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A constraint satisfaction problem (CSP),\(\textsf {Max-CSP}(\mathcal {F})\), is specified by a finite set of constraints\(\mathcal {F}\subseteq \lbrace [q]^k \rightarrow \lbrace 0,1\rbrace \rbrace\)for positive integersqandk. An instance of the problem onnvariables is given bymapplications of constraints from\(\mathcal {F}\)to subsequences of thenvariables, and the goal is to find an assignment to the variables that satisfies the maximum number of constraints. In the (γ ,β)-approximation version of the problem for parameters 0 ≤ β ≤ γ ≤ 1, the goal is to distinguish instances where at least γ fraction of the constraints can be satisfied from instances where at most β fraction of the constraints can be satisfied. In this work, we consider the approximability of this problem in the context of sketching algorithms and give a dichotomy result. Specifically, for every family\(\mathcal {F}\)and every β < γ, we show that either a linear sketching algorithm solves the problem in polylogarithmic space or the problem is not solvable by any sketching algorithm in\(o(\sqrt {n})\)space. In particular, we give non-trivial approximation algorithms using polylogarithmic space for infinitely many constraint satisfaction problems. We also extend previously known lower bounds for general streaming algorithms to a wide variety of problems, and in particular the case ofq=k=2, where we get a dichotomy, and the case when the satisfying assignments of the constraints of\(\mathcal {F}\)support a distribution on\([q]^k\)with uniform marginals. Prior to this work, other than sporadic examples, the only systematic classes of CSPs that were analyzed considered the setting of Boolean variablesq= 2, binary constraintsk=2, and singleton families\(|\mathcal {F}|=1\)and only considered the setting where constraints are placed on literals rather than variables. Our positive results show wide applicability of bias-based algorithms used previously by [47] and [41], which we extend to include richer norm estimation algorithms, by giving a systematic way to discover biases. Our negative results combine the Fourier analytic methods of [56], which we extend to a wider class of CSPs, with a rich collection of reductions among communication complexity problems that lie at the heart of the negative results. In particular, previous works used Fourier analysis over the Boolean cube to initiate their results and the results seemed particularly tailored to functions on Boolean literals (i.e., with negations). Our techniques surprisingly allow us to get to generalq-ary CSPs without negations by appealing to the same Fourier analytic starting point over Boolean hypercubes.
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Width Helps and Hinders Splitting Flows
Minimum flow decomposition (MFD) is the NP-hard problem of finding a smallest decomposition of a network flow/circulationXon a directed graphGinto weighted source-to-sink paths whose weighted sum equalsX. We show that, for acyclic graphs, considering thewidthof the graph (the minimum number of paths needed to cover all of its edges) yields advances in our understanding of its approximability. For the version of the problem that uses only non-negative weights, we identify and characterise a new class ofwidth-stablegraphs, for which a popular heuristic is aO(logVal(X))-approximation (Val(X) being the total flow ofX), and strengthen its worst-case approximation ratio from\(\Omega (\sqrt {m})\)to Ω (m/logm) for sparse graphs, wheremis the number of edges in the graph. We also study a new problem on graphs with cycles, Minimum Cost Circulation Decomposition (MCCD), and show that it generalises MFD through a simple reduction. For the version allowing also negative weights, we give a (⌈ log ‖ X ‖ ⌉ +1)-approximation (‖X‖ being the maximum absolute value ofXon any edge) using a power-of-two approach, combined with parity fixing arguments and a decomposition of unitary circulations (‖X‖ ≤ 1), using a generalised notion of width for this problem. Finally, we disprove a conjecture about the linear independence of minimum (non-negative) flow decompositions posed by Kloster et al. [2018], but show that its useful implication (polynomial-time assignments of weights to a given set of paths to decompose a flow) holds for the negative version.
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- Award ID(s):
- 2309902
- PAR ID:
- 10497154
- Publisher / Repository:
- Association for Computing Machinery
- Date Published:
- Journal Name:
- ACM Transactions on Algorithms
- Volume:
- 20
- Issue:
- 2
- ISSN:
- 1549-6325
- Page Range / eLocation ID:
- 1 to 20
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1145/3641820
