A Stronger LP Bound for Formula Size Lower Bounds via Clique Constraints
We introduce a new technique proving formula size lower bounds based on the linear programming bound originally introduced by Karchmer, Kushilevitz and Nisan [KKN95] and the theory of stable set polytope. We apply it to majority functions and prove their formula size lower bounds improved from the classical result of Khrapchenko [Khrapchenko71]. Moreover, we introduce a notion of unbalanced recursive ternary majority functions motivated by a decomposition theory of monotone self-dual functions and give integrally matching upper and lower bounds of their formula size. We also show monotone formula size lower bounds of balanced recursive ternary majority functions improved from the quantum adversary bound of Laplante, Lee and Szegedy [LLS06].
Key words and phrases:Computational and Structural Complexity
Kenya Ueno \@ifemptyUT
section1[Introduction]Introduction Proving formula size lower bounds is a fundamental problem in complexity theory and also an extremely tough problem to resolve. A super-polynomial lower bound of a function in implies . There are a lot of techniques to prove formula size lower bounds, e.g. [Hastad98, HLS07, KKN95, Khrapchenko71, Koutsoupias93, LLS06, Lee07]. Laptente, Lee and Szegedy [LLS06] introduced a technique based on the quantum adversary method [Ambainis02] and gave a comparison with known techniques. In particular, they showed that their technique subsumes several known techniques such as Khrapchenko [Khrapchenko71] and its extension [Koutsoupias93]. The current best formula size lower bound is by Håstad [Hastad98] and a key lemma used in the proof is also subsumed by the quantum adversary bound [LLS06]. Karchmer, Kushilevitz and Nisan [KKN95] introduced a technique proving formula size lower bounds called the linear programming (or LP) bound and showed that it cannot prove a lower bound larger than for non-monotone formula size in general. Lee [Lee07] proved that the LP bound [KKN95] subsumes the quantum adversary bound [LLS06] and Høyer, Lee and Špalek [HLS07] introduced a stronger version of the quantum adversary bound.
Motivated by the result of Lee [Lee07], we devise a stronger version of the LP bound by using an idea from the theory of stable set polytope, known as clique constraints [Padberg73]. Suggesting a stronger technique compared to the original LP bound [KKN95] has possibilities to improve the best formula size lower bound because it subsumes many techniques including the key lemma of Håstad [Hastad98]. Moreover, our technique has various possibilities of extensions such as rank constraints discussed in Section LABEL:brec_section and orthonormal constraints [GLS88], each of which subsume clique constraints. Due to this extendability, it is difficult to show the limitation of our new technique.
To study the relative strength of our technique, we apply it to some families of Boolean functions. For each family, we have distinct motivation to investigate their formula size. Three kinds of Boolean functions treated in this paper are defined as follows. All of them are called monotone self-dual Boolean functions defined in the next section.
A majority function outputs if the number of 1’s in the input bits is greater than or equal to and otherwise. We define unbalanced recursive ternary majority functions as
with . We also define balanced recursive ternary majority functions as
with . Through the paper, means the number of input bits. Formula size and monotone formula size of a Boolean function are denoted by and , respectively.
Although our improvements of lower bounds seem to be slight, it breaks a stiff barrier (known as the certificate complexity barrier [LLS06]) of previously known proof techniques. The best monotone upper and lower bounds of majority functions are [Valiant84] and [Rad97], respectively. In the non-monotone case, the best formula size upper and lower bounds of majority functions are [PPZ92] and ( when ), respectively, which can be proven by the classical result of Khrapchenko [Khrapchenko71]. In this paper, we slightly improve the non-monotone formula size lower bound while no previously known techniques has been able to improve it since 1971. In Section A Stronger LP Bound for Formula Size Lower Bounds via Clique Constraints, we will prove where . Here, denotes . Since formula size takes an integral value, it implies a lower bound.
It is known that the class of monotone self-dual Boolean functions is closed under compositions (equivalently, in so-called Post’s lattice [BCRV03, Post41]). Any monotone self-dual Boolean functions can be decomposed into compositions of 3-bit majority functions [IK93]. A key observation for our proofs is that a communication matrix (defined in the next section) of a monotone self-dual Boolean function contains those of the 3-bit majority function as its submatrices. Ibaraki and Kameda [IK93] developed a decomposition theory of monotone self-dual Boolean functions in the context of mutual exclusions in distributed systems. The theory has been further investigated by [BI95, BIM99]. Given a monotone self-dual Boolean function , we can decompose it as after decomposing into a conjunction of monotone self-dual functions . It holds in its internal structure. To determine its formula size is of particular interest because it is related with efficiency of the decomposition scheme. In Section LABEL:urec_section, we will prove .
Balanced recursive ternary majority functions have been studied in several contexts [JKS03, LLS06, MO03, O04, RS08, SW86], see [LLS06] and [RS08] for details. Ambainis et al. [ACRSZ07] showed a quantum algorithm which evaluates a monotone formula of size (or called AND-OR formula) in time even if it is not balanced. This result implies can be evaluated in time by the quantum algorithm because we have a formula size upper bound as noted in [LLS06]. Improving this result, Reichardt and Spalek [RS08] gave a quantum algorithm which evaluates in time. From this context, seeking the true bound of the monotone formula size of is a very interesting research question. The quantum adversary bound [LLS06] has a quite nice property written as . It directly implies a formula size lower bound . In Section LABEL:brec_section, we will prove and . This gives a slight improvement of the lower bound and means that the lower bound is at least not optimal in the monotone case.
section1[Preliminaries]Preliminaries We define a total order between the two Boolean values. For Boolean vectors and , we define if for all . A Boolean function is called monotone if implies for all . For a monotone Boolean function , a Boolean vector is called minterm if and ( implies for any and called maxterm if and ( implies for any . Sets of all minterms and maxterms of a monotone Boolean function are denoted by and , respectively. A Boolean function is called self-dual if where is the negation of . Remark that, if a Boolean function is self-dual, its communication matrix (see below) has some nice properties, e.g. .
