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Greedy Δ-Approximation Algorithm for Covering with Arbitrary Constraints and Submodular Cost

Abstract

This paper describes a simple greedy Δ-approximation algorithm for any covering problem whose objective function is submodular and non-decreasing, and whose feasible region can be expressed as the intersection of arbitrary (closed upwards) covering constraints, each of which constrains at most Δ variables of the problem. (A simple example is Vertex Cover, with Δ=2.) The algorithm generalizes previous approximation algorithms for fundamental covering problems and online paging and caching problems.

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Notes

  1. 1.

    Formally, c(x)+c(y)≥c(xy)+c(xy), where xy (and xy) are the component-wise minimum (and maximum) of x and y. Intuitively, there is no positive synergy between the variables: let j c(x) denote the rate at which increasing x j would increase c(x); then, increasing x i (for ij) does not increase j c(x). Any separable function c(x)=∑ j c j (x j ) is submodular, the product c(x)=∏ j x j is not. The maximum c(x)=max j x j is submodular, the minimum c(x)=min j x j is not.

  2. 2.

    If yx and xS, then yS, perhaps the minimal requirement for a constraint to be called a “covering” constraint.

  3. 3.

    One way to extend c from U to \(\bar{\mathbb{R}}_{\ge0}\): take the cost of \(x\in\bar{\mathbb{R}}_{\ge0}^{n}\) to be the expected cost of \(\tilde{x}\), where \(\tilde{x}_{j}\) is rounded up or down to its nearest elements a,b in U j such that ax j b: take \(\tilde{x}_{j} = b\) with probability \(\frac{b-x_{j}}{b-a}\), otherwise take \(\tilde{x}_{j} = a\). If a or b doesn’t exist, let \(\tilde{x}_{j}\) be the one that does. As long as c is non-decreasing, sub-modular, and (where appropriate) continuous over U, this extension will have these properties over \(\bar{\mathbb{R}}_{\ge0}^{n}\).

  4. 4.

    Changed from “Monotone Covering” in the conference version [44] due to name conflicts.

  5. 5.

    Set Multicover is CIP-UB restricted to A ij ∈{0,1}; Set Cover is Set Multicover restricted to b i =1; Vertex Cover is Set Cover restricted to Δ=2.

  6. 6.

    The standard LP relaxation has arbitrarily large gap (e.g. min{x 1∣10x 1+10x 2≥11;x 2≤1} has gap 10).

    Carr et al. [16] state (without details) that their CIP-01 result extends CIP-UP, but it is not clear how (see [54, 55]).

  7. 7.

    If some c j =0, then x j is raised instantaneously to ∞ at cost 0, after which the cost of x increases at rate less than \(|\operatorname{\mathsf{vars}}(S)|\).

  8. 8.

    Readers may recognize a similarity to the local-ratio method. This is explored in Sect. 5.

  9. 9.

    To see this, consider the variables x j for \(j\in\operatorname{\mathsf{vars}}(S)\) one at a time, in at most Δ steps; by submodularity of c, in a step that increases a given x j , the increase in c(x) is at most what it would have been if x j had been increased first, i.e., at most β.

  10. 10.

    If the cost function is linear, in responding to S this algorithm needs to know only S and the values of variables in S and their cost coefficients. In general, the algorithm needs to know S, the entire cost function, and all variables’ values.

  11. 11.

    We assume the last request must stay cached. If not, don’t subtract r t from Q in each constraint. The competitive ratio is k+1.

  12. 12.

    This definition assumes that the request sequence and cacheability requirements are independent of the responses of the algorithm. In practice, even for standard paging, this assumption might not hold. For example, a fault incurred by one process may cause another process’s requests to come earlier. In this case, the optimal offline strategy would choose responses that take into account the effects on inputs at subsequent times (possibly leading to a lower cost). Modeling this accurately seems difficult.

  13. 13.

    Here is one of many ways to modify Algorithm 3 to handle arbitrary closed U j ’s. In each step, take β small enough so that for each \(j\in\operatorname {\mathsf{vars}}(S)\), either U j contains the entire interval [x j ,x j +β], or U j contains just x j from that interval. For the latter type of j, take β j and \(\hat{x}_{j}\) as described in Algorithm 3. For the former type of j, take β j =β and take \(\hat{x}_{j}\) to be the smallest value such that increasing x j to \(\hat{x}_{j}\) would increase c(x) by β. Then proceed as above. (Taking β infinitesimally small gives the following process. For each \(j\in \operatorname{\mathsf{vars}}(S)\) simultaneously, x j increases continuously at rate inversely proportional to its contribution to the cost, if it is possible to do so while maintaining x j U j , and otherwise x j increases to its next allowed value randomly according to a Poisson process whose intensity is inversely proportional to the resulting expected increase in the cost.)

  14. 14.

    The online solution is not x, but rather x′≤x defined from x by \(x'_{j} = \max\{\alpha\in U_{j}\mid\alpha\le x_{j}\}\) or something similar, so the algorithms maintain state other than the current online solution x′. For example, for paging problems, the algorithms maintain x t ∈[0,1] as they proceed, where a requested item r s is currently evicted only once x s =1. To be stateless, they should maintain each x t ∈{0,1}, where x s =0 iff page r s is still in the cache.

  15. 15.

    If xS and \(y\in\overline{S}^{*}\), then y is the limit of some sequence {y t} of points in \(\overline{S}\). Each y t has \(x^{t}_{j(t)} \ge y^{t}_{j(t)}\) for some \(j(t)\in \operatorname{\mathsf{vars}}(S)\). Since \(|\operatorname{\mathsf{vars}}(S)|\) is finite, for some \(j\in \operatorname{\mathsf{vars}}(S)\), the infinite subsequence {y tj(t)=j} also has y as a limit point. Then y j is the limit of the \(y^{t}_{j}\)’s in this subsequence, each of which is at most x j , so y j is at most x j .

