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

📝 Abstract

This paper describes a simple greedy D-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 D variables of the problem. (A simple example is Vertex Cover, with D = 2.) The algorithm generalizes previous approximation algorithms for fundamental covering problems and online paging and caching problems.

💡 Analysis

This paper describes a simple greedy D-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 D variables of the problem. (A simple example is Vertex Cover, with D = 2.) The algorithm generalizes previous approximation algorithms for fundamental covering problems and online paging and caching problems.

📄 Content

Greedy ∆-Approximation Algorithm for Covering with Arbitrary Constraints and Submodular Cost Christos Koufogiannakis Neal E. Young ∗ October 26, 2018 Abstract This paper describes a simple greedy ∆-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 ∆variables of the problem. (A simple example is VERTEX COVER, with ∆= 2.) The algorithm generalizes previous approximation algorithms for fundamental covering problems and online paging and caching problems. 1 Introduction and summary The classification of general techniques is an important research program within the field of approximation algorithms. What abstractions are useful for capturing a wide variety of problems and analyses? What are the scopes of, and the relationships between, the various algorithm-design techniques such as the primal- dual method, the local-ratio method [5], and randomized rounding? Which problems admit optimal and fast greedy approximation algorithms [26, 11, 12]? What general techniques are useful for designing online algorithms? What is the role of locality among constraints and variables [53, 46, 5]? We touch on these topics, exploring a simple greedy algorithm for a general class of covering problems. The algorithm has approximation ratio ∆provided each covering constraint in the instance constrains only ∆variables. Throughout the paper, ¯R≥0 denotes R≥0 ∪{∞} and ¯Z≥0 denotes Z≥0 ∪{∞}. The conference version of this paper is [44]. The journal version has been accepted to Algorithmica. Definition 1 (Submodular-Cost Covering). An instance is a triple (c, C, U), where • The cost function c : ¯Rn ≥0 →¯R≥0 is submodular,1 continuous, and non-decreasing. • The constraint set C ⊆2 ¯Rn ≥0 is a collection of covering constraints, where each constraint S ∈C is a subset of ¯Rn ≥0 that is closed upwards2 and under limit. We stress that each S may be non-convex. ∗Department of Computer Science and Engineering, University of California, Riverside. 1Formally, c(x) + c(y) ≥c(x ∧y) + c(x ∨y), where x ∧y (and x ∨y) are the component-wise minimum (and maximum) of x and y. Intuitively, there is no positive synergy between the variables: let ∂jc(x) denote the rate at which increasing xj would increase c(x); then, increasing xi (for i ̸= j) does not increase ∂jc(x). Any separable function c(x) = P j cj(xj) is submodular, the product c(x) = Q j xj is not. The maximum c(x) = maxj xj is submodular, the minimum c(x) = minj xj is not. 2If y ≥x and x ∈S, then y ∈S, perhaps the minimal requirement for a constraint to be called a “covering” constraint. 1 arXiv:0807.0644v4 [cs.DS] 30 Dec 2011 • For each j ∈[n], the domain Uj (for variable xj) is any subset of ¯R≥0 that is closed under limit. The problem is to find x ∈¯Rn ≥0, minimizing c(x) subject to xj ∈Uj (∀j ∈[n]) and x ∈S (∀S ∈C). The definition assumes the objective function c(x) is defined over all x ∈¯Rn ≥0, even though the so- lution space is constrained to x ∈U. This is unnatural, but any c that is properly defined on U extends appropriately3 to ¯Rn ≥0. In the cases discussed here c extends naturally to ¯Rn ≥0 and this issue does not arise. We call this problem SUBMODULAR-COST COVERING.4 For intuition, consider the well-known FACILITY LOCATION problem. An instance is specified by a collection of customers, a collection of facilities, an opening cost fj ≥0 for each facility, and an assignment cost dij ≥0 for each customer i and facility j ∈N(i). The problem is to open a subset F of the facilities so as to minimize the cost to open facilities in F (that is, P j∈F fj) plus the cost for each customer to reach its nearest open, admissible facility (that is, P i min{dij | j ∈F}). This is equivalent to SUBMODULAR-COST COVERING instance (c, C, U), with • a variable xij for each customer i and facility j, with domain Uij = {0, 1}, • for each customer i, (non-convex) constraint maxj∈N(i) xij ≥1 (the customer is assigned a facility), • and (submodular) cost c(x) = P j fj maxi xij + P i,j dijxij (opening cost plus assignment cost). A greedy algorithm for Submodular-Cost Covering (Section 2). The core contribution of the paper is a greedy ∆-approximation algorithm for the problem, where ∆is the maximum number of variables that any individual covering constraint S in C constrains. For S ∈C, let vars(S) contain the indices of variables that S constrains (i.e, j ∈vars(S) if membership of x in S depends on xj). The algorithm is roughly as follows. Start with an all-zero solution x, then repeat the following step until all constraints are satisfied: Choose any not-yet-satisfied constraint S. To satisfy S, raise each xj for j ∈vars(S) (i.e., raise the variables that S constrains), so that each raised variable’s increase contributes the same amount to the increase in the cost. Section 2 gives the full algorithm and its analys

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