

A\i

:|B|>1

(B) +

∑

∈2

A\i

)







− q



∑

B∈2

A\i

:|B|>1

(B).

Now assume that q is not a local maximizer, i.e.

W (p) − W (q

−i

) < 0. Since p

− q

> 0 (because

= 1 and q

∈ ∆

is a deviation from p

), then

∑

B∈2

A\i

:|B|>1

(B) = w(A) − w(A\i) −w({i}) < 0.

Hence q cannot be the output of Second Loop.

In local search methods, the chosen initial can-

ditate solution determines what neighborhoods shall

be visited. The range of the objective function in a

neighborhood is a set of real values. In a neighbor-

hood N (p) of a p ∈ ∆

or partition P only those

∑

A∈P:|A|>1

|A| elements i ∈ A in non-sigleton blocks

A ∈ P,|A| > 1 can modify global worth by reallocating

their membership. In view of (the proof of) proposi-

tion 13, the only admissible variations obtain by de-

viating from p

= 1 with an alternative membership

distribution q

∈ [0,1), with W (q

−i

) −W (p) equal

to (q

− 1)

∑

B∈2

A\i

,|B|>1

(B) + (1 − q

)w({i}).

Hence, choosing partitions as initial candidate solu-

tions of LOCALSEARCH is evidently poor. A sen-

sible choice should conversely allow the search to

explore different neighborhoods where the objective

function may range widely. A simplest example of

such an initial candidate solution is q

= 2

1−n

for all

A ∈ 2

and all i ∈ N, i.e. the uniform distribution.

On the other hand, the input of local search algo-

rithms is commonly desired to be close to a global

optimum, i.e. a maximizer in the present setting.

ICPRAM 2016 - International Conference on Pattern Recognition Applications and Methods

This translates here into the idea of deﬁning the in-

put by means of set function w. In this view, consider

= w(A)/

∑

B∈2

w(B), yielding

w(A)

w(B)

for

all A,B ∈ 2

∩ 2

and all i, j ∈ N (see lines (c), (d)).

With a suitable initial candidate solution, the

search may be restricted to explore only a maximum

number of fuzzy partitions, thereby containing the

computational burden. In particular, if q(0) is the

ﬁnest partition {{1},...,{n}} or q

{i}

(0) = 1 for all

i ∈ N, then the search does not explore any neigh-

borhood at all, and such an input coincides with the

output. More reasonably, let A

max

= {A

,. .. ,A

} de-

note the collection of ⊇-maximal subsets where input

memberships are strictly positive. That is, q

> 0

for all i ∈ A

,1 ≤ k

≤ k as well as q

= 0 for all

B ∈ 2



∪ ··· ∪ 2



and all j ∈ B. Then, the out-

put shall be a partition P each of whose blocks A ∈ P

satisﬁes A ⊆ A

for some 1 ≤ k

≤ k. Hence, by suit-

ably choosing the input q, LOCALSEARCH outputs

a partition with no less than some maximum desired

number k(q) blocks.

3 LOWER-DIMENSIONAL CASE

If F ⊂ 2

, then 2

∩ F 6=

0 for every i ∈ N, other-

wise the problem reduces to packing the proper sub-

set N\{i : F ∩ 2

0} of elements contained in at

least one feasible subset. As outlined in section 1,

without additional notation simply let {

0} ∈ F 3 {i}

for all i ∈ N with null weigths w(

0) = 0 = w({i})

if {i} /∈ F . Thus (F ,⊇) is a poset (partially or-

dered set) with bottom element

0, and weight func-

tion w : F → R

has well-deﬁned M

obius inversion

: F → R (Rota, 1964b). Memberships q

distribute

over F

= 2

∩ F = {A

,. .. ,A

}, with lower(|F

|)-

dimensional unit simplices

∆









,. .. ,q



∈ R

∑

1≤k≤A

= 1







and corresponding fuzzy covers q ∈

∆

= ×

1≤i≤n

∆

Note that a fuzzy cover now may maximally con-

sist of |F | − 1 points in the unit n-dimensional hy-

percube [0,1]

. Accordingly, hypercube [0,1]

is re-

placed with C (F ) = co({χ

: A ∈ F }) ⊆ [0,1]

, i.e.

the convex hull of feasible characteristic functions, re-

garded as n-vectors (Gr

unbaum, 2001). Recursively

(with w(

0) = 0), M

obius inversion µ

: F → R is

(A) = w(A) −

∑

B∈F :B⊂A

(B),

while the MLE f

: C (F ) → R of w is

) =

∑

B∈F ∩2

∏

i∈B

(B).

