We study the rank of the Sum of Squares (SoS) hierarchy over the Boolean hypercube for Symmetric Quadratic Functions (SQFs) in n variables with roots placed in points k - 1 and k. Functions of this type have played a central role in deepening the understanding of the performance of the SoS method for various unconstrained Boolean hypercube optimization problems, including the Max Cut problem. Recently, Lee, Prakash, de Wolf, and Yuen proved a lower bound on the SoS rank for SQFs of Ω(√k(n - k)) and conjectured the lower bound of Ω(n) by similarity to a polynomial representation of the n-bit OR function. Leveraging recent developments on Chebyshev polynomials, we refute the Lee-Prakash-de Wolf- Yuen conjecture and prove that the SoS rank for SQFs is at most O(√nk log(n)). We connect this result to two constrained Boolean hypercube optimization problems. First, we provide a degree O(√n) SoS certificate that matches the known SoS rank lower bound for an instance of Min Knapsack, a problem that was intensively studied in the literature. Second, we study an instance of the Set Cover problem for which Bienstock and Zuckerberg conjectured an SoS rank lower bound of n/4. We refute the Bienstock-Zuckerberg conjecture and provide a degree O(√n log(n)) SoS certificate for this problem.
We study the rank of the Sum of Squares (SoS) hierarchy over the Boolean hypercube for Symmetric Quadratic Functions (SQFs) in n variables with roots placed in points k - 1 and k. Functions of this type have played a central role in deepening the understanding of the performance of the SoS method for various unconstrained Boolean hypercube optimization problems, including the Max Cut problem. Recently, Lee, Prakash, de Wolf, and Yuen proved a lower bound on the SoS rank for SQFs of Ω(√k(n - k)) and conjectured the lower bound of Ω(n) by similarity to a polynomial representation of the n-bit OR function. Leveraging recent developments on Chebyshev polynomials, we refute the Lee-Prakash-de Wolf- Yuen conjecture and prove that the SoS rank for SQFs is at most O(√nk log(n)). We connect this result to two constrained Boolean hypercube optimization problems. First, we provide a degree O(√n) SoS certificate that matches the known SoS rank lower bound for an instance of Min Knapsack, a problem that was intensively studied in the literature. Second, we study an instance of the Set Cover problem for which Bienstock and Zuckerberg conjectured an SoS rank lower bound of n/4. We refute the Bienstock-Zuckerberg conjecture and provide a degree O(√n log(n)) SoS certificate for this problem.
We study how well functions over the boolean hypercube of the form fk(x)=(∣x∣−k)(∣x∣−k−1) can be approximated by sums of squares of low-degree polynomials, obtaining good bounds for the case of approximation in ℓ∞-norm as well as in ℓ1-norm. We describe three complexity-theoretic applications: (1) a proof that the recent breakthrough lower bound of Lee, Raghavendra, and Steurer on the positive semidefinite extension complexity of the correlation and TSP polytopes cannot be improved further by showing better sum-of-squares degree lower bounds on ℓ1-approximation of fk; (2) a proof that Grigoriev's lower bound on the degree of Positivstellensatz refutations for the knapsack problem is optimal, answering an open question from his work; (3) bounds on the query complexity of quantum algorithms whose expected output approximates such functions.
Chvátal--Gomory cuts (CG-cuts) and the Bienstock--Zuckerberg hierarchy capture useful linear programs that the standard bounded degree sum-of-squares (SoS) hierarchy fails to capture. In this paper we present a novel polynomial time SoS hierarchy for 0/1 problems with a custom subspace of high degree polynomials (not the standard subspace of low degree polynomials). We show that the new SoS hierarchy recovers the Bienstock--Zuckerberg hierarchy. Our result implies a linear program that reproduces the Bienstock--Zuckerberg hierarchy as a polynomial-sized, efficiently constructible extended formulation that satisfies all constant pitch inequalities. The construction is also very simple, and it is fully defined by giving the supporting polynomials. Moreover, for a class of polytopes (e.g., set cover and packing problems), the resulting SoS hierarchy optimizes in polynomial time over the polytope resulting from any constant rounds of CG-cuts, up to an arbitrarily small error in the solution value. Arguably, this is the first example where different basis functions can be useful in asymmetric situations to obtain a hierarchy of relaxations.
