When a small pattern graph does not occur inside a larger target graph, we can ask how to find "as much of the pattern as possible" inside the target graph. In general, this is known as the maximum common subgraph problem, which is much more computationally challenging in practice than subgraph isomorphism. We introduce a restricted alternative, where we ask if all but k vertices from the pattern can be found in the target graph. This allows for the development of slightly weakened forms of certain invariants from subgraph isomorphism which are based upon degree and number of paths.  We show that when k is small, weakening the invariants still retains much of their effectiveness. We are then able to solve this problem on the standard problem instances used to benchmark subgraph isomorphism algorithms, despite these instances being too large for current maximum common subgraph algorithms to handle. Finally, by iteratively increasing k, we obtain an algorithm which is also competitive for the maximum common subgraph

The study of broken-triangles is becoming increasingly ambitious, by both solving constraint satisfaction problems (CSPs) in polynomial time and reducing search space size through value merging or variable elimination. Considerable progress has been made in extending this important concept, such as dual broken-triangle and weakly broken-triangle, in order to maximize the number of captured tractable CSP instances and/or the number of merged values. Specifically, m -wBTP allows to merge more values than BTP. k -BTP, WBTP and m -BTP permit to capture more tractable instances than BTP. Here, we introduce a new weaker form of BTP, which will be called m -fBTP for flexible broken-triangle property. m -fBTP allows on the one hand to eliminate more variables than BTP while preserving satisfiability and on the other to define new bigger tractable class for which arc consistency is a decision procedure. Likewise, m -fBTP permits to merge more values than BTP but less than m -wBTP.

Potential heuristics for state-space search are defined as weighted sums over simple state features. Atomic features consider the value of a single state variable in a factored state representation, while binary features consider joint assignments to two state variables. Previous work showed that the set of all admissible and consistent potential heuristics using atomic features can be characterized by a compact set of linear constraints. We generalize this result to binary features and prove a hardness result for features of higher dimension. Furthermore, we prove a tractability result based on the treewidth of a new graphical structure we call the context-dependency graph . Finally, we study the relationship of potential heuristics to transition cost partitioning . Experimental results show that binary potential heuristics are significantly more informative than the previously considered atomic ones.

Multi-agent planning problems with constraints on global resource consumption occur in several domains. Existing algorithms for solving Multi-agent Markov Decision Processes can compute policies that meet a resource constraint in expectation, but these policies provide no guarantees on the probability that a resource constraint violation will occur. We derive a method to bound constraint violation probabilities using Hoeffding's inequality. This method is applied to two existing approaches for computing policies satisfying constraints: the Constrained MDP framework and a Column Generation approach. We also introduce an algorithm to adaptively relax the bound up to a given maximum violation tolerance. Experiments on a hard toy problem show that the resulting policies outperform static optimal resource allocations to an arbitrary level. By testing the algorithms on more realistic planning domains from the literature, we demonstrate that the adaptive bound is able to efficiently trade off violation probability with expected value, outperforming state-of-the-art planners.

Consider the situation where your favorite clustering algorithm applied to a data set returns a good clustering but there are a few undesirable properties. One adhoc way to fix this is to re-run the clustering algorithm and hope to find a better variation. Instead, we propose to not run the algorithm again but minimally modify the existing clustering to remove the undesirable properties. We formulate the minimal clustering modification problem where we are given an initial clustering produced from any algorithm. The clustering is then modified to: i) remove the undesirable properties and ii) be minimally different to the given clustering. We show the underlying feasibility sub-problem can be intractable and demonstrate the flexibility of our constraint programming formulation. We empirically validate its usefulness through experiments on social network and medical imaging data sets.

We study the problem of consistency checking for constraint networks over combined qualitative formalisms. We propose a framework which encompasses loose integrations and a form of spatio-temporal reasoning. In particular, we identify sufficient conditions ensuring the polynomiality of consistency checking, and we use them to find tractable subclasses.

Logic-based Benders decomposition (LBBD) is a powerful hybrid optimisation technique that can combine the strong dual bounds of mixed integer programming (MIP) with the combinatorial search strengths of constraint programming (CP). A major drawback of LBBD is that it is a far more involved process to implement an LBBD solution to a problem than the "model-and-run" approach provided by both CP and MIP. We propose an automated approach that accepts an arbitrary MiniZinc model and solves it using LBBD with no additional intervention on the part of the modeller. The design of this approach also reveals an interesting duality between LBBD and large neighborhood search (LNS). We compare our implementation of this approach to CP and MIP solvers on 4 different problem classes where LBBD has been applied before.

We present a new game-theoretic framework in which Bayesian players with bounded rationality engage in a Markov game and each has private but incomplete information regarding other players' types. Instead of utilizing Harsanyi's abstract types and a common prior, we construct intentional player types whose structure is explicit and induces a {\em finite-level} belief hierarchy. We characterize an equilibrium in this game and establish the conditions for existence of the equilibrium. The computation of finding such equilibria is formalized as a constraint satisfaction problem and its effectiveness is demonstrated on two cooperative domains.

Given an agricultural watershed containing a set of spatial units, and a set of land management practices, the Geodesign Optimization (GOP) aims to find a land management practice for each spatial unit that optimizes overall water quality improvements in the watershed under both budget constraint and spatial constraints (e.g., minimum contiguous area, shape) arising from farm equipment operation practicalities. GOP is important for redesign of agricultural watersheds in Midwestern US to mitigate soil and water quality degradation and loss of habitat. The problem is computationally challenging as a large-scale combinatorial problem (NP-hard) under spatial constraints. Existing optimization techniques do not address spatial constraints, and lead to impractical solutions requiring frequent farm equipment reconfiguration. In this paper, we formalize the spatially-constrained GOP and propose a novel spatial optimizer which explores optimal solution without constraint violations. Our approach is further validated through a Geodesign case study at Seven Mile Creek watershed in Midwestern US.

This paper deals with the constrained DAG shortest path problem (CDSP), which finds the shortest path on a given directed acyclic graph (DAG) under any logical constraints posed on taken edges. There exists a previous work that uses binary decision diagrams (BDDs) to represent the logical constraints, and traverses the input DAG and the BDD simultaneously. The time complexity of this BDD-based method is derived from BDD size, and tends to be fast only when BDDs are small. However, since it does not prioritize the search order, there is considerable room for improvement, particularly for large BDDs. We combine the well-known A* search with the BDD-based method synergistically, and implement several novel heuristic functions. The key insight here is that the ‘shortest path’ in the BDD is a solution of a relaxed problem, just as the shortest path in the DAG is. Experiments, particularly practical machine learning applications, show that the proposed method deceases search time by up to 2 orders of magnitude, with the specific result that it is 2,000 times faster than a commercial solver.