Rank-Deficient Quadratic-Form Maximization Over M-Phase Alphabet: Polynomial-Complexity Solvability And Algorithmic Developments

The maximization of a positive (semi) definite complex quadratic form over a finite alphabet is NP-hard and achieved through exhaustive search when the form has full rank. However, if the form is rank-deficient, the optimal solution can be computed with only polynomial complexity in the length N of the maximizing vector. In this work, we consider the general case of a rank-D positive (semi) definite complex quadratic form and develop a method that maximizes the form with respect to a M-phase vector with polynomial complexity. The proposed method efficiently reduces the size of the feasible set from exponential to polynomial. We also develop an algorithm that constructs the polynomial-size candidate set in polynomial time and observe that it is fully parallelizable and rank-scalable.

Fixed-Rank Rayleigh Quotient Maximization by an MPSK Sequence

Anastasios Kyrillidis

Certain optimization problems in communication systems, such as limited-feedback constant-envelope beamforming or noncoherent M-ary phase-shift keying (MPSK) sequence detection, result in the maximization of a fixed-rank positive semidefinite quadratic form over the MPSK alphabet. This form is a special case of the Rayleigh quotient of a matrix and, in general, its maximization by an MPSK sequence is NP-hard. However, if the rank of the matrix is not a function of its size, then the optimal solution can be computed with polynomial complexity in the matrix size. In this work, we develop a new technique to efficiently solve this problem by utilizing auxiliary continuous-valued angles and partitioning the resulting continuous space of solutions into a polynomial-size set of regions, each of which corresponds to a distinct MPSK sequence. The sequence that maximizes the Rayleigh quotient is shown to belong to this polynomial-size set of sequences, thus efficiently reducing the size of the feasible set from exponential to polynomial. Based on this analysis, we also develop an algorithm that constructs this set in polynomial time and show that it is fully parallelizable, memory efficient, and rank scalable. The proposed algorithm compares favorably with other solvers for this problem that have appeared recently in the literature.

Institute of Electrical and Electronics Engineers2014

Strengths and Limitations of Linear Programming Relaxations

Abbas Bazzi

Many of the currently best-known approximation algorithms for NP-hard optimization problems are based on Linear Programming (LP) and Semi-definite Programming (SDP) relaxations. Given its power, this class of algorithms seems to contain the most favourable candidates for outperforming the current state-of-the-art approximation guarantees for NP-hard problems, for which there still exists a gap between the inapproximability results and the approximation guarantees that we know how to achieve in polynomial time. In this thesis, we address both the power and the limitations of these relaxations, as well as the connection between the shortcomings of these relaxations and the inapproximability of the underlying problem. In the first part, we study the limitations of LP relaxations of well-known graph problems such as the Vertex Cover problem and the Independent Set problem. We prove that any small LP relaxation for the aforementioned problems, cannot have an integrality gap strictly better than

2

and

\omega(1)

, respectively. Furthermore, our lower bound for the Independent Set problem also holds for any SDP relaxation. Prior to our work, it was only known that such LP relaxations cannot have an integrality gap better than

1.5

for the Vertex Cover Problem, and better than

2

for the Independent Set problem. In the second part, we study the so-called knapsack cover inequalities that are used in the current best relaxations for numerous combinatorial optimization problems of covering type. In spite of their widespread use, these inequalities yield LP relaxations of exponential size, over which it is not known how to optimize exactly in polynomial time. We address this issue and obtain LP relaxations of quasi-polynomial size that are at least as strong as that given by the knapsack cover inequalities. In the last part, we show a close connection between structural hardness for k-partite graphs and tight inapproximability results for scheduling problems with precedence constraints. This connection is inspired by a family of integrality gap instances of a certain LP relaxation. Assuming the hardness of an optimization problem on k-partite graphs, we obtain a hardness of

2-\varepsilon

for the problem of minimizing the makespan for scheduling with preemption on identical parallel machines, and a super constant inapproximability for the problem of scheduling on related parallel machines. Prior to this result, it was only known that the first problem does not admit a PTAS, and the second problem is NP-hard to approximate within a factor strictly better than 2, assuming the Unique Games Conjecture.

EPFL2017

New Results in Integer and Lattice Programming

Christoph Hunkenschröder

An integer program (IP) is a problem of the form

\min \{f(x) : \, Ax = b, \ l \leq x \leq u, \ x \in \Z^n\}

, where

A \in \Z^{m \times n}

b \in \Z^m

l,u \in \Z^n

, and

f: \Z^n \rightarrow \Z

is a separable convex objective function. The problem of finding an optimal solution for an integer program is known as integer programming. Integer programming is NP-hard in general, though several algorithms exist: Lenstra provided an algorithm that is polynomial if the dimension

n

is fixed. For variable dimension, the best known algorithm depends linearly on

n

, and exponentially on the number of equalities as well as the largest absolute value of an entry in the matrix

A

. The first part of this thesis considers integer programming for variable dimensions and sparse matrices. We measure the sparsity of a matrix by the tree-depth of the dual graph of

A

. A typical example for these integer programs are

N

-fold IPs, used for scheduling and social choice problems. We obtain the currently fastest fixed-parameter tractable algorithm with parameters tree-depth and the largest absolute value of the entries in

A

. The running time we achieve is near-linear in the dimension. With a slightly worse running time, we are able to show that

N

-fold integer programs of constant block size can be solved in strongly polynomial time. Assuming the exponential time hypothesis, we complement these results with a lower bound on the parameter dependency that almost matches the parameter dependency of the running time. As a consequence, we provide the currently strongest lower bound for

N

-fold integer programs. Another problem closely related to integer programming is the closest vector problem. A lattice is a discrete additive subgroup of

\R^n

. The closest vector problem (CVP) asks for a lattice point closest to a given target vector. An important tool for solving the closest vector problem is the Voronoi cell

\vc

of a lattice

\Lambda \subseteq \R^n

, which is the set of all points for which

0

is a closest lattice point. It is a polytope whose facets are induced by a set of lattice vectors, the Voronoi relevant vectors. A generic lattice has exponentially many Voronoi relevant vectors, leading to exponential space for certain CVP algorithms. In the second part of this thesis, we introduce the notion of a

c

-compact lattice basis

B \in \R^{n \times n}

that facilitates to represent the Voronoi relevant vectors with coefficients bounded by

c

. Such a basis allows to reduce the space requirement of Micciancio's & Voulgaris' algorithm for the closest vector problem from exponential to polynomial, while the running time becomes exponential in

c

. We show that for every lattice an

n^2

-compact basis exists, but there are lattices for which we cannot choose

c \in o (n)

. If the Voronoi cell is a zonotope, we can choose

c=1

, providing a single-exponential time and polynomial space algorithm for CVP, assuming a

1

-compact basis is known. Deciding whether a given lattice has a certain structure that helps to solve the closest vector problem more efficiently is a reappearing and non-trivial problem. The third part of this thesis is concerned with the specific structure of having an orthonormal basis. We show that this problem belongs to NP

\cap

co-NP. Moreover, it can be reduced to solving a single closest vector problem. We also show that if a separation oracle for the Voronoi cell is provided, CVP is solvable in polynomial time.

EPFL2020