We analyze the sum-of-squares rank of unweighted instances of the Minimum Knapsack (MK) problem, i.e., minimization of for 0/1 variables under the constraint, with. Such instances have long served as a testbed for understanding the limitations of lift-And-project methods in Boolean optimization. For example, both the Lovász-Schrijver and Sherali-Adams hierarchies require (maximal) rank n to solve them, already when q = 1/2 is constant. The SOS hierarchy requires only sublinear rank to solve unweighted MK when q = 1/2. On the other hand, when q is allowed to vary with n, the SOS rank of the problem may become linear. Interestingly, this is known to happen both when q is large, and when q is very small (0 < q ≤ 2-n). This raises the question of whether we should think of hard instances of unweighted MK as being typical for the SOS hierarchy, or as a consequence of very specific choices of the threshold parameter q. In this paper, we address this question by showing new upper and lower bounds on the SOS rank of unweighted MK in the whole regime of the parameter q. For n-q ≤ O(1), we show that the SOS rank is constant. In contrast, when q ≤ O(1), a linear rank is needed if q is exponentially close to an integer. As our main positive result, we show that linear rank is very rare for q ≤ O(1). This can be expressed in the language of smoothed analysis: After perturbing q by a Gaussian with mean 0 and variance σ2, the expected SOS rank of MK is.
We analyze the sum-of-squares rank of unweighted instances of the Minimum Knapsack (MK) problem, i.e., minimization of ∑i=1nxi for 0/1 variables under the constraint ∑i=1nxi≥q, with q∈R. Such instances have long served as a testbed for understanding the limitations of lift-and-project methods in Boolean optimization. For example, both the Lovász–Schrijver and Sherali–Adams hierarchies require (maximal) rank n to solve them, already when q = 1/2 is constant. The SOS hierarchy requires only sublinear rank O(n) to solve unweighted MK when q = 1/2. On the other hand, when q is allowed to vary with n, the SOS rank of the problem may become linear. Interestingly, this is known to happen both when q is large, and when q is very small (0 < q ≤ 2− n). This raises the question of whether we should think of hard instances of unweighted MK as being typical for the SOS hierarchy, or as a consequence of very specific choices of the threshold parameter q.
We analyze the sum-of-squares rank of unweighted instances of the Minimum Knapsack (MK) problem, i.e., minimization of ∑i=1nxi for 0/1 variables under the constraint ∑i=1nxi≥q, with q∈R. Such instances have long served as a testbed for understanding the limitations of lift-and-project methods in Boolean optimization. For example, both the Lovász-Schrijver and Sherali-Adams hierarchies require (maximal) rank n to solve them, already when q=1/2 is constant. The SOS hierarchy requires only \emph{sublinear} rank O(n) to solve unweighted MK when q=1/2. On the other hand, when q is allowed to vary with~n, the SOS rank of the problem may become linear. Interestingly, this is known to happen both when q is large, and when q is very small ($0
We prove lower bounds on complexity measures, such as the approximate degree of a Boolean function and the approximate rank of a Boolean matrix, using quantum arguments. We prove these lower bounds using a quantum query algorithm for the combinatorial group testing problem. We show that for any function f, the approximate degree of computing the OR of n copies of f is Omega(sqrt n) times the approximate degree of f, which is optimal. No such general result was known prior to our work, and even the lower bound for the OR of ANDs function was only resolved in 2013. We then prove an analogous result in communication complexity, showing that the logarithm of the approximate rank (or more precisely, the approximate gamma-2 norm) of F: X x Y to 0,1 grows by a factor of Omega (sqrtn) when we take the OR of n copies of F, which is also essentially optimal. As a corollary, we give a new proof of Razborov's celebrated Omega(sqrtn) lower bound on the quantum communication complexity of the disjointness problem. Finally, we generalize both these results from composition with the OR function to composition with arbitrary symmetric functions, yielding nearly optimal lower bounds in this setting as well.