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We develop a unified framework to characterize the power of higher-level algorithms for the constraint satisfaction problem (CSP), such as $k$-consistency, the Sherali-Adams LP hierarchy, and the affine IP hierarchy. As a result,…
We study the Sherali-Adams linear programming hierarchy in the context of promise constraint satisfaction problems (PCSPs). We characterise when a level of the hierarchy accepts an instance in terms of a homomorphism problem for an…
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…
In this paper we study the interactions between so-called fractional relaxations of the integer programs (IPs) which encode homomorphism and isomorphism of relational structures. We give a combinatorial characterization of a certain natural…
Following the success of the so-called algebraic approach to the study of decision constraint satisfaction problems (CSPs), exact optimization of valued CSPs, and most recently promise CSPs, we propose an algebraic framework for valued…
We introduce a method for proving lower bounds on the efficacy of semidefinite programming (SDP) relaxations for combinatorial problems. In particular, we show that the cut, TSP, and stable set polytopes on $n$-vertex graphs are not the…
Semidefinite programs (SDPs) often arise in relaxations of some NP-hard problems, and if the solution of the SDP obeys certain rank constraints, the relaxation will be tight. Decomposition methods based on chordal sparsity have already been…
The Promise Constraint Satisfaction Problem (PCSP for short) is a generalization of the well-studied Constraint Satisfaction Problem (CSP). The PCSP has its roots in such classic problems as the Approximate Graph Coloring and the…
We study the use of local consistency methods as reductions between constraint satisfaction problems (CSPs), and promise version thereof, with the aim to classify these reductions in a similar way as the algebraic approach classifies gadget…
We consider the general polynomial optimization problem $P: f^*=\min \{f(x)\,:\,x\in K\}$ where $K$ is a compact basic semi-algebraic set. We first show that the standard Lagrangian relaxation yields a lower bound as close as desired to the…
Promise CSPs are a relaxation of constraint satisfaction problems where the goal is to find an assignment satisfying a relaxed version of the constraints. Several well-known problems can be cast as promise CSPs including approximate graph…
This paper studies a class of so-called linear semi-infinite polynomial programming (LSIPP) problems. It is a subclass of linear semi-infinite programming problems whose constraint functions are polynomials in parameters and index sets are…
The semiring-based constraint satisfaction problems (semiring CSPs), proposed by Bistarelli, Montanari and Rossi \cite{BMR97}, is a very general framework of soft constraints. In this paper we propose an abstraction scheme for soft…
The presence of symmetries is one of the central structural features that make some integer programs challenging for state-of-the-art solvers. In this work, we study the efficacy of Linear Programming (LP) hierarchies in the presence of…
We develop an analytical framework for Boolean Promise Constraint Satisfaction Problems (PCSPs) that studies polymorphisms through the notion of influence from Fourier analysis of Boolean functions. Extending the work of Brakensiek,…
This paper investigates the minimization of the expectation of piecewise polynomial loss functions over Wasserstein balls. This optimization problem often appears as a key sub-problem of distributionally robust optimization problems. We…
We prove super-polynomial lower bounds on the size of linear programming relaxations for approximation versions of constraint satisfaction problems. We show that for these problems, polynomial-sized linear programs are exactly as powerful…
We introduce multiple symmetric LP relaxations for minimum cut problems. The relaxations give optimal and approximate solutions when the input is a Hamiltonian cycle. We show that this leads to one of two interesting results. In one case,…
The so-called algebraic approach to the constraint satisfaction problem (CSP) has been a prevalent method of the study of complexity of these problems since early 2000's. The core of this approach is the notion of polymorphisms which…
Given a pair of graphs $\textbf{A}$ and $\textbf{B}$, the problems of deciding whether there exists either a homomorphism or an isomorphism from $\textbf{A}$ to $\textbf{B}$ have received a lot of attention. While graph homomorphism is…