Therefore, every fuzzy cover q ∈

∆

has global worth

W (q) =

∑

A∈F

∑

B∈F ∩2

∏

i∈B

(B).

For all i ∈ N,q

∈

∆

, and q

−i

∈

∆

N\i

= ×

j∈N\i

∆

−i

) =

∑

A∈F

∑

B∈F ∩2

A\i

∏

j∈B

(B)

∑

∈F \F





∑

∈F ∩2

∏

∈B

)





−i

) =

∑

A∈F

∑

B∈F

∩2

∏

j∈B\i

(B)

yielding again

W (q) = W

−i

) +W

−i

). (6)

From (4) above, w

−i

: F

→ R now is

−i

(A) =

∑

B∈F

∩2

∏

j∈B\i

(B) (7)

for all i ∈ N, all A ∈ F

and all q

−i

∈

∆

N\i

For each i ∈ N, denote by ex(

∆

) the set of |F

extreme points of simplex

∆

. Like in the full-

dimensional case, at any fuzzy cover

q ∈

∆

every

i ∈ N such that ˆq

6∈ ex(

∆

) may deviate by concen-

trating its whole membership on some A ∈ F

such

that w

−i

(A) ≥ w

−i

(B) for all B ∈ F

. This yields a

non-decreasing variation W (q

−i

) ≥ W (

q) in global

worth, with q

∈ ex(

∆

). When all n elements do

so, one after the other while updating w

−i

(t)

as in

ROUNDUP above, i.e. t = 0, 1,. .., then the ﬁnal

q = (q

,. .. ,q

) satisﬁes q ∈ ×

i∈N

ex(

∆

). Yet cases

F ⊂ 2

and F = 2

are different in terms of exact-

ness. Speciﬁcally, consider any

0 6= A ∈ F such that

|{i : q

= 1}| 6∈ {0,|A|} or A

= {i : q

= 1} ⊂ A, with

) =

∑

B∈F ∩2

(B). Then, F ∩ 2

is likely

to admit no shrinking (see above) yielding an exact

fuzzy cover with same global worth as (non-exact) q.

Proprosition 14. The values taken on exact fuzzy

covers do not saturate the range of W :

∆

→ R

Continuous Set Packing and Near-Boolean Functions

Proof. Consider this example: N = {1, 2,3, 4}

and F = {N,{4}, {1,2}, {1,3}, {2,3}}, with worth

w(N) = 3, w({4}) = 2, w({i, j}) = 1 for 1 ≤ i < j ≤ 3.

Deﬁne q = (q

,. .. ,q

) by q

{4}

= 1 = q

,i = 1,2,3,

with non-exactness |{i : q

> 0}| = 3 < 4 = |N|. As

W (q) = w({4}) +

∑

1≤i< j≤3

w({i, j}) = 2 +1 + 1 + 1

and A

= {1,2,3}, for all distributions ˆq

, ˆq

placing membership only over feasible B ∈ F ∩ 2

global worth is W ( ˆq

, ˆq

) < W (q).

This observation simply indicates that an arbitrary

search for optimal fuzzy covers may yield a maxi-

mizer (global or local) which is not reducible to any

feasible solution of the original set packing problem.

On the other hand, such feasible solutions are parti-

tions P all of whose blocks are feasible, and where

singleton blocks with worth 0 are not included in the

packing. In fact, similarly to the full-dimensional

case, fairly simple conditions may be shown to be suf-

ﬁcient for a partition to be a local maximizer.

Deﬁnition 15. Any ˆq

−i

q ∈

∆

is a local maxi-

mizer of W :

∆

→ R

if W

( ˆq

−i

) ≥ W

−i

) for

all i ∈ N and all q

∈

∆

(see (6) above).

The neighborhood N (q) ⊂

∆

of q ∈

∆

thus is

N (

q) =

[

i∈N

q : q = q

−i

∈

∆

Any partition P with A ∈ F for each block A ∈ P

clearly has associated p such that p ∈ ×

i∈N

ex(

∆

) ⊂

∆

Proprosition 16. Any partition P with associated p

such that p ∈

∆

is a local maximizer if for all A ∈ P

w(A) ≥ w({i}) +

∑

B∈F ∩2

A\i

(

B).

Proof. Firstly note that for all blocks A ∈ P, if any,

such that |A| = 1 there is nothing to prove, as the

summation reduces to w(

0) = 0, and thus there only

remains w({i}) ≥ w({i}). Accordingly, let A ∈ P

and |A| > 1. For every i ∈ A, any membership re-

allocation q

∈

∆

deviating from p

(i.e. p

= 1),

given memberships p

−i

of other elements j ∈ N\i (i.e.

