arXiv subjects

Monaldo Mastrolilli

Publications and source records attributed to Monaldo Mastrolilli.

At least 19 recordsLinked to original sources

arXiv2025

On the Bit Size of Sum-of-Squares Proofs for Symmetric Formulations

The Sum-of-Squares (SoS) hierarchy is a powerful framework for polynomial optimization and proof complexity, offering tight semidefinite relaxations that capture many classical algorithms. Despite its broad applicability, several works have revealed fundamental limitations to SoS automatability. (i) While low-degree SoS proofs are often desirable for tractability, recent works have revealed they may require coefficients of prohibitively large bit size, rendering them computationally infeasible. (ii) Prior works have shown that SoS proofs for seemingly easy problems require high-degree. In particular, this phenomenon also arises in highly symmetric problems. Instances of symmetric problems-particularly those with a small number of constraints-have repeatedly served as benchmarks for establishing high-degree lower bounds in the SoS hierarchy. It has remained unclear whether symmetry can also lead to large bit sizes in SoS proofs, potentially making low-degree proofs computationally infeasible even in symmetric settings. In this work, we resolve this question by proving that symmetry alone does not lead to large bit size SoS proofs. Focusing on symmetric Archimedean instances, we show that low-degree SoS proofs for such systems admit compact, low bit size representations. Together, these results provide a conceptual separation between two sources of SoS hardness-degree and bit size-by showing they do not necessarily align, even in highly symmetric instances. This insight guides future work on automatability and lower bounds: symmetry may necessitate high-degree proofs, but it does not by itself force large coefficients.

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arXiv2025

The Integrality Gap of the Traveling Salesman Problem is $4/3$ if the LP Solution Has at Most $n+6$ Non-zero Components

We address the classical Dantzig - Fulkerson - Johnson formulation of the symmetric metric Traveling Salesman Problem and study the integrality gap of its linear relaxation, namely the Subtour Elimination Problem (SEP). This integrality gap is conjectured to be 4/3. We prove that, when solving a problem on n nodes, if the optimal SEP solution has at most n + 6 non-zero components, then the conjecture is true. To establish this result, we devise a new methodology that combines theoretical analysis and computational verification.

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arXiv2025

On the integrality Gap of Small Asymmetric Traveling Salesman Problems: A Polyhedral and Computational Approach

In this paper, we investigate the integrality gap of the Asymmetric Traveling Salesman Problem (ATSP) with respect to the linear relaxation given by the Asymmetric Subtour Elimination Problem (ASEP) for instances with $n$ nodes, where $n$ is small. In particular, we focus on the geometric properties and symmetries of the ASEP polytope ($P^{n}_{ASEP}$) and its vertices. The polytope's symmetries are exploited to design a heuristic pivoting algorithm to search vertices where the integrality gap is maximized. Furthermore, a general procedure for the extension of vertices from $P^{n}_{ASEP}$ to $P^{n + 1}_{ASEP}$ is defined. The generated vertices improve the known lower bounds of the integrality gap for $ 16 \leq n \leq 22$ and, provide small hard-to-solve ATSP instances.

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arXiv2025

On the Degree Automatability of Sum-of-Squares Proofs

The Sum-of-Squares (SoS) hierarchy, also known as Lasserre hierarchy, has emerged as a promising tool in optimization. However, it remains unclear whether fixed-degree SoS proofs can be automated [O'Donnell (2017)]. Indeed, there are examples of polynomial systems with bounded coefficients that admit low-degree SoS proofs, but these proofs necessarily involve numbers with an exponential number of bits, implying that low-degree SoS proofs cannot always be found efficiently. A sufficient condition derived from the Nullstellensatz proof system [Raghavendra and Weitz (2017)] identifies cases where bit complexity issues can be circumvented. One of the main problems left open by Raghavendra and Weitz is proving any result for refutations, as their condition applies only to polynomial systems with a large set of solutions. In this work, we broaden the class of polynomial systems for which degree-$d$ SoS proofs can be automated. To achieve this, we develop a new criterion and we demonstrate how our criterion applies to polynomial systems beyond the scope of Raghavendra and Weitz's result. In particular, we establish a separation for instances arising from Constraint Satisfaction Problems (CSPs). Moreover, our result extends to refutations, establishing that polynomial-time refutation is possible for broad classes of polynomial time solvable constraint problems, highlighting a first advancement in this area.

