Submodular problems - approximations and algorithms

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📝 Original Info

  • Title: Submodular problems - approximations and algorithms
  • ArXiv ID: 1010.1945
  • Date: 2017-05-01
  • Authors: 원문에 저자 정보가 제공되지 않았습니다. —

📝 Abstract

We show that any submodular minimization (SM) problem defined on a linear constraint set with constraints having up to two variables per inequality, are 2-approximable in polynomial time. If the constraints are monotone (the two variables appear with opposite sign coefficients) then the problems of submodular minimization or supermodular maximization are polynomial time solvable. The key idea is to link these problems to a submodular s,t-cut problem defined here. This framework includes the problems: SM-vertex cover; SM-2SAT; SM-min satisfiability; SM-edge deletion for clique, SM-node deletion for biclique and others. We also introduce here the submodular closure problem and and show that it is solvable in polynomial time and equivalent to the submodular cut problem. All the results are extendible to multi-sets where each element of a set may appear with a multiplicity greater than 1. For all these NP-hard problems 2-approximations are the best possible in the sense that a better approximation factor cannot be achieved in polynomial time unless NP=P. The mechanism creates a relaxed "monotone" problem, solved as a submodular closure problem, the solution to which is mapped to a half integral super-optimal solution to the original problem. That half-integral solution has the persistency property meaning that integer valued variables retain their value in an optimal solution. This permits to delete the integer valued variables, and restrict the search of an optimal solution to the smaller set of remaining variables.

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We demonstrate here that constrained submodular minimization problems, where each constraint has at most two variables, (SM2), are 2-approximable in polynomial time. This approximation factor of 2 for SM2 is provably best possible unless NP=P. Furthermore, if the coefficients of the two variables in each constraint have opposite signs, then the submodular minimization problem is solved in (strongly) polynomial time. All these results extend to multi-sets submodular minimization as well.

A nonnegative function f defined on the subsets of a set V is said to be submodular if it satisfies for all X, Y ⊆ V , f (X) + f (Y ) ≥ f (X ∩ Y ) + f (X ∪ Y ). A submodular function f is said to be monotone if f (S) ≤ f (T ) for any S ⊆ T . A binary vector of dimension n = |V |, x = {x i } n i=1 , is associated with a corresponding subset of V , X = {i ∈ V |x i = 1}. The vector x is then said to be the characteristic vector of the set X.

The SM2 problem addressed here is, min f (X) (SM2) subject to a ij x i + b ij x j ≥ c ij for all (i, j) ∈ A

x j ∈ {0, 1} for all j ∈ V, where a ij , b ij and c ij are any real numbers and A is a set of pairs (including singletons, and also allowing multiple copies of the same pair) defining the constraints. Our main results are that any SM2 problem, with constraints that satisfy the round up property or with monotone submodular objective function, is 2-approximable in polynomial time and this approximation factor cannot be improved unless NP=P. A set of constraints satisfies the round up property if any feasible half integer solution can be rounded up to an integer feasible vector. Vertex cover and all covering constraints satisfy the round up property, but non-covering problems, such as minimum (weighted) node deletion so remaining graph is a maximum clique, satisfy the round up property as well. The formulations and discussion of the properties of these and other SM2 problems is given in Section 3.1. An inequality constraint in up to two variables, a ij x ib ij x j ≥ c ij is called monotone if a ij and b ij have the same signs. (This concept of monotonicity is unrelated to the monotonicity of a submodular function.) The problem of submodular minimization or supermodular maximization on monotone constraints is shown here to be polynomial time solvable. This is in stark contrast to submodular minimization or supermodular maximization over constraints with totally unimodular constraints matrix which is proved here to be NP-hard. This demonstrates that monotone constraints form a more significant structure than totally unimodular constraints, in terms of complexity, for submodular (supermodular) minimization (maximization).

The results here all apply to submodular minimization on multi-sets, (SM2-multi). These are submodular functions defined on sets containing elements with multiplicity greater than 1. A nonnegative integer vector x ∈ Z n is the characteristic vector of a multiset X = {(i, q i )|x i = q i }, where (i, q i ) ∈ X means that X contains element i q i times, for positive integers q i . All properties of submodular functions extend to multi-sets, with the generalized definition of containment, X 1 ⊆ X 2 to mean that for all (i, q i ) ∈ X 1 , (i, q ′ i ) ∈ X 2 with q i ≤ q ′ i . The problem of constrained submodular minimization on multi-sets is min{f (X)|Ax ≥ b, 0 ≤ x ≤ u, x ∈ Z n }. Let the upper bound on the multiplicity of element i be u i . The formulation of SM2-multi is then,

The respective 2-approximations or polynomial time algorithms for multi-sets are attained in time polynomial in U = max j=1,…n u j . The dependence of the run time on U cannot be removed (to, say, logarithmic dependence) unless NP=P.

A prominent example of SM2 is the submodular vertex cover, SM-vertex cover, where the constraint matrix A contains exactly two 1s per row and b is a vector of 1s.

Approximating SM-vertex cover has been a subject of previous research work. Three different 2approximation algorithms were devised for the problem: Koufogiannakis and Young [KY09] devised approximations for SM-“covering” problems with monotone submodular objective function. The approximation algorithm is based on the frequency technique (called maximal dual feasible technique in [Hoc97] Ch. 3). Their algorithm is a 2-approximation for the SM-vertex cover for monotone submodular objective function. Goel et al. [GKTW09] devised a 2-approximation algorithm for SM-vertex cover with monotone submodular function which involves solving a relaxation with the Ellipsoid method with a separation algorithm equivalent to a submodular minimization problem. Goel et al. further proved that submodular vertex cover is inapproximable within a factor better than 2. Iwata and Nagano in [IN09] presented a 2-approximation algorithm for the SM-vertex cover, and addressed the SM-set cover and the SM-edge cover. Their algorithm does not require the submodular function to be monotone. Iwata and Nagano’s technique relies on using Lo

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