∆

3 p

= 1 for all A

∈ P and all j ∈ A

), yields

q = (q

−i

) ∈

∆

which is best analyzed by distin-

guishing between F

\A and A. In particular,

w(A) = w({i}) +

∑

B∈F

∩2

|B|>1

(B) +

∑

B∈F ∩2

A\i

(

B).

All membership mass

∑

B∈F

> 0 collapses on

singleton {i}, because

∏

∈B\i

= 0 for all B ∈ F

by the deﬁnition of p

−i

(see example 2 above). Thus,

W (p) −W (q

−i

) = w(A) − w({i})+

−







∑

B∈F

∩2

|B|>1

(B) +

∑

B∈F ∩2

A\i

(









− q



∑

B∈F

∩2

|B|>1

(B).

Now assume that p is not a local maximizer, i.e.

W (p) −W(q

−i

) < 0. Since p

− q

> 0 because

= 1 and q

∈

∆

is a deviation from p

, then

∑

B∈F

∩2

|B|>1

(B) = w(A) − w({i}) −

∑

B∈F ∩2

A\i

(

B) < 0

must hold. This contradicts precisely the premise

w(A) ≥ w({i}) +

∑

B∈F ∩2

A\i

(

B) for all A ∈ P and

i ∈ A, thus completing the proof.

4 LOCAL SEARCH WITH COST

In order to design a gradient-based local search for

this lower-dimensional case, the only tool still miss-

ing is the derivative, which clearly shall reproduce

deﬁnition 11 above with F

in place of 2

. Before

that, as outlined in section 1, let c : F → N count the

number c(A) = |{B : B ∈ F ,B ∩ A 6=

0}| of feasible

subsets with which each A ∈ F has non-empty inter-

section, itself included, i.e. c(A) ∈ {1, .. ., |F |} is the

cost of including A in the packing. Accordingly, the

underlying poset function ˆw : F → R

now used (still

taking positive values only) incorporates both weights

w(A),A ∈ F (used thus far) and costs by means of ra-

tio ˆw(A) =

w(A)

c(A)

. The result is quasi-objective function

W :

∆

→ R

, obtained via MLE f

ˆw

: C (F ) → R

ˆw, i.e.

W (q) =

∑

A∈F

ˆw

), and all of the above ap-

plies invariately simply replacing W with

W .

Deﬁnition 17. The (i,A)-derivative of

W at q ∈

∆

A ∈ F

, is ∂

W (q)/∂q

W (q(i,A)) −

W (q(i,A)) =



(i,A)|q

−i

(i,A)



−



(i,A)|q

−i

(i,A)



with q(i,A) =



(i,A), .. ., q

(i,A)



given by

(i,A) =







for all j ∈ N\i, B ∈ F

1 for j = i, B = A

0 for j = i, B 6= A

and q(i,A) =



(i,A), .. ., q

(i,A)



given by

(i,A) =



for all j ∈ N\i, B ∈ F

0 for j = i and all B ∈ F

ICPRAM 2016 - International Conference on Pattern Recognition Applications and Methods

The (full) gradient of

W at q ∈

∆

∇

W (q) =



∂

W (q)/∂q

: i ∈ N, A ∈ F



∈ R

∑

i∈N

as well as the i-gradient ∇

W (q) ∈ R

W at

q = (q

−i

) ∈

∆

is poset function ∇

W (q) : F

→ R

deﬁned by ∇

W (q)(A) = ˆw

−i

(A) for all A ∈ F

, where

ˆw

−i

is given by (7) with ˆw in place of w. Again, mem-

bership distribution q

(i,A) is the null one: its |F

entries are all 0, hence q

(i,A) 6∈

∆

LS-WITHCOST( ˆw, q)

Initialize: Set t = 0 and q(0) = q, with require-

ment |{i : q

> 0}| ∈ {0,|A|} for all A ∈ F , ˆw(A) > 0.

Loop 1: While 0 <

∑

i∈A

(t) < |A| for a A ∈ F ,

set t = t + 1 and:

(a) select a A

∗

(t) ∈ F such that

min

i∈A

∗

(t)

ˆw

−i

(t−1)

∗

(t)) ≥ min

j∈B

ˆw

− j

(t−1)

(B)

for all B ∈ 2

such that 0 <

∑

i∈B

(t) < |B|,

(b) for i ∈ A

∗

(t) and A ∈ F

, deﬁne

(t) =



1 if A = A

∗

(t),

0 if A 6= A

∗

(t),

∗

(t) and A ∈ F

with A ∩ A

∗

(t) =

deﬁne q

(t) = q

(t − 1)+







ˆw(A)

∑

B∈F

B∩A

∗

(t)6=

(t − 1)













∑

∈F

∩A

∗

(t)=

ˆw(B

)







−1

(d) for j ∈ N\A

∗

(t) and A ∈ F

with A ∩ A

∗

(t) 6=

deﬁne

(t) = 0.