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arXiv2025

Branch-and-Bound Algorithms as Polynomial-time Approximation Schemes

Branch-and-bound algorithms (B&B) and polynomial-time approximation schemes (PTAS) are two seemingly distant areas of combinatorial optimization. We intend to (partially) bridge the gap between them while expanding the boundary of theoretical knowledge on the B\&B framework. Branch-and-bound algorithms typically guarantee that an optimal solution is eventually found. However, we show that the standard implementation of branch-and-bound for certain knapsack and scheduling problems also exhibits PTAS-like behavior, yielding increasingly better solutions within polynomial time. Our findings are supported by computational experiments and comparisons with benchmark methods. This paper is an extended version of a paper accepted at ICALP 2025

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arXiv2025

Computational complexity of sum-of-squares bounds for copositive programs

In recent years, copositive programming has received significant attention for its ability to model hard problems in both discrete and continuous optimization. Several relaxations of copositive programs based on semidefinite programming (SDP) have been proposed in the literature, meant to provide tractable bounds. However, while these SDP-based relaxations are amenable to the ellipsoid algorithm and interior point methods, it is not immediately obvious that they can be solved in polynomial time (even approximately). In this paper, we consider the sum-of-squares (SOS) hierarchies of relaxations for copositive programs introduced by Parrilo (2000), de Klerk & Pasechnik (2002) and Pe\~na, Vera & Zuluaga (2006), which can be formulated as SDPs. We establish sufficient conditions that guarantee the polynomial-time computability (up to fixed precision) of these relaxations. These conditions are satisfied by copositive programs that represent standard quadratic programs and their reciprocals. As an application, we show that the SOS bounds for the (weighted) stability number of a graph can be computed efficiently. Additionally, we provide pathological examples of copositive programs (that do not satisfy the sufficient conditions) whose SOS relaxations admit only feasible solutions of doubly-exponential size.

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arXiv2024

Ideal Membership Problem for Boolean Minority and Dual Discriminator

We consider the polynomial Ideal Membership Problem (IMP) for ideals encoding combinatorial problems that are instances of CSPs over a finite language. In this paper, the input polynomial $f$ has degree at most $d=O(1)$ (we call this problem IMP$_d$). We bridge the gap in \cite{MonaldoMastrolilli2019} by proving that the IMP$_d$ for Boolean combinatorial ideals whose constraints are closed under the minority polymorphism can be solved in polynomial time. This completes the identification of the tractability for the Boolean IMP$_d$. We also prove that the proof of membership for the IMP$_d$ for problems constrained by the dual discriminator polymorphism over any finite domain can be found in polynomial time. Our results can be used in applications such as Nullstellensatz and Sum-of-Squares proofs.

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arXiv2024

Lower bounds for the integrality gap of the bi-directed cut formulation of the Steiner Tree Problem

In this work, we study the metric Steiner Tree problem on graphs focusing on computing lower bounds for the integrality gap of the bi-directed cut (BCR) formulation and introducing a novel formulation, the Complete Metric (CM) model, specifically designed to address the weakness of the BCR formulation on metric instances. A key contribution of our work is extending the Gap problem, previously explored in the context of the Traveling Salesman problems, to the metric Steiner Tree problem. To tackle the Gap problem for Steiner Tree instances, we first establish several structural properties of the CM formulation. We then classify the isomorphism classes of the vertices within the CM polytope, revealing a correspondence between the vertices of the BCR and CM polytopes. Computationally, we exploit these structural properties to design two complementary heuristics for finding nontrivial small metric Steiner instances with a large integrality gap. We present several vertices for graphs with a number of nodes <=10, which realize the best-known lower bounds on the integrality gap for the CM and the BCR formulations. We conclude the paper by presenting two new conjectures on the integrality gap of the BCR and CM formulations for small graphs.

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arXiv2023

On the generation of Metric TSP instances with a large integrality gap by branch-and-cut

This paper introduces a computational method for generating metric Travelling Salesman Problem (TSP) instances having a large integrality gap. The method is based on the solution of an integer programming problem, called IH-OPT, that takes as input a fractional solution of the Subtour Elimination Problem (SEP) on a TSP instance and computes a TSP instance having an integrality gap larger than or equal to the integrality gap of the first instance. The decision variables of IH-OPT are the entries of the TSP cost matrix, and the constraints are defined by the intersection of the metric cone with an exponential number of inequalities, one for each possible TSP tour. Given the very large number of constraints, we have implemented a branch-and-cut algorithm for solving IH-OPT. Then, by sampling cost vectors over the metric polytope and by solving the corresponding SEP, we can generate random fractional vertices of the SEP polytope. If we solve the IH-OPT problem for every sampled vertex using our branch-and-cut algorithm, we can select the generated TSP instance (i.e., cost vector), yielding the longest runtime for Concorde, the state-of-the-art TSP solver. Our computational results show that our method is very effective in producing challenging instances. As a by-product, we release the Hard-TSPLIB, a library of 41 small metric TSP instances which have a large integrality gap and are challenging in terms of runtime for Concorde.

