A Note on Submodular Function Minimization by Chubanov s LP Algorithm

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1 A Note on Submodular Function Minimization by Chubanov s LP Algorithm Satoru FUJISHIGE September 25, 2017 Abstract Recently Dadush, Végh, and Zambelli (2017) has devised a polynomial submodular function minimization (SFM) algorithm based on their LP algorithm. In the present note we also show a weakly polynomial algorithm for SFM based on the recently developed linear programming feasibility algorithm of Chubanov (2017). Our algorithm is different from Dadush, Végh, and Zambelli s but is similar to theirs in ideas because both algorithms are based on recent geometric rescaling LP algorithms. Keywords: Submodular function minimization; base polyhedra; Chubanov s algorithm. 1. Introduction Since the advent of polynomial combinatorial algorithms for submodular function minimization (SFM) in 1999 [9, 11] substantial improvements over the algorithms have been made till recently. On the other hand, important developments in linear programming algorithms have recently been made by Chubanov [2, 3, 4], and Dadush, Végh, Zambelli [5] among others. The aim of this note is to give still another approach to SFM which should stimulate further research on SFM and hopefully lead us to some efficient algorithms both practically and theoretically. From the point of view of the computational complexity the currently best SFM algorithm is obtained by Lee, Sidford, and Wong [10], which is much faster than ours, though. Research Institute for Mathematical Sciences, Kyoto University, Kyoto , Japan. fujishig@kurims.kyoto-u.ac.jp. Research supported by JSPS KAKENHI Grant Number JP

2 The present note shows a weakly polynomial algorithm for submodular function minimization (SFM) based on the recently developed linear programming feasibility algorithm of Chubanov [4]. Recently Dadush, Végh, and Zambelli [6] has also devised a polynomial SFM algorithm based on their LP algorithms. Our algorithm is different from theirs [4] but is similar to theirs in ideas because both algorithms are based on recent geometric rescaling LP algorithms [4, 6]. We give some definitions and preliminaries in Section 2. In Section 3 we first consider the problem of testing membership in the base polyhedron associated with a given submodular function and show a weakly polynomial algorithm for the membership problem based on Chubanov s algorithm [4]. In Section 4 we show how to make the membership testing algorithm to compute a minimizer of a given submodular function. 2. Definitions and Preliminaries Let V be a finite nonempty set of cardinality V = n and f : 2 V R be a submodular function, i.e., f(x) + f(y ) f(x Y ) + f(x Y ) ( X, Y V ), (2.1) where we assume f( ) = 0. (See [7] for more information about relevant definitions and basic results on submodular functions.) The submodular polyhedron P(f) R V associated with the submodular function f : 2 V R is given by and the base polyhedron B(f) by P(f) = {x R V X V : x(x) f(x)} (2.2) B(f) = {x R V x P(f), x(v ) = f(v )}, (2.3) where for any X V we define x(x) = v X x(v) and x( ) = 0. We consider the membership problem that is to discern whether 0 B(f) or not and to find a set X V with f(x) < 0 if 0 / B(f), where 0 is the zero vector of appropriate dimension determined by the context. Since f(v ) 0 implies 0 / B(f), we assume f(v ) = 0 when we consider the membership problem, unless otherwise stated. We assume without loss of generality that B(f) is full-dimensional, i.e., the dimension of B(f) is equal to V 1(= n 1), where note that B(f) lies within the hyperplane x(v ) = 0. 1 We thus consider the relative topology within the hyperplane x(v ) = 0. Denote by Int(B(f)) the relative interior of B(f) within the hyperplane x(v ) = 0. 1 Note that we can find connected components of the submodular system (2 V, f) in O(n 2 EO) time by using the algorithm of Bixby, Cunningham, and Topkis [1] and each connected component has a fulldimensional base polyhedron. See [7, Section 3.3]. Here EO denotes the time required for the function evaluation of f. 2