(e) for A ∈ F with A∩A

∗

(t) =

0, update cost function

by c(A) = |{B : B ∈ F ,B∩A 6=

0 = B∩A

∗

(t)}| and

plug it into ˆw.

Loop 2: While q

(t) = 1, |A| > 1 for a i ∈ N and

w(A) < w({i}) +

∑

B∈F ∩2

A\i

(

B),

set t = t + 1 and deﬁne:

(t) =



1 if |

A| = 1

0 otherwise

for all

A ∈ F

(t) =



1 if B = A\i

0 otherwise

for all j ∈ A\i, B ∈ F

(t) = q

(t − 1) for all j

∈ A

B ∈ F

Output: Set q

∗

= q(t).

Both LOCALSEARCH and LS-WITHCOST gen-

erate in |P| ≤ n iterations of Loop 1 a partition

∗

(1),. .. ,A

∗

(|P|)} = P. Now selected blocks

∗

(t) ∈ F , 1 ≤ t ≤ |P| are any of those feasible

subsets where the minimum over elements i ∈ A

∗

(t)

of (i,A

∗

(t))-derivatives ∂

W (q(t − 1))/∂q

∗

(t)

(t − 1)

is maximal. The following Loop 2 again checks

whether the partition generated by Loop 1 may be

improved by exctracting some elements from exist-

ing blocks and putting them in singleton blocks of the

ﬁnal output, which thus allows for the following.

Proprosition 18. LS-WITHCOST(W,q) outputs a lo-

cal maximizer q

∗

Proof. Follows from proposition 16 since Loop 2

deals with w, not with ˆw.

Concerning input q = q(0), consider again setting

ˆw(A)

∑

B∈F

ˆw(B)

for all A ∈ F

,i ∈ N, which entails

w(A)c(B)

w(B)c(A)

for all A,B ∈ F

∩ F

,i, j ∈ N.

Evidently, Loop 1 may take exactly the same form

as in LOCALSEARCH, that is with selected blocks

∗

(t) ∈ F ,t = 1,...,|P| of the generated partition P

being any of those feasible subsets where the sum,

rather than the minimum, over elements i ∈ A

∗

(t)

of (i,A

∗

(t))-derivatives ∂

W (q(t − 1))/∂q

∗

(t)

(t − 1)

is maximal. This possibility may be useful in those

settings where set packing appears in its weighted

version, while using the minimum in place of the

sum may be interesting for k-uniform set packing

problems (see section 1). In fact, for the k-uniform

case M

obius inversion is µ

ˆw

(A) =

c(A)

if |A| = k and

ˆw

(A) = 0 if |A| ∈ {0, 1} for all A ∈ F (recall the con-

vention {

0} ∈ F 3 {i} for all i ∈ N), with the cost

function iteratively updated in line (e). It is also plain

that in k-uniform set packing Loop 2 is ineffective.

5 NEAR-BOOLEAN FUNCTIONS

Boolean functions (Crama and Hammer, 2011) pro-

vide key analytical tools and methods with a variety of

important applications. Beyond set packing problems

that here constitute the main benchmark, this section

further develops the full-dimensional case detailed in

section 2 with the aim to indicate additional oppor-

tunities obtained from expanding the standard frame-

work where pseudo-Boolean models are traditionally

exploited. Recall that Boolean functions of n vari-

ables have form f : {0, 1}

→ {0,1}, and constitute

Continuous Set Packing and Near-Boolean Functions

a subclass of pseudo-Boolean functions f : {0,1}

→

R, which in turn admit the unique MLE

f : [0,1]

→ R

over the whole n-dimensional unit hypercube exten-

sively employed thus far. The n variables thus range

each in the unit interval [0,1]. Such a setting is here

expanded by letting each variable i = 1,...,n range

in a 2

n−1

− 1-dimensional simplex ∆

, with the goal

to evaluate collections of fuzzy subsets of a n-set

through the MLE given by (1) and (2).

Deﬁnition 19. Near-Boolean functions of n variables

have form F : ×

1≤i≤n

ex(∆

) → R.

Following (Hammer and Holzman, 1992, p. 4),

denote by N = {1,...,n} the set of indices of vari-

ables (i.e., the ground set in previous sections).