  16. 16.

    The relaxation gap is the maximum, over all instances \((c,\mathcal{C})\) of Linear-Cost Covering, of the ratio \([\mbox{\textit{optimal cost for}}\ (c,\mathcal{C})]/[\mbox{\textit{optimal cost for its relaxation}}\ (c,\mathcal{R}^{1})]\).

  17. 17.

    Briefly, run the ellipsoid method to solve \((c,\mathcal{R}^{1})\) using a separation oracle that, given x, checks whether ΔxS for all \(S\in\mathcal{C}\), and, if not, returns an inequality that x violates for \(\mathcal{R}^{1}\) (from the proof of Observation 5). Either the oracle finds, for some x, that ΔxS for all S, in which case x′=Δx is a Δ-approximate solution for \((c,\mathcal{C})\), or the oracle returns to the ellipsoid method a sequence of violated inequalities that, collectively, prove that \((c,\mathcal{R}^{1})\) (and thus \((c,\mathcal{C})\)) is infeasible.

  18. 18.

    The instance \((c,\mathcal{C})\) defined by \(\min\{x_{1} + x_{2} \mid x\in\mathbb{R}_{\ge0}^{2}; x_{1}+x_{2} \ge1\}\) has optimum cost 1. In its first relaxation \((c,\mathcal{R}^{1})\), x 1=x 2=1/4 with cost 1/2 is feasible. But one can show (via duality) that \((c',\mathcal{R}^{2})\) has optimal cost at least 1.

  19. 19.

    In fact this dual variable must be 0 before this, because \(x'_{j}>x_{j}\) for some j, so this dual variable has not been raised before.

  20. 20.

    Here is an example in ℝ2. For v∈ℝ2, let |v| denote the 1-norm ∑ i |v i |. For each \(v\in\mathbb{R}_{\ge0}^{2}\) such that |v|=1, define constraint set \(S_{v} = \{ x\in\mathbb{R}_{\ge0}^{2} : (\exists j) x_{j} \ge v_{j}\}\). Consider the covering problem min{|x|:(∀v)xS v }.

    Each constraint xS v excludes points dominated by v, so the intersection of all S v ’s is \(\{x \in\mathbb{R}_{\ge0}^{2} :|x| \ge1\}\). On the other hand, since S v contains the points (v 1,0) and (0,v 2), \(\operatorname{\mathsf{conv}}(S_{v})\) must contain x=v 2(v 1,0)+v 1(0,v 1)=(v 1 v 2,v 1 v 2), where v 1 v 2≤(1/2)2=1/4. Thus, each \(\operatorname{\mathsf{conv}}(S_{v})\) contains x=(1/4,1/4), with |x|=1/2. Thus, the relaxation gap of \((c,\mathcal{L})\) for this instance is at least 2.

    Another example with Δ=2, this time in \(\mathbb{R}_{\ge0}^{n}\). Consider the sets \(S_{ij} = \{x\in\mathbb{R}_{\ge0}^{n} : \max (x_{i},x_{j}) \ge1\}\). Consider the covering problem min{|x|:(∀i,j)xS ij }. Each point x∈⋂ ij S ij has |x |≥(n−1)/n, but x=(1/2,1/2,1/2,…,1/2) is in each \(\operatorname{\mathsf {conv}}(S)\), and |x|=n/2, so the relaxation gap of \((c,\mathcal{L})\) is at least 2.

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Acknowledgements

The authors gratefully acknowledge Marek Chrobak for useful discussions, and two anonymous reviewers for careful and constructive reviews that helped improve the presentation.

This work was partially supported by National Science Foundation (NSF) grants CNS-0626912 and CCF-0729071.

Author information

Correspondence to Neal E. Young.

Appendix

Appendix

Proof of Observation 1 (reduction to canonical form)

Here is the reduction: Let \((c,U,\mathcal{C})\) be any instance of Submodular-Cost Covering. Construct its canonical form \((c,\mathcal{C}')\) as follows. First, assume without loss of generality that minU j =0 for each j. (If not, let j =minU j , then apply the translation xx′+ to the cost and feasible region: rewrite the cost c(x) as c′(x′)=c(x′+); rewrite each constraint “xS” as “x′∈S”; replace each domain U j by \(U'_{j} = U_{j} - \ell_{j}\).)

Next, define μ j (x)=max{αU j αx j } (that is, μ(x) is x with each coordinate lowered into U j ). For each constraint S in \(\mathcal{C}\), put a corresponding constraint “μ(x)∈S” in \(\mathcal{C}'\). The new constraint is closed upwards and closed under limit because S is and μ is non-decreasing. It is not hard to verify that any solution x to the canonical instance \((c,\mathcal{C}')\) gives a corresponding solution μ(x) to the original instance \((c,U,\mathcal{C})\), and that this reduction preserves Δ-approximation. □

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Koufogiannakis, C., Young, N.E. Greedy Δ-Approximation Algorithm for Covering with Arbitrary Constraints and Submodular Cost. Algorithmica 66, 113–152 (2013). https://doi.org/10.1007/s00453-012-9629-3

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Keywords

  • Covering
  • Linear programming
  • Approximation algorithms
  • Local ratio
  • Primal-dual
  • Vertex cover
  • Set cover
  • Integer linear programming
  • Online algorithms
  • Competitive analysis
  • Submodular cost
  • Paging
  • Caching