In this paper, we propose new efficient algorithms to verify the null space condition in compressed sensing (CS). Given an $(n-m) \times n$ ($m>0$) CS matrix $A$ and a positive $k$, we are interested in computing $\displaystyle \alpha_k = \max_{\{z: Az=0,z\neq 0\}}\max_{\{K: |K|\leq k\}}$ ${\|z_K \|_{1}}{\|z\|_{1}}$, where $K$ represents subsets of $\{1,2,...,n\}$, and $|K|$ is the cardinality of $K$. In particular, we are interested in finding the maximum $k$ such that $\alpha_k < {1}{2}$. However, computing $\alpha_k$ is known to be extremely challenging. In this paper, we first propose a series of new polynomial-time algorithms to compute upper bounds on $\alpha_k$. Based on these new polynomial-time algorithms, we further design a new sandwiching algorithm, to compute the \emph{exact} $\alpha_k$ with greatly reduced complexity. When needed, this new sandwiching algorithm also achieves a smooth tradeoff between computational complexity and result accuracy. Empirical results show the performance improvements of our algorithm over existing known methods; and our algorithm outputs precise values of $\alpha_k$, with much lower complexity than exhaustive search.

In this paper, we propose the first heuristic approach for the vertex separator problem (VSP), based on Breakout Local Search (BLS). BLS is a recent meta-heuristic that follows the general framework of the popular Iterated Local Search (ILS) with a particular focus  on the perturbation strategy. Based on some relevant information on search history, it tries to introduce the most suitable degree of diversification by determining adaptively the number and type of moves for the next perturbation phase. The proposed heuristic is highly competitive with the exact state-of-art approaches from the literature on the current VSP benchmark. Moreover, we present for the first time computational results for a set of large graphs with up to 3000 vertices, which constitutes a new challenging benchmark for VSP approaches.

Heuristic search with reachability-based heuristics is arguably the most successful paradigm in Automated Planning to date. In its earlier stages of development, heuristic search was proposed as both forward and backward search. Due to the disadvantages of backward search, in the last decade researchers focused mainly on forward search, and backward search was abandoned for the most part as a valid alternative. In the last years, important advancements regarding both the theoretical understanding and the performance of heuristic search have been achieved, applied mainly to forward search planners. In this work we revisit regression in planning with reachability-based heuristics, trying to extrapolate to backward search current lines of research that were not as well understood as they are now.

Random walks have become a popular component of recent planning systems. The increased exploration is a valuable addition to more exploitative search methods such as Greedy Best First Search (GBFS). A number of successful planners which incorporate random walks have been built. The work presented here aims to exploit the experience gained from building those systems. It begins a systematic study of the design space and alternative choices for building such a system, and develops a new random walk planner from scratch, with careful experiments along the way. Four major insights are: 1. a high state evaluation frequency is usually superior to the endpoint-only evaluation used in earlier systems, 2. adjusting the restarting parameter according to the progress speed in the search space performs better than any fixed setting, 3. biasing the action selection towards preferred operators of only the current state is better than Monte Carlo Helpful Actions, which depend on the number of times an action has been a preferred operator in previous walks, and 4. even simple forms of random walk planning can compete with GBFS.

There are many hard shortest-path search problems that cannot be solved, because best-first search runs out of memory space and depth-first search runs out of time. We propose Forward Perimeter Search (FPS), a heuristic search with controlled use of memory. It builds a perimeter around the root node and tests each perimeter node for a shortest path to the goal. The perimeter is adaptively extended towards the goal during the search process. We show that FPS expands in random 24-puzzles 50% fewer nodes than BF-IDA* while requiring several orders of magnitude less memory. Additionally, we present a hard problem instance of the 24-puzzle that needs at least 140 moves to solve; i.e. 26 more moves than the previously published hardest instance.