As already observed, any pseudo-Boolean function

has a unique expression as a multilinear polynomial

f (x

,. .. ,x

) =

∑

A⊆N



∏

i∈A



in n variables (Boros

and Hammer, 2002, p. 162), since α

,A ∈ 2

is in fact

the M

obius inversion (Rota, 1964b) of a unique set

function w : 2

→ R such that w(A) = f (χ

), where

is the characteristic function deﬁned in section 2.

Deﬁnition 20. The MLE

F of near-Boolean

functions F has polynomial form

F : ×

1≤i≤n

∆

→ R

given by expression (2) in section 2, that is

F(q) =

∑

A∈2

∑

B⊇A

∏

i∈A

(A), with (see above)

q = (q

,. .. ,q

) and q

= (q

,. .. ,q

n−1

) ∈ ∆

5.1 Bounded-degree LS

Approximations

In line with (Hammer and Holzman, 1992), the issue

of approximating a given near-Boolean function F by

means of the least squares LS criterion amounts to

determine a near-Boolean function F

such that

∑

q∈ ×

i∈N

ex(∆

)

[F(q) −F

(q)]

(8)

attains its minimum over all near-Boolean functions

with polynomial MLE

of degree k, that is

(q) =

∑

A∈2

|A|≤k

∑

B⊇A

∏

i∈A

(A),

or, equivalently stated in terms of the underlying set

function w, such that µ

(A) = 0 if |A| > k.

Near-Boolean functions F take their values on n-

product ×

i∈N

ex(∆

), and |ex(∆

)| = 2

n−1

for each i ∈ N.

They might thus be regarded as points F ∈ R

n(n−1)

a 2

n(n−1)

-dimensional vector space. In view of propo-

sition 4 formalizing exactness, this seems conceptu-

ally incorrect and with useless enumerative demand.

Speciﬁcally, for every partition P ∈ P

with associ-

ated p ∈ ×

i∈N

ex(∆

), there clearly exist many non-exact

q ∈ ×

i∈N

ex(∆

) such that F(q) = F(p) (see corollary 8

above). Counting these redundant extreme points of

simplices appears wothless. Hence k-degree approxi-

mation is to be dealt with by replacing expression (8)

with the following, applying to partitions p ↔ P only:

∑

p∈ ×

i∈N

ex(∆

)

with p↔P∈P

[F(p) −F

(p)]

. (9)

The number |P

| of partitions of a n-set is given by

Bell number B

(Rota, 1964a; Aigner, 1997). Ac-

cordingly, near-Boolean functions might be regarded

as points F ∈ R

in a B

-dimensional vector space.

Still, this also is far too large, as points in such a vec-

tor space correspond in fact to generic partition func-

tions, i.e. with M

obius inversion free to live on every

partition P ∈ P

. Conversely, near-Boolean functions

factually involve only partition functions h : P

→ R

such that h(P) = h

(P) =

∑

A∈P

w(A) for some set

function w : 2

→ R. The M

obius inversion of these

partition functions lives only on the 2

−n modular el-

ements (Stanley, 1971) of lattice (P

,∧, ∨), namely

on those partitions with a number of non-sigleton

blocks ≤ 1. When regarded as points in a vector space

(i.e. expressed as a linear combination of a basis, see

above) these functions may be seen as h

∈ R

−n

This is shown below via recursion through the M

obius

inversion of additively separable partition functions

(Gilboa and Lehrer, 1990; Gilboa and Lehrer, 1991).

It seems crucial emphasizing that while pseudo-

Boolean functions admit a unique set function pro-

viding their best k-degree approximation, 0 ≤ k ≤ n

(Hammer and Holzman, 1992), every near-Boolean

function admits a continuum of set functions w deter-

mining their unique best k-degree approximation. In

particular, consider ﬁrst the linear (i.e. k = 1) case:

the issue is to ﬁnd a best (least squares) approxima-

tion F

of any given F. That is, the set function w

determining F

has to satisfy w(A) =

∑

i∈A

w({i}) for

all A ∈ 2

. Then,

(P) =

∑

A∈P

w(A) =

∑

A∈P

∑

i∈A

w({i}) = w(N)

for all P ∈ P

. Thus h

is a constant partition func-

tion, or a valuation (Aigner, 1997) of partition lat-

tice (P

,∧, ∨). Also, any further linear v : 2

→ R

ICPRAM 2016 - International Conference on Pattern Recognition Applications and Methods

such that v(N) = w(N) also satisﬁes h

(P) = h

(P)

for all P ∈ P

. In other terms, there is a contin-

uum of equivalent linear v 6= w such that h

= h

obtained each by distributing arbitrarily the whole of

w(N) over the n singletons {i} ∈ 2

. Cases k > 1

maintain this same feature: consider a set function w

such that µ

(A) 6= 0 for one or more (possibly all





)

subsets A ∈ 2

such that |A| = k. Now ﬁx arbitrarily n

values v({i}),i ∈ N with

∑

i∈N

w({i}) =

∑

i∈N

v({i}).

