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- W4300786943 abstract "Given an n-vertex digraph D = (V, A) the Max-k-Ordering problem is to compute a labeling $ell : V to [k]$ maximizing the number of forward edges, i.e. edges (u,v) such that $ell$(u) < $ell$(v). For different values of k, this reduces to Maximum Acyclic Subgraph (k=n), and Max-Dicut (k=2). This work studies the approximability of Max-k-Ordering and its generalizations, motivated by their applications to job scheduling with soft precedence constraints. We give an LP rounding based 2-approximation algorithm for Max-k-Ordering for any k={2,..., n}, improving on the known 2k/(k-1)-approximation obtained via random assignment. The tightness of this rounding is shown by proving that for any k={2,..., n} and constant $varepsilon > 0$, Max-k-Ordering has an LP integrality gap of 2 - $varepsilon$ for $n^{Omegaleft(1/loglog kright)}$ rounds of the Sherali-Adams hierarchy. A further generalization of Max-k-Ordering is the restricted maximum acyclic subgraph problem or RMAS, where each vertex v has a finite set of allowable labels $S_v subseteq mathbb{Z}^+$. We prove an LP rounding based $4sqrt{2}/(sqrt{2}+1) approx 2.344$ approximation for it, improving on the $2sqrt{2} approx 2.828$ approximation recently given by Grandoni et al. (Information Processing Letters, Vol. 115(2), Pages 182-185, 2015). In fact, our approximation algorithm also works for a general version where the objective counts the edges which go forward by at least a positive offset specific to each edge. The minimization formulation of digraph ordering is DAG edge deletion or DED(k), which requires deleting the minimum number of edges from an n-vertex directed acyclic graph (DAG) to remove all paths of length k. We show that both, the LP relaxation and a local ratio approach for DED(k) yield k-approximation for any $kin [n]$." @default.
- W4300786943 created "2022-10-04" @default.
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- W4300786943 date "2015-07-02" @default.
- W4300786943 modified "2023-09-23" @default.
- W4300786943 title "On the Approximability of Digraph Ordering" @default.
- W4300786943 doi "https://doi.org/10.48550/arxiv.1507.00662" @default.
- W4300786943 hasPublicationYear "2015" @default.
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