In this paper, we revisit the idea of splitting a planning problem into subproblems hopefully easier to solve with the help of landmark analysis. While this technique initially proposed in the first approaches related to landmarks has been outperformed by landmark-based heuristics, we believe that it is still a promising research direction. To this end, we propose a new method for problem splitting based on landmarks which has two advantages over the original technique: it is complete (if a solution exists, the algorithm finds it), and it uses the precedence relation over the landmarks in a more flexible way. We lay in this paper the foundations of a meta best-first search algorithm, which explores the landmark orderings to create subproblems and can use any embedded planner to solve subproblems. It opens up avenues for future research: among them are new heuristics for guiding the meta search towards the most promising orderings, different policies for generating subproblems, and influence of the embedded subplanner.

Constructing a strong heuristic function is a central problem in heuristic search. A common approach is to combine a number of heuristics by maximizing over the values from each. If a limit is placed on this number, then a subset selection problem arises. We treat this as an optimization problem, and proceed by translating a natural loss function into a submodular and monotonic utility function under which greedy selection is guaranteed to be near-optimal. We then extend this approach with a sampling scheme that retains provable optimality. Our empirical results show large improvements over existing methods, and give new insight into building heuristics for directed domains.

The recent proliferation of smart-phones and other wearable devices has lead to a surge of new mobile applications. Partially observable Markov decision processes provide a natural framework to design applications that continuously make decisions based on noisy sensor measurements. However, given the limited battery life, there is a need to minimize the amount of online computation. This can be achieved by compiling a policy into a finite state controller since there is no need for belief monitoring or online search. In this paper, we propose a new branch and bound technique to search for a good controller. In contrast to many existing algorithms for controllers, our search technique is not subject to local optima. We also show how to reduce the amount of search by avoiding the enumeration of isomorphic controllers and by taking advantage of suitable upper and lower bounds. The approach is demonstrated on several benchmark problems as well as a smart-phone application to assist persons with Alzheimer's to wayfind.

Combinatorial Optimization is an important area of computer science that has many theoretical and practical applications. In the thesis, we present important contributions to several different areas of combinatorial optimization, including nogood learning, symmetry breaking, dominance, relaxations and parallelization. We develop a new nogood learning technique based on constraint projection that allows us to exploit subproblem dominances that arise when two different search paths lead to subproblems which are identical on the remaining unfixed variables. We present a new symmetry breaking technique called SBDS-1UIP, which extends Symmetry Breaking During Search (SBDS) by using the more powerful 1UIP nogoods generated by Lazy Clause Generation (LCG) solvers. We present two new general methods for exploiting almost symmetries by modifying SBDS-1UIP and by using conditional symmetry breaking constraints. We solve the Minimization of Open Stacks Problem, the Talent Scheduling Problem (CSPLib prob039), and the Maximum Density Still Life Problem (CSPLib prob032) many orders of magnitude faster than the previous state of the art by applying various powerful techniques such as nogood learning, dynamic programming, dominance and relaxations. We present cache aware data structures for SAT solvers which allows sequential and parallel versions of SAT solvers to run more quickly. And we present a new load balancing scheme for parallel search called confidence based work stealing, which allows the parallel search to make use of the information contained in the branching heuristic.

This paper introduces Monte Carlo *-Minimax Search (MCMS), a Monte Carlo search algorithm for turned-based, stochastic, two-player, zero-sum games of perfect information. The algorithm is designed for the class of of densely stochastic games; that is, games where one would rarely expect to sample the same successor state multiple times at any particular chance node. Our approach combines sparse sampling techniques from MDP planning with classic pruning techniques developed for adversarial expectimax planning. We compare and contrast our algorithm to the traditional *-Minimax approaches, as well as MCTS enhanced with the Double Progressive Widening, on four games: Pig, EinStein Wurfelt Nicht!, Can't Stop, and Ra. Our results show that MCMS can be competitive with enhanced MCTS variants in some domains, while consistently outperforming the equivalent classic approaches given the same amount of thinking time.