On Some Combinatorial Optimization Problems : Algorithms and Complexity

Uppman, Hannes

doi:10.3384/diss.diva-116859

Cited by 2 publications

(1 citation statement)

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“…The following lemma is a generalisation of [66, Lemma 2.9] from arity one to arbitrary arity and from finite-valued to valued constraint languages, but the proof is analogous. A special case has also been observed, in the context of Min-Sol problems [68], by Hannes Uppman [69]. Lemma 2.9.…”

Section: Fractional Polymorphismsmentioning

confidence: 78%

The Power of Sherali--Adams Relaxations for General-Valued CSPs

Thapper¹,

Živný²

2017

SIAM J. Comput.

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We give a precise algebraic characterisation of the power of Sherali-Adams relaxations for solvability of valued constraint satisfaction problems to optimality. The condition is that of bounded width which has already been shown to capture the power of local consistency methods for decision CSPs and the power of semidefinite programming for robust approximation of CSPs.Our characterisation has several algorithmic and complexity consequences. On the algorithmic side, we show that several novel and many known valued constraint languages are tractable via the third level of the Sherali-Adams relaxation. For the known languages, this is a significantly simpler algorithm than the previously obtained ones. On the complexity side, we obtain a dichotomy theorem for valued constraint languages that can express an injective unary function. This implies a simple proof of the dichotomy theorem for conservative valued constraint languages established by Kolmogorov andŽivný [JACM'13], and also a dichotomy theorem for the exact solvability of Minimum-Solution problems. These are generalisations of Minimum-Ones problems to arbitrary finite domains. Our result improves on several previous classifications by Khanna et al. [SICOMP'00], Jonsson et al. [SICOMP'08], and Uppman [ICALP'13]. Union is not liable for any use that may be made of the information contained therein.unless we settle the CSP dichotomy conjecture [30], some additional requirement on the unary weighted relations (such as injectivity) is necessary.As a corollary of our characterisation, we give, in Section 3.4, a complete complexity classification of exact solvability of Minimum-Solution problems over arbitrary finite domains, thus improving on previous partial classifications for domains of size two [42] and three [68], homogeneous and maximal (under a certain algebraic conjecture) languages [39] and on graphs with few vertices [41]. Theorem 3.19 shows that the Minimum-Solution problem is NP-hard unless it satisfies the bounded width condition. Previous partial results included ad-hoc algorithms for various special cases. Our result shows that one algorithm, the third level of the Sherali-Adams relaxation, solves all tractable cases and is thus universal. As a matter of fact, we actually prove a complexity classification for a larger class of problems that includes Minimum-Solutions problems as a special case, as described in detail in Section 3.4. Related workThe first level of the Sherali-Adams hierarchy is known as the basic linear programming (BLP) relaxation [16]. In [63], the authors gave a precise algebraic characterisation of Γ for which any instance of VCSP(Γ) is solved to optimality by BLP, see also [44]. The characterisation proved important not only in the study of VCSPs [36] and other classes of problems [34], but also in the design of fixed-parameter algorithms [37]. In [66], it was then shown that for finite-valued CSPs, the BLP solves all tractable cases; i.e. if BLP fails to solve any instance of some finite-valued constraint language then this la...

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Section: Fractional Polymorphismsmentioning

confidence: 78%