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arXiv2021

The Complexity of the Ideal Membership Problem for Constrained Problems Over the Boolean Domain

Given an ideal $I$ and a polynomial $f$ the Ideal Membership Problem is to test if $f\in I$. This problem is a fundamental algorithmic problem with important applications and notoriously intractable. We study the complexity of the Ideal Membership Problem for combinatorial ideals that arise from constrained problems over the Boolean domain. As our main result, we identify the borderline of tractability. By using Gröbner bases techniques, we extend Schaefer's dichotomy theorem [STOC, 1978] which classifies all Constraint Satisfaction Problems over the Boolean domain to be either in P or NP-hard. Moreover, our result implies necessary and sufficient conditions for the efficient computation of Theta Body SDP relaxations, identifying therefore the borderline of tractability for constraint language problems. This paper is motivated by the pursuit of understanding the recently raised issue of bit complexity of Sum-of-Squares proofs [O'Donnell, ITCS, 2017]. Raghavendra and Weitz [ICALP, 2017] show how the Ideal Membership Problem tractability for combinatorial ideals implies bounded coefficients in Sum-of-Squares proofs.

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arXiv2020

Ideal Membership Problem for Boolean Minority

The Ideal Membership Problem (IMP) tests if an input polynomial $f\in \mathbb{F}[x_1,\dots,x_n]$ with coefficients from a field $\mathbb{F}$ belongs to a given ideal $I \subseteq \mathbb{F}[x_1,\dots,x_n]$. It is a well-known fundamental problem with many important applications, though notoriously intractable in the general case. In this paper we consider the IMP for polynomial ideals encoding combinatorial problems and where the input polynomial $f$ has degree at most $d=O(1)$ (we call this problem IMP$_d$). A dichotomy result between ``hard'' (NP-hard) and ``easy'' (polynomial time) IMPs was recently achieved for Constraint Satisfaction Problems over finite domains [Bulatov FOCS'17, Zhuk FOCS'17] (this is equivalent to IMP$_0$) and IMP$_d$ for the Boolean domain [Mastrolilli SODA'19], both based on the classification of the IMP through functions called polymorphisms. The complexity of the IMP$_d$ for five polymorphisms has been solved in [Mastrolilli SODA'19] whereas for the ternary minority polymorphism it was incorrectly declared to have been resolved by a previous result. As a matter of fact the complexity of the IMP$_d$ for the ternary minority polymorphism is open. In this paper we provide the missing link by proving that the IMP$_d$ for Boolean combinatorial ideals whose constraints are closed under the minority polymorphism can be solved in polynomial time. This result, along with the results in [Mastrolilli SODA'19], completes the identification of the precise borderline of tractability for the IMP$_d$ for constrained problems over the Boolean domain. This paper is motivated by the pursuit of understanding the issue of bit complexity of Sum-of-Squares proofs raised by O'Donnell [ITCS'17]. Raghavendra and Weitz [ICALP'17] show how the IMP$_d$ tractability for combinatorial ideals implies bounded coefficients in Sum-of-Squares proofs.

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arXiv2019

High Degree Sum of Squares Proofs, Bienstock-Zuckerberg hierarchy and Chvatal-Gomory cuts

Chvatal-Gomory (CG) cuts and the Bienstock-Zuckerberg hierarchy capture useful linear programs that the standard bounded degree Lasserre/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 constructive 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 covering 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. Arguably, this is the first example where different basis functions can be useful in asymmetric situations to obtain a hierarchy of relaxations.

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arXiv2018

On bounded pitch inequalities for the min-knapsack polytope

In the min-knapsack problem one aims at choosing a set of objects with minimum total cost and total profit above a given threshold. In this paper, we study a class of valid inequalities for min-knapsack known as bounded pitch inequalities, which generalize the well-known unweighted cover inequalities. While separating over pitch-1 inequalities is NP-hard, we show that approximate separation over the set of pitch-1 and pitch-2 inequalities can be done in polynomial time. We also investigate integrality gaps of linear relaxations for min-knapsack when these inequalities are added. Among other results, we show that, for any fixed $t$, the $t$-th CG closure of the natural linear relaxation has the unbounded integrality gap.