For all A ∈ 2

,|A| > 1 M

obius inversion µ

: 2

→ R

can always be determined uniquely via recursion by

v(A) +

∑

i∈A

v({i}) =

∑

B⊆A

(B) +

∑

i∈A

v({i}) =

= w(A) +

∑

i∈A

w({i}) =

∑

B⊆A

(B) +

∑

i∈A

w({i}).

If set function w additively separates partition func-

tion h, i.e. h = h

, and v

= w − v is a linear set

function, then v + v

also additively separates h, i.e.

= h

v+v

. Hence, there is a continuum of equivalent

set functions w and v + v

available for the sought k-

degree approximation F

, but still the B

values taken

by F

(more precisely, the 2

−n values taken by F

the modular elements of partition lattice (P

,∧, ∨))

are unique and independent from the chosen set func-

tion in the continuum of set functions w,v + v

avail-

able, each determining an equivalent MLE

F of F.

5.2 A Continuum of Polynomials

In this work, two lattices play a central role, namely

the Boolean lattice (2

,∩, ∪) of subsets of N ordered

by inclusion ⊇, and the geometric lattice (P

,∧, ∨)

of partitions of N ordered by coarsening > (Aigner,

1997; Stern, 1999). Both, of course, are posets (par-

tially ordered sets), and M

obius inversion applies to

any (locally ﬁnite) poset, provided a bottom element

exists (Rota, 1964b). The bottom subset is

0, while

the bottom partition P

⊥

= {{1},...,{n}} is the ﬁnest

one. For a lattice (L, ∧,∨) ordered by > and with

generic elements x,y,z ∈ L, any lattice function f :

L → R has M

obius inversion µ

: L → R given by

(x) =

∑

⊥

6y6x

(y,x) f (y), where x

⊥

is the bottom

element and µ

is the M

obius function, deﬁned recur-

sively on ordered pairs (y,x) ∈ L × L by µ

(y,x) =

−

∑

y6z<x

(z,x) if y < x (i.e. y 6 x and y 6= x) as

well as µ

(y,x) = 1 if y = x, while µ

(y,x) = 0 if

y 66 x. The M

obius function of the subset lattice im-

plicitly appears since the beginning of this work, and

is µ

(B,A) = (−1)

|A\B|

, with B ⊆ A. Concerning

the M

obius function of P

, given any two partitions

P,Q ∈ P

, if Q < P = {A

,. .. ,A

|P|

}, then for every

block A ∈ P there are blocks B

,. .. ,B

∈ Q such

that A = B

∪ ··· ∪ B

, with k

> 1 for at least one

A ∈ P. Segment [Q, P] = {P

: Q 6 P

6 P} is thus

isomorphic to product ×

A∈P

P (k

), where P (k) de-

notes the lattice of partitions of a k-set. Accordingly,

let m

= |{A : k

= k}| for k = 1,. .. ,n. Then (Rota,

1964b, pp. 359-360),

(Q,P) = (−1)

−n+

∑

1≤k≤n

∏

1<k<n

(k!)

k+1

If a partition function h : P

→ R admits a set func-

tion v : 2

→ R to satisfy h(P) =

∑

A∈P

v(A) for all

P ∈ P

, then it may be said to be additively separable

(Gilboa and Lehrer, 1990; Gilboa and Lehrer, 1991),

with the notation h = h

. As already outlined, any

such an additively separable partition function h = h

has M

obius inversion µ

that lives only on the mod-

ular elements of the partition lattice, i.e. where only

one block, at most, has cardinality > 1. That is to

say, together with the bottom P

⊥

and top P

= {N},

all other modular elements are those partitions of the

form {A} ∪ P

⊥

for A ∈ 2

such that 1 < |A| < n,

where P

⊥

is the ﬁnest partition of A

(Aigner, 1997,

Ex. 13, p. 71). The total number of these modular

partitions thus is 2

− n. The M

obius inversion of an

additively separable partition function h

is detailed

hereafter; see (Gilboa and Lehrer, 1990, Prop. 4.4, p.

138 and Appendix, p. 144) and (Gilboa and Lehrer,

1991, Prop. 3.3, p. 452).

Proprosition 21. If h = h

, then h = h

for a contin-

uuum of set functions w : 2

→ R,w 6= v.