A formula is a binary tree with leaves labeled by literals and internal nodes labeled by and . A literal is either a variable or the negation of a variable. A formula is called monotone if it does not have negations. It is known that all (monotone) Boolean functions can be represented by a (monotone) formula. The size of a formula is its number of leaves. We define the (monotone) formula size of a Boolean function as the size of the smallest formula computing .
Karchmer and Wigderson [KW90] characterize formula size of any Boolean function in terms of a communication game called the Karchmer-Wigderson game. In the game, given a Boolean function , Alice gets an input such that and Bob gets an input such that . The goal of the game is to find an index such that . They also characterize monotone formula size by a monotone version of the Karchmer-Wigderson game. In the monotone game, Alice gets a minterm and Bob gets a maxterm . The goal of the monotone game is to find an index such that and . The number of leaves in a best communication protocol for the (monotone) Karchmer-Wigderson game is equal to the (monotone) formula size of . From these characterizations, we consider communication matrices derived from the games.
Definition 0.2 (Communication Matrix).
Given a Boolean function , we define its communication matrix as a matrix whose rows and columns are indexed by and , respectively. Each cell of the matrix contains indices such that . In a monotone case, given a monotone Boolean function , we define its monotone communication matrix as a matrix whose rows and columns are indexed by and , respectively. Each cell of the matrix contains indices such that and . A combinatorial rectangle is a direct product where and . A combinatorial rectangle is called monochromatic if every cell contains the same index . We call a cell singleton if it contains just one index.
The minimum number of disjoint monochromatic rectangles which exactly cover all cells in the (monotone) communication matrix gives a lower bound for the number of leaves of a best communication protocol for the (monotone) Karchmer-Wigderson game. Thus, we obtain the following bound.
Theorem 0.3 (Rectangle Bound [Kw90]).
The minimum size of an exact cover by disjoint monochromatic rectangles for the communication matrix (or monotone communication matrix) associated with a Boolean function gives a lower bound of (or ).
section1[A Stronger Linear Programming Bound via Clique Constraints]A Stronger Linear Programming Bound via Clique Constraints In this study, we devise a new technique proving formula size lower bounds based on the LP bound [KKN95] with clique constraints. We assume that readers are familiar with the basics of the linear and integer programming theory. Karchmer, Kushilevitz and Nisan [KKN95] formulate the rectangle bound as an integer programming problem and give its LP relaxation. Given a (monotone) communication matrix, it can be written as such that for each cell in the matrix and for each monochromatic rectangle . The dual problem can be written as such that for each monochromatic rectangle . Here, each variable is indexed by a cell in the matrix. From the duality theorem, showing a feasible solution of the dual problem gives a formula size lower bound.
Now, we introduce our stronger LP bound using clique constraints from the theory of stable set polytope. We assume that each monochromatic rectangle is a node of a graph. We connect two nodes by an edge if the two corresponding monochromatic rectangles intersect. If a set of monochromatic rectangles compose a clique in the graph, we add a constraint to the primal problem of the LP relaxation. This constraint is valid for all integral solutions since we consider the disjoint cover problem. That is, we can assign the value 1 to at most 1 rectangle in a clique for all integral solutions under the condition of disjointness. The dual problem can be written as such that for each monochromatic rectangle and for each clique . Intuitively, this formulation can be interpreted as follows. Each cell is assigned a weight . The summation of weights over all cells in a monochromatic rectangle is limited to 1. This limit is relaxed by 1 if it is contained by a clique. Thus, the limit of the total weight for a monochromatic rectangle contained by distinct cliques is .
By using clique constraints, we obtain the following matching lower bound for the formula size of the 3-bit majority function while the original LP bound cannot prove a lower bound larger than 4.5. In our proofs, we utilize the following property of combinatorial rectangles which is trivial from the definition. If a rectangle contains two cells and , it also contains both and . A notion of singleton cells also occupies an important role for our proofs because there are no monochromatic rectangles which contain different kinds of singleton cells.
We have a monotone formula for . From the definition, . To prove , we consider a communication matrix of the 3-bit majority function whose rows and columns are restricted to minterms and maxterms, respectively.
In the dual problem, we assign weights 1 for all singleton cells and 0 for other cells. There are 6 singleton cells and hence the total weight is 6. We take a clique composed of monochromatic rectangles containing two singleton cells. It is clear that every pair of monochromatic rectangles contained by intersect at some cell. We assign . Then, the objective function of the dual problem becomes .
Now, we show that all constraints of the dual problem are satisfied. First, we consider a monochromatic rectangle which contains at most one singleton cell. In this case, the constraint is clearly satisfied because the summation of weights in the monochromatic rectangle is less than or equal to 1. Then, we consider a monochromatic rectangle which contains two singleton cells. In this case, the summation of weights in the monochromatic rectangle is 2. However, it is contained by the clique . It implies that the limit of the total weight is relaxed by 1. Thus, the constraint is satisfied. There are no monochromatic rectangles which contain more than 3 singleton cells because a rectangle which contains more than two kinds of singleton cells is not monochromatic.
section1[Formula Size of Majority Functions]Formula Size of Majority Functions In this section, we show a non-monotone formula size lower bound of majority functions improved from the classical result of Khrapchenko [Khrapchenko71].