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arXiv2016

Tight Sum-of-Squares lower bounds for binary polynomial optimization problems

We give two results concerning the power of the Sum-of-Squares(SoS)/Lasserre hierarchy. For binary polynomial optimization problems of degree $2d$ and an odd number of variables $n$, we prove that $\frac{n+2d-1}{2}$ levels of the SoS/Lasserre hierarchy are necessary to provide the exact optimal value. This matches the recent upper bound result by Sakaue, Takeda, Kim and Ito. Additionally, we study a conjecture by Laurent, who considered the linear representation of a set with no integral points. She showed that the Sherali-Adams hierarchy requires $n$ levels to detect the empty integer hull, and conjectured that the SoS/Lasserre rank for the same problem is $n-1$. We disprove this conjecture and derive lower and upper bounds for the rank.

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arXiv2016

Sum-of-squares hierarchy lower bounds for symmetric formulations

We introduce a method for proving Sum-of-Squares (SoS)/ Lasserre hierarchy lower bounds when the initial problem formulation exhibits a high degree of symmetry. Our main technical theorem allows us to reduce the study of the positive semidefiniteness to the analysis of "well-behaved" univariate polynomial inequalities. We illustrate the technique on two problems, one unconstrained and the other with constraints. More precisely, we give a short elementary proof of Grigoriev/Laurent lower bound for finding the integer cut polytope of the complete graph. We also show that the SoS hierarchy requires a non-constant number of rounds to improve the initial integrality gap of 2 for the Min-Knapsack linear program strengthened with cover inequalities.

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arXiv2015

A Lasserre Lower Bound for the Min-Sum Single Machine Scheduling Problem

The Min-sum single machine scheduling problem (denoted 1||sum f_j) generalizes a large number of sequencing problems. The first constant approximation guarantees have been obtained only recently and are based on natural time-indexed LP relaxations strengthened with the so called Knapsack-Cover inequalities (see Bansal and Pruhs, Cheung and Shmoys and the recent 4+ε-approximation by Mestre and Verschae). These relaxations have an integrality gap of 2, since the Min-knapsack problem is a special case. No APX-hardness result is known and it is still conceivable that there exists a PTAS. Interestingly, the Lasserre hierarchy relaxation, when the objective function is incorporated as a constraint, reduces the integrality gap for the Min-knapsack problem to 1+ε. In this paper we study the complexity of the Min-sum single machine scheduling problem under algorithms from the Lasserre hierarchy. We prove the first lower bound for this model by showing that the integrality gap is unbounded at level Ω(\sqrt{n}) even for a variant of the problem that is solvable in O(n log n) time by the Moore-Hodgson algorithm, namely Min-number of tardy jobs. We consider a natural formulation that incorporates the objective function as a constraint and prove the result by partially diagonalizing the matrix associated with the relaxation and exploiting this characterization.

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arXiv2015

On the Hardest Problem Formulations for the 0/1 Lasserre Hierarchy

The Lasserre/Sum-of-Squares (SoS) hierarchy is a systematic procedure for constructing a sequence of increasingly tight semidefinite relaxations. It is known that the hierarchy converges to the 0/1 polytope in n levels and captures the convex relaxations used in the best available approximation algorithms for a wide variety of optimization problems. In this paper we characterize the set of 0/1 integer linear problems and unconstrained 0/1 polynomial optimization problems that can still have an integrality gap at level n-1. These problems are the hardest for the Lasserre hierarchy in this sense.

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arXiv2014

The Lasserre Hierarchy in Almost Diagonal Form

The Lasserre hierarchy is a systematic procedure for constructing a sequence of increasingly tight relaxations that capture the convex formulations used in the best available approximation algorithms for a wide variety of optimization problems. Despite the increasing interest, there are very few techniques for analyzing Lasserre integrality gaps. Satisfying the positive semi-definite requirement is one of the major hurdles to constructing Lasserre gap examples. We present a novel characterization of the Lasserre hierarchy based on moment matrices that differ from diagonal ones by matrices of rank one (almost diagonal form). We provide a modular recipe to obtain positive semi-definite feasibility conditions by iteratively diagonalizing rank one matrices. Using this, we prove strong lower bounds on integrality gaps of Lasserre hierarchy for two basic capacitated covering problems. For the min-knapsack problem, we show that the integrality gap remains arbitrarily large even at level $n-1$ of Lasserre hierarchy. For the min-sum of tardy jobs scheduling problem, we show that the integrality gap is unbounded at level $Ω(\sqrt{n})$ (even when the objective function is integrated as a constraint). These bounds are interesting on their own, since both problems admit FPTAS.

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