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Binary Integer Programming (BIP) problems are of interest due in part to the difficulty they pose and because of their various applications, including those in graph theory, combinatorial optimization and network optimization. In this note,…
Constraint satisfaction problems (CSPs) consist of a set of variables taking values from some finite domain and a set of local constraints on these variables. The objective is to find an assignment to the variables that maximizes the…
This thesis explores algorithmic applications and limitations of convex relaxation hierarchies for approximating some discrete and continuous optimization problems. - We show a dichotomy of approximability of constraint satisfaction…
Semidefinite programs (SDPs) are a framework for exact or approximate optimization that have widespread application in quantum information theory. We introduce a new method for using reductions to construct integrality gaps for SDPs. These…
These lectures provide a simple introduction to supersymmetry breaking. After presenting the basics of the subject and illustrating them in tree-level examples, we discuss dynamical supersymmetry breaking, emphasizing the role of holomorphy…
Exploitation of symmetries is an indispensable approach to solve certain classes of difficult SAT instances. Numerous techniques for the use of symmetry in SAT have evolved over the past few decades. But no matter how symmetries are used…
Polynomial optimization problems are infinite-dimensional, nonconvex, NP-hard, and are often handled in practice with the moment-sums of squares hierarchy of semidefinite programming bounds. We consider problems where the objective function…
A sparse linear programming (SLP) problem is a linear programming problem equipped with a sparsity (or cardinality) constraint, which is nonconvex and discontinuous theoretically and generally NP-hard computationally due to the…
We show that for every homogeneous polynomial of degree $d$, if it has determinantal complexity at most $s$, then it can be computed by a homogeneous algebraic branching program (ABP) of size at most $O(d^5s)$. Moreover, we show that for…
The secret protection problem (SPP) seeks to synthesize a minimum-cost policy ensuring that every execution from an initial state to a secret state includes a sufficient number of protected events. Previous work showed that the problem is…
In Verification and in (optimal) AI Planning, a successful method is to formulate the application as boolean satisfiability (SAT), and solve it with state-of-the-art DPLL-based procedures. There is a lack of understanding of why this works…
The Maximum Balanced Biclique Problem (MBBP) is a prominent model with numerous applications. Yet, the problem is NP-hard and thus computationally challenging. We propose novel ideas for designing effective exact algorithms for MBBP.…
The Pancharatnam-Berry (PB) phase in metasurfaces obeys the symmetry restriction, according to which the PB phases of two orthogonal circularly polarized waves are the same but with opposite signs. Here, we reveal a general mechanism to…
We advocate a new approach of addressing hidden structure problems and finding efficient quantum algorithms. We introduce and investigate the Hidden Symmetry Subgroup Problem (HSSP), which is a generalization of the well-studied Hidden…
Answer set programming (ASP) is a form of declarative programming that allows to succinctly formulate and efficiently solve complex problems. An intuitive extension of this formalism is communicating ASP, in which multiple ASP programs…
We describe simple algebraic and combinatorial characterisations of finite relational core structures admitting finitely many obstructions. As a consequence, we show that it is decidable to determine whether a constraint satisfaction…
Joint object matching, also known as multi-image matching, namely, the problem of finding consistent partial maps among all pairs of objects within a collection, is a crucial task in many areas of computer vision. This problem subsumes…
The constrained linear representability problem (CLRP) for polymatroids determines whether there exists a polymatroid that is linear over a specified field while satisfying a collection of constraints on the rank function. Using a computer…
We study the structure of solutions to linear programming formulations for the traveling salesperson problem (TSP). We perform a detailed analysis of the support of the subtour elimination linear programming relaxation, which leads to…
We consider the bilevel minimum spanning tree (BMST) problem where the leader and the follower choose a spanning tree together, according to different objective functions. By showing that this problem is NP-hard in general, we answer an…