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This work aims to introduce the framework of polynomial optimization theory to solve fractional polynomial problems (FPPs). Unlike other widely used optimization frameworks, the proposed one applies to a larger class of FPPs, not…
We consider the hardness of approximation of optimization problems from the point of view of definability. For many NP-hard optimization problems it is known that, unless P = NP, no polynomial-time algorithm can give an approximate solution…
We study the almost-sure termination problem for probabilistic programs. First, we show that supermartingales with lower bounds on conditional absolute difference provide a sound approach for the almost-sure termination problem. Moreover,…
The problem of high-dimensional path-dependent optimal stopping (OS) is important to multiple academic communities and applications. Modern OS tasks often have a large number of decision epochs, and complicated non-Markovian dynamics,…
The Monniaux Problem in abstract interpretation asks, roughly speaking, whether the following question is decidable: given a program $P$, a safety (\emph{e.g.}, non-reachability) specification $\varphi$, and an abstract domain of invariants…
Indefinite quadratic programs (QPs) are known to be very difficult to be solved to global optimality, so are linear programs with linear complementarity constraints. Treating the former as a subclass of the latter, this paper presents a…
In this paper, we propose a generalized successive approximation method (SAM), called invariantly admissible policy iteration (PI), for finding the solution to a class of input-affine nonlinear optimal control problems by iterations. Unlike…
Unit two-variable-per-inequality (UTVPI) constraints form one of the largest class of integer constraints which are polynomial time solvable (unless P=NP). There is considerable interest in their use for constraint solving, abstract…
Nonuniform families of polynomial-size finite automata, which are series of indexed finite automata having polynomially many inner states, are used in the past literature to solve nonuniform families of promise decision problems. Among such…
A subset of Q^n is called semilinear (or piecewise linear) if it is Boolean combination of linear half-spaces. We study the computational complexity of the constraint satisfaction problem (CSP) over the rationals when all the constraints…
We give polynomial time algorithms for quantitative (and qualitative) reachability analysis for Branching Markov Decision Processes (BMDPs). Specifically, given a BMDP, and given an initial population, where the objective of the controller…
Datalogo is an extension of Datalog that allows for aggregation and recursion over an arbitrary commutative semiring. Like Datalog, Datalogo programs can be evaluated via the natural iterative algorithm until a fixed point is reached.…
Scheduling problems are fundamental in combinatorial optimization. Much work has been done on approximation algorithms for NP-hard cases, but relatively little is known about exact solutions when some part of the input is a fixed parameter.…
Modified policy iteration (MPI) is a dynamic programming algorithm that combines elements of policy iteration and value iteration. The convergence of MPI has been well studied in the context of discounted and average-cost MDPs. In this…
Loop invariants play a very important role in proving correctness of programs. In this paper, we address the problem of generating invariants of polynomial loop programs. We present a new approach, for generating polynomial equation…
Modified policy iteration (MPI) is a dynamic programming (DP) algorithm that contains the two celebrated policy and value iteration methods. Despite its generality, MPI has not been thoroughly studied, especially its approximation form…
Permutation Pattern Matching (or PPM) is a decision problem whose input is a pair of permutations $\pi$ and $\tau$, represented as sequences of integers, and the task is to determine whether $\tau$ contains a subsequence order-isomorphic to…
We study time-varying semidefinite programs (TV-SDPs), which are semidefinite programs whose data (and solutions) are functions of time. Our focus is on the setting where the data varies polynomially with time. We show that under a strict…
In this paper we give a detailed analysis of deterministic and randomized algorithms that enumerate any number of irreducible polynomials of degree $n$ over a finite field and their roots in the extension field in quasilinear where $N=n^2$…
Provably correct software is one of the key challenges in our softwaredriven society. While formal verification establishes the correctness of a given program, the result of program synthesis is a program which is correct by construction.…