Simulation of many-qubit quantum computation with matrix product states

Matrix product states provide a natural entanglement basis to represent a quantum register and operate quantum gates on it. This scheme can be materialized to simulate a quantum adiabatic algorithm solving hard instances of a NP-Complete problem. Errors inherent to truncations of the exact action of interacting gates are controlled by the size of the matrices in the representation. The property of finding the right solution for an instance and the expected value of the energy are found to be remarkably robust against these errors. As a symbolic example, we simulate the algorithm solving a 100-qubit hard instance, that is, finding the correct product state out of ~ 10^30 possibilities. Accumulated statistics for up to 60 qubits point at a slow growth of the average minimum time to solve hard instances with highly-truncated simulations of adiabatic quantum evolution.Comment: 5 pages, 4 figures, final versio

Semi-autonomous Intersection Collision Avoidance through Job-shop Scheduling

In this paper, we design a supervisor to prevent vehicle collisions at intersections. An intersection is modeled as an area containing multiple conflict points where vehicle paths cross in the future. At every time step, the supervisor determines whether there will be more than one vehicle in the vicinity of a conflict point at the same time. If there is, then an impending collision is detected, and the supervisor overrides the drivers to avoid collision. A major challenge in the design of a supervisor as opposed to an autonomous vehicle controller is to verify whether future collisions will occur based on the current drivers choices. This verification problem is particularly hard due to the large number of vehicles often involved in intersection collision, to the multitude of conflict points, and to the vehicles dynamics. In order to solve the verification problem, we translate the problem to a job-shop scheduling problem that yields equivalent answers. The job-shop scheduling problem can, in turn, be transformed into a mixed-integer linear program when the vehicle dynamics are first-order dynamics, and can thus be solved by using a commercial solver.Comment: Submitted to Hybrid Systems: Computation and Control (HSCC) 201

Integer Polynomial Optimization in Fixed Dimension

We classify, according to their computational complexity, integer optimization problems whose constraints and objective functions are polynomials with integer coefficients and the number of variables is fixed. For the optimization of an integer polynomial over the lattice points of a convex polytope, we show an algorithm to compute lower and upper bounds for the optimal value. For polynomials that are non-negative over the polytope, these sequences of bounds lead to a fully polynomial-time approximation scheme for the optimization problem.Comment: In this revised version we include a stronger complexity bound on our algorithm. Our algorithm is in fact an FPTAS (fully polynomial-time approximation scheme) to maximize a non-negative integer polynomial over the lattice points of a polytop

Approximately coloring graphs without long induced paths

It is an open problem whether the 3-coloring problem can be solved in polynomial time in the class of graphs that do not contain an induced path on $t$ vertices, for fixed $t$. We propose an algorithm that, given a 3-colorable graph without an induced path on $t$ vertices, computes a coloring with $\max\{5,2\lceil{\frac{t-1}{2}}\rceil-2\}$ many colors. If the input graph is triangle-free, we only need $\max\{4,\lceil{\frac{t-1}{2}}\rceil+1\}$ many colors. The running time of our algorithm is $O((3^{t-2}+t^2)m+n)$ if the input graph has $n$ vertices and $m$ edges

Complexity and Inapproximability Results for Parallel Task Scheduling and Strip Packing

We study the Parallel Task Scheduling problem $Pm|size_j|C_{\max}$ with a constant number of machines. This problem is known to be strongly NP-complete for each $m \geq 5$, while it is solvable in pseudo-polynomial time for each $m \leq 3$. We give a positive answer to the long-standing open question whether this problem is strongly $NP$-complete for $m=4$. As a second result, we improve the lower bound of $\frac{12}{11}$ for approximating pseudo-polynomial Strip Packing to $\frac{5}{4}$. Since the best known approximation algorithm for this problem has a ratio of $\frac{4}{3} + \varepsilon$, this result narrows the gap between approximation ratio and inapproximability result by a significant step. Both results are proven by a reduction from the strongly $NP$-complete problem 3-Partition

Coloring random graphs

We study the graph coloring problem over random graphs of finite average connectivity $c$. Given a number $q$ of available colors, we find that graphs with low connectivity admit almost always a proper coloring whereas graphs with high connectivity are uncolorable. Depending on $q$, we find the precise value of the critical average connectivity $c_q$. Moreover, we show that below $c_q$ there exist a clustering phase $c\in [c_d,c_q]$ in which ground states spontaneously divide into an exponential number of clusters and where the proliferation of metastable states is responsible for the onset of complexity in local search algorithms.Comment: 4 pages, 1 figure, version to app. in PR

Quadratic equations over free groups are NP-complete

We prove that the problems of deciding whether a quadratic equation over a free group has a solution is NP-complete

Testing the bounds on quantum probabilities

Bounds on quantum probabilities and expectation values are derived for experimental setups associated with Bell-type inequalities. In analogy to the classical bounds, the quantum limits are experimentally testable and therefore serve as criteria for the validity of quantum mechanics.Comment: 9 pages, Revte

Phase transition for cutting-plane approach to vertex-cover problem

We study the vertex-cover problem which is an NP-hard optimization problem and a prototypical model exhibiting phase transitions on random graphs, e.g., Erdoes-Renyi (ER) random graphs. These phase transitions coincide with changes of the solution space structure, e.g, for the ER ensemble at connectivity c=e=2.7183 from replica symmetric to replica-symmetry broken. For the vertex-cover problem, also the typical complexity of exact branch-and-bound algorithms, which proceed by exploring the landscape of feasible configurations, change close to this phase transition from "easy" to "hard". In this work, we consider an algorithm which has a completely different strategy: The problem is mapped onto a linear programming problem augmented by a cutting-plane approach, hence the algorithm operates in a space OUTSIDE the space of feasible configurations until the final step, where a solution is found. Here we show that this type of algorithm also exhibits an "easy-hard" transition around c=e, which strongly indicates that the typical hardness of a problem is fundamental to the problem and not due to a specific representation of the problem.Comment: 4 pages, 3 figure

Approximation Algorithms for Scheduling Parallel Jobs: Breaking the Approximation Ratio of 2

In this paper we study variants of the non-preemptive parallel job scheduling problem in which the number of machines is polynomially bounded in the number of jobs. For this problem we show that a schedule with length at most (1 + ε)OPT can be calculated in polynomial time. Unless P = NP, this is the best possible result (in the sense of approximation ratio), since the problem is strongly NP-hard. For the case, where all jobs must be allotted to a subset of consecutive machines, a schedule with length at most (1.5 + ε)OPT can be calculated in polynomial time. The previously best known results are algorithms with absolute approximation ratio 2. Furthermore, we extend both algorithms to the case of malleable jobs with the same approximation ratios