Proof. Firstly, by direct substitution, for all P ∈ P

(P) =

∑

A∈P

∑

B⊆A

v(B)

∑

Q6P:B∈Q

(Q,P).

Secondly, if P 6= {B} ∪ P

⊥

, then the recursive deﬁni-

tion of M

obius function µ

yields

∑

Q6P:B∈Q

(Q,P) = 0.

obius inversion µ

thus takes non-zero values only

on modular elements, where it obtains recursively by

⊥

) =

∑

i∈N

v({i}), µ

) = µ

(N) as well as

({A} ∪ P

⊥

) = µ

(A) for 1 < |A| < n. In other

terms, any w 6= v satisfying

∑

i∈N

v({i}) =

∑

i∈N

w({i})

and µ

(A) = µ

(A) for all A ∈ 2

,|A| > 1 also addi-

tively separates h, i.e. h

= h

In view of corollary 8, the setting considered in

this work deals precisely with additively separable

partition functions, and thus the polynomial expres-

sion (2) in section 2 is not unique. Speciﬁcally, recall

that the degree of a polynomial is the highest degree

of its terms. Hence in (2), for any chosen set function

w additively separating partiton function h = h

, the

degree is max{|A| : µ

(A) 6= 0}, while every non-zero

Continuous Set Packing and Near-Boolean Functions

value of M

obius inversion µ

: 2

→ R is a coefﬁcient

of the polynomial. The only degree k such that there

is a unique set function available for polynomial ex-

pression (2) is k = 0, in which case the unique addi-

vitely separating set function w is trivial: w(A) = 0

for all A ∈ 2

. On the other hand, for any degree

k, 0 < k ≤ n there exists a continuum of set func-

tions available for additive separability and such that

max{|A| : µ

(A) 6= 0} = k, each deﬁning alternative

but equivalent coefﬁcients of the polynomial.

6 NEAR-BOOLEAN GAMES

In view of the above deﬁnition of local maximiz-

ers relying on equilibrium conditions for strategic n-

player games, and having mentioned additive separa-

blity of partition functions or global games (Gilboa

and Lehrer, 1990), it seems now useful to regard vari-

ables as players in near-Boolean coalition formation

games (for pseudo-Boolean functions and coalitional

games see (Hammer and Holzman, 1992, section 3)).

Deﬁnition 22. A near-Boolean n-player game is a

triple (N,F,π) such that N = {1,.. ., n} is the player

set and F is a near-Boolean function taking real

values on proﬁles q = (q

,. .. ,q

) ∈ ×

i∈N

ex(∆

) of

strategies, while π : ×

i∈N

ex(∆

) → R

efﬁciently as-

signs payoffs π(q) = (π

(q),. .. ,π

(q)) to players, i.e.

∑

i∈N

(q) = F(q) at all q ∈ ×

i∈N

ex(∆

Deﬁnition 23. A fuzzy near-Boolean n-player game is

a triple (N,

F, π) such that N = {1,...,n} is the player

set and

F is the MLE of a near-Boolean function tak-

ing values on proﬁles q = (q

,. .. ,q

) ∈ ×

i∈N

∆

, while

π : ×

i∈N

∆

→ R

efﬁciently assigns payoffs to players,

i.e.

∑

i∈N

(q) =

F(q) at all q ∈ ×

i∈N

∆

In both near-Boolean games and fuzzy ones the

player set is ﬁnite. Given this, a main distinction

is between games where players have either ﬁnite

or else inﬁnite sets of strategies, with near-Boolean

games in the former class and fuzzy ones in the lat-

ter. In addition, players may play either determinis-

tic (i.e. pure) or else random (i.e. mixed) strategies.

In the latter case equilibrium conditions are stated

in terms of expected payoffs, and by means of ﬁxed

point arguments for upper hemicontinuous correspon-

dences such conditions are commonly fulﬁlled (Mas-

Colell et al., 1995, p. 260). The sets of deterministic

strategies in fuzzy near-Boolean games are precisely

the sets of random strategies in near-Boolean games.

Nevertheless, the payoffs for the fuzzy setting are not,

in general, expectations.

The main framework where these games seem in-

teresting is coalition formation, which combines both

strategic and cooperative games. A generic strat-

egy proﬁle q ∈ ×

i∈N

ex(∆

) of near-Boolean (non-fuzzy)

games may well fail to be exact (see proposition 4),

but it seems plain that there is a unique partition P

of N with associated p ∈ ×

i∈N

ex(∆

) obtained through

shrinkings of q and such that F(p) = F(q). Let

p(q) be such a unique p. In view of the above dis-

cussion, it is also evident that for every p there are

many (non-exact) q such that p=p(q). In these terms,

near-Boolean games model stategic coalition forma-

tion in a very handy manner, in that they totally by-

pass the need to deﬁne a mechanism mapping strategy

proﬁles into partitions of players or coalition struc-

tures (Slikker, 2001). More precisely, a mechanism

is a mapping M : ×

i∈N

ex(∆

) → P

such that when

each player i ∈ N speciﬁes a coalition A

∈ 2

, then

M(A

,. .. ,A

) = P is a resulting coalition structure.

