A clique model is one of the most important techniques on the cohesive subgraph detection; however, its applications are rather limited due to restrictive conditions of the model. Hence much research resorts to k -plex — a graph in which any vertex is adjacent to all but at most k vertices — which is a relaxation model of the clique. In this paper, we study the maximum k -plex problem and propose a fast algorithm to compute maximum k -plexes by exploiting structural properties of the problem. In an n -vertex graph, the algorithm computes optimal solutions in c n n O(1) time for a constant c < 2 depending only on k . To the best of our knowledge, this is the first algorithm that breaks the trivial theoretical bound of 2 n for each k ≥ 3. We also provide experimental results over multiple real-world social network instances in support.

One common pattern database compression technique is to merge adjacent database entries and store the minimum of merged entries to maintain heuristic admissibility. In this paper we propose a compression technique that preserves every entry, but reduces the number of bits used to store each entry, therefore limiting the values that can be represented. Even when this technique throws away low values in the heuristic, it can still have better performance than the traditional approach. We develop a theoretical basis for selecting which values to keep and show improved performance in both unidirectional and bidirectional search.

Bounding the partition function is a key inference task in many graphical models. In this paper, we develop an anytime anyspace search algorithm taking advantage of AND/OR tree structure and optimized variational heuristics to tighten deterministic bounds on the partition function. We study how our priority-driven best-first search scheme can improve on state-of-the-art variational bounds in an anytime way within limited memory resources, as well as the effect of the AND/OR framework to exploit conditional independence structure within the search process within the context of summation. We compare our resulting bounds to a number of existing methods, and show that our approach offers a number of advantages on real-world problem instances taken from recent UAI competitions.

The Minimum Sum Coloring Problem (MSCP) is an NP-Hard problem derived from the graph coloring problem (GCP) and has practical applications in different domains such as VLSI design, distributed resource allocation, and scheduling. There exist few exact solutions for MSCP, probably due to its search space much more elusive than that of GCP. On the contrary, much effort is spent in the literature to develop upper and lower bounds for MSCP. In this paper, we borrow a notion called motif, that was used in a recent work for upper bounding the minimum number of colors in an optimal solution of MSCP, to develop a new algebraic lower bound called for MSCP. Experiments on standard benchmarks for MSCP and GCP show that this new lower bound is substantially better than the existing lower bounds for several families of graphs.

We investigate the potential of exhaustively exploring larger neighborhoods in local search algorithms for Minimum Vertex Cover. More precisely, we study whether, for moderate values of k , it is feasible and worthwhile to determine, given a graph G with vertex cover C , if there is a k -swap S such that ( C ∖ S ) ∪ ( S ∖ C ) is a smaller vertex cover of G . First, we describe an algorithm running in ∆ O(k) ⋅ n time for searching the k -swap neighborhood on n -vertex graphs with maximum degree ∆. Then, we demonstrate that, by devising additional pruning rules that decrease the size of the search space, this algorithm can be implemented so that it solves the problem quickly for k ≈ 20. Finally, we show that it is worthwhile to consider moderately-sized k -swap neighborhoods. For our benchmark data set, we show that when combining our algorithm with a hill-climbing approach, the solution quality improves quickly with the radius k of the local search neighborhood and that in most cases optimal solutions can be found by setting k =21.

Black-box domains where the successor states generated by applying an action are generated by a completely opaque simulator pose a challenge for domain-independent planning. The main computational bottleneck in search-based planning for such domains is the number of calls to the black-box simulation. We propose a method for significantly reducing the number of calls to the simulator by the search algorithm by detecting and pruning sequences of actions which are dominated by others. We apply our pruning method to Iterated Width and breadth-first search in domain-independent black-box planning for Atari 2600 games in the Arcade Learning Environment (ALE), adding our pruning method significantly improves upon the baseline algorithms.

We describe an exact branch-and-bound algorithm for the maximum weight clique problem (MWC), called WLMC, that is especially suited for large vertex-weighted graphs. WLMC incorporates two original contributions: a preprocessing to derive an initial vertex ordering and to reduce the size of the graph, and incremental vertex-weight splitting to reduce the number of branches in the search space. Experiments on representative large graphs from real-world applications show that WLMC greatly outperforms relevant exact and heuristic MWC algorithms, and refute the prevailing hypothesis that exact MWC algorithms are less adequate for large graphs than heuristic algorithms.

A common strategy for improving optimization algorithms is to restart the algorithm when it is believed to be trapped in an inferior part of the search space. However, while specific restart strategies have been developed for specific problems (and specific algorithms), restarts are typically not regarded as a general tool to speed up an optimization algorithm. In fact, many optimization algorithms do not employ restarts at all. Recently, "bet-and-run" was introduced in the context of mixed-integer programming, where first a number of short runs with randomized initial conditions is made, and then the most promising run of these is continued. In this article, we consider two classical NP-complete combinatorial optimization problems, traveling salesperson and minimum vertex cover, and study the effectiveness of different bet-and-run strategies. In particular, our restart strategies do not take any problem knowledge into account, nor are tailored to the optimization algorithm. Therefore, they can be used off-the-shelf. We observe that state-of-the-art solvers for these problems can benefit significantly from restarts on standard benchmark instances.

In the recent years, there has been significant work on the difficulty of heuristic search problems, identifying different problem instance characteristics that can have a significant impact on search effort. Phase transitions in the solubility of random problem instances have proved useful in the study of problem difficulty for other classes of computational problems, notably SAT and CSP, and it has been shown that the hardest problems typically occur during this rapid transition. In this work, we perform the first empirical investigation of the phase transition phenomena for heuristic search. We establish the existence of a rapid transition in the solubility of an abstract model of heuristic search problems and show that, for greedy best first search, the hardest instances are associated with the phase transition region. We then perform a novel investigation of the behavior of heuristics of different strength across the solubility spectrum. Finally, we demonstrate that the behavior of our abstract model carries over to commonly used benchmark problems including the Pancake Problem, Grid Navigation, TopSpin, and the Towers of Hanoi. An interesting deviation is observed and explained in the Sliding Puzzle.

Metaheuristics have been developed to provide general purpose approaches for solving hard combinatorial problems. While these frameworks often serve as the starting point for the development of problem-specific search procedures, they very rarely work efficiently in their default state. We combine the ideas of reactive search, which adjusts key parameters during search, and algorithm configuration, which fine-tunes algorithm parameters for a given set of problem instances, for the automatic compilation of a portfolio of highly reactive dialectic search heuristics for MaxSAT. Even though the dialectic search metaheuristic knows nothing more about MaxSAT than how to evaluate the cost of a truth assignment, our automatically generated solver defines a new state of the art for random weighted partial MaxSAT instances. Moreover, when combined with an industrial MaxSAT solver, the self-assembled reactive portfolio was able to win four out of nine gold medals at the recent 2016 MaxSAT Evaluation on random, crafted, and industrial partial and weighted-partial MaxSAT instances.