522 research outputs found
Guarantees and Limits of Preprocessing in Constraint Satisfaction and Reasoning
We present a first theoretical analysis of the power of polynomial-time
preprocessing for important combinatorial problems from various areas in AI. We
consider problems from Constraint Satisfaction, Global Constraints,
Satisfiability, Nonmonotonic and Bayesian Reasoning under structural
restrictions. All these problems involve two tasks: (i) identifying the
structure in the input as required by the restriction, and (ii) using the
identified structure to solve the reasoning task efficiently. We show that for
most of the considered problems, task (i) admits a polynomial-time
preprocessing to a problem kernel whose size is polynomial in a structural
problem parameter of the input, in contrast to task (ii) which does not admit
such a reduction to a problem kernel of polynomial size, subject to a
complexity theoretic assumption. As a notable exception we show that the
consistency problem for the AtMost-NValue constraint admits a polynomial kernel
consisting of a quadratic number of variables and domain values. Our results
provide a firm worst-case guarantees and theoretical boundaries for the
performance of polynomial-time preprocessing algorithms for the considered
problems.Comment: arXiv admin note: substantial text overlap with arXiv:1104.2541,
arXiv:1104.556
A Branch-and-Reduce Algorithm for Finding a Minimum Independent Dominating Set
An independent dominating set D of a graph G = (V,E) is a subset of vertices
such that every vertex in V \ D has at least one neighbor in D and D is an
independent set, i.e. no two vertices of D are adjacent in G. Finding a minimum
independent dominating set in a graph is an NP-hard problem. Whereas it is hard
to cope with this problem using parameterized and approximation algorithms,
there is a simple exact O(1.4423^n)-time algorithm solving the problem by
enumerating all maximal independent sets. In this paper we improve the latter
result, providing the first non trivial algorithm computing a minimum
independent dominating set of a graph in time O(1.3569^n). Furthermore, we give
a lower bound of \Omega(1.3247^n) on the worst-case running time of this
algorithm, showing that the running time analysis is almost tight.Comment: Full version. A preliminary version appeared in the proceedings of WG
200
Fair assignment of indivisible objects under ordinal preferences
We consider the discrete assignment problem in which agents express ordinal
preferences over objects and these objects are allocated to the agents in a
fair manner. We use the stochastic dominance relation between fractional or
randomized allocations to systematically define varying notions of
proportionality and envy-freeness for discrete assignments. The computational
complexity of checking whether a fair assignment exists is studied for these
fairness notions. We also characterize the conditions under which a fair
assignment is guaranteed to exist. For a number of fairness concepts,
polynomial-time algorithms are presented to check whether a fair assignment
exists. Our algorithmic results also extend to the case of unequal entitlements
of agents. Our NP-hardness result, which holds for several variants of
envy-freeness, answers an open question posed by Bouveret, Endriss, and Lang
(ECAI 2010). We also propose fairness concepts that always suggest a non-empty
set of assignments with meaningful fairness properties. Among these concepts,
optimal proportionality and optimal weak proportionality appear to be desirable
fairness concepts.Comment: extended version of a paper presented at AAMAS 201
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