If the n speciﬁed coalitions A

,i ∈ N are such that

for some partition P it holds A

= A for all i ∈ A

and all A ∈ P, then M(A

,. .. ,A

) = P. Otherwise,

the generated partition P

= M(A

,. .. ,A

) depends

on what mechanism is chosen, and generally may be

a rather ﬁne one, i.e. possibly consisting of many

small blocks. Conversely, near-Boolean games do not

need any mechanism, in that even if players’ strate-

gies (q

,. .. ,q

) = q are such that q does not corre-

spond to a partition, still the global worth F(q) is that

attained at the partition P with corresponding p(q),

i.e. whose blocks A ∈ P each include maximal sub-

sets of players choosing the same superset A

⊇ A.

Given coalitional game v : 2

→ R

,v(

0) = 0,

with F(p) =

∑

A∈P

v(A) for all partitions P ↔ p, let

payoffs be deﬁned, for i ∈ A and A ∈ P and p = p(q),

by π

(q) =

∑

B∈2

A\i

(B)

|B|

for all q ∈ ×

j∈N

ex(∆

where P thus is the partition with associated p(q).

Apart from the absence of any coalition structure gen-

eration mechanism (see above), this is in fact a well-

known coalition formation game (Slikker, 2001), with

payoffs given by the Shapley value (Roth, 1988).

Deﬁnition 24. A local maximizer of near-Boolean

function F is any q ∈ ×

i∈N

ex(∆

) such that for all i ∈ N

and all q

∈ ex(∆

) inequality F(q) ≥ F(q

−i

) holds.

Remark 25. If payoffs are given by π

(q) =

F(q)

∑

j∈N

for all i ∈ N, with ω

,. .. ,ω

> 0, then near-Boolean

games are (pure) common interest potential games

(Monderer and Shapley, 1996; Bowles, 2004). That

is to say, the set of equilibria of (N,F,π) coincides

ICPRAM 2016 - International Conference on Pattern Recognition Applications and Methods

with the set of local maximizers of F, and players’

preferences all agree on the set of strategy proﬁles.

In artiﬁcial intelligence, these games may model

coalition formation in multiagent systems (Rahwan

and Jenning, 2007; Conitzer and Sandholm, 2006).

7 CONCLUSIONS AND FUTURE

WORK

Via polynomial MLE, near-Boolean functions of

n variables take values on the n-product of high-

dimensional unit simplices, thus enabling to approach

discrete optimization problems, namely set pack-

ing/partitioning, through an objective function de-

ﬁned over a continuous domain, and with feasible so-

lutions found at extreme points of the simplices. Least

squares approximations with polynomials of bounded

degree are discussed in terms of additive separabil-

ity of partition functions, while near-Boolean n-player

games ﬂexibly model strategic coalition formation.

Apart from settings in artiﬁcial intelligence such

as combinatorial auctions and coalition structure gen-

eration in multiagent systems discussed above, near-

Boolean optimization seems generally interesting for

objective function-based clustering (Rossi, 2015), and

for graph clustering in particular (Schaeffer, 2007).

Speciﬁcally, for a given weighted graph, edge weights

and vertices’ weighted degrees may be used (in alter-

native ways) to obtain a quadratic MLE in expression

(2), i.e. a polynomial with degree 2. Then, optimiza-

tion with respect to such an objective function pro-

vides a method for partitioning the vertex set. This de-

serves separate investigation in a forthcoming work.

Along an alternative route, through the core con-

cept in cooperative game theory, it may be interesting

to consider the following problem: for given generic

(i.e. non-additively separable) global game or parti-

tion function h ∈ R

, determine a quadratic MLE in

(2), i.e. a set function w such that µ

(A) = 0 for all

A ∈ 2

,|A| > 2, satisfying h

(P) ≥ h(P) for all parti-

tions P ∈ P

of players, where h

(P) =

∑

A∈P

w(A),

and with h

) = h(P

) for the coarsest partition.

The idea behind this problem comes from a novel ap-

proach to the solution of global games h (Gilboa and

Lehrer, 1990), and shall be explored in future work.

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ICPRAM 2016 - International Conference on Pattern Recognition Applications and Methods