3 For any α > 0 define f (α) : 2 V R by f (α) (X) = { f(x) + α if X 2 V \ {, V } f(x)(= 0) if X {, V }. (2.4) We can easily see that f (α) is a submodular function, which is called the α-enlargement 2 of f. It should be noted that B(f (α) ) is full-dimensional since B(f) is full-dimensional and B(f (α) ) B(f). Lemma 2.1: Suppose that f is integer-valued. Then we have 0 Int(B(f (1) )) if and only if 0 B(f). (Proof) Note that for any X 2 V \ {, V } we have f(x) + 1 = f (1) (X) > 0 if and only if f(x) 0, since f is integer-valued. Therefore, the following three are equivalent Int(B(f (1) )). 2. X 2 V \ {, V } : f (1) (X) > X 2 V \ {, V } : f(x) 0. This completes the proof. It should be noted that the operation of enlargement is also employed by Dadush, Végh, and Zambelli [6] and plays an important rôle in their algorithm as well. Define vectors ᾱ, α R V by ᾱ(v) = f({v}), α(v) = f(v ) f(v \ {v}) ( v V ). (2.5) (See [7, (3.89),(3.95)]). Then we have α(x) x(x) f(x) ᾱ(x) ( x B(f), X V ). (2.6) Note that for any base b B(f) we must have α b ᾱ, i.e., B(f) [α, ᾱ] (a standard box between α and ᾱ). Lemma 2.2: We have 0 min{f(x) X V } v V min{0, f(v ) f(v \ {v})}. (2.7) Moreover, for any base b B(f) we have b(v) max{f({v}), f(v ) + f(v \ {v})} ( v V ). (2.8) (Proof) The present lemma follows from (2.5) and (2.6). 2 See [7, Section 3.1(d)]. 3

4 3. Membership in Base Polyhedra and Chubanov s LP Algorithm In the sequel we assume that f is an integer-valued submodular function. Because of Lemma 2.1 we consider the membership problem to discern whether 0 Int(B(f (1) )) or not. For any x, y R V define x, y = v V x(v)y(v), and let 1 be the vector of all 1 s of appropriate dimension. Lemma 3.1: We have 0 / Int(B(f (1) )) if and only if the system of linear inequalities b, x 0 for all extreme bases b of B(f (1) ) (3.1) has a feasible solution x R V such that x α1 for all α R. (Proof) Suppose that 0 / Int(B(f (1) )). Then there exists a hyperplane w, x = 0 such that (i) w α1 for all α R and (ii) the closed halfspace w, x 0 contains B(f (1) ). Hence w is a desired feasible solution of (3.1). Conversely, suppose that 0 Int(B(f (1) )). Then for any x α1 for all α R there exists some extreme base b B(f (1) ) such that b, x < 0. Lemma 3.1 can be rewritten in the space (hyperplane) of x(v )(= 1, x ) = 0 as follows. Lemma 3.2: We have 0 / Int(B(f (1) )) if and only if the system of linear inequalities b, x 0 for all extreme bases b of B(f (1) ) (3.2) has a non-zero feasible solution x R V such that x(v ) = 0. It follows from Lemma 3.2 that we can adapt Chubanov s LP algorithm [4] to find a non-zero feasible solution x of (3.2) such that x(v ) = 0 or to decide the infeasibility of (3.2) with the required property. 3 It should be noted that given any non-zero w R V such that w(v ) = 0, w is a feasible solution of (3.2) if and only if letting b be the Edmonds greedy solution that minimizes w, x subject to x B(f), we have w, b 0. This gives a separation oracle required in Chubanov s algorithm. Calling the separation oracle invokes the function evaluation oracle n 1 times for f, due to the greedy algorithm. It should be noted that the polynomial complexity of Chubanov s algorithm does not depend on the number of inequalities, so that we can apply his algorithm to (3.2) having possibly n! inequalities. This is a crucial advantage of Chubanov s algorithm, which 3 Choosing any v V and projecting B(f (1) ) along the axis v into the one-dimensional lower coordinate space R V \{v}, we have a generalized polymatroid Q (see [7, Section 3.5(a)]) and we may consider the equivalent membership problem of 0 Int(Q), where Q is full-dimensional in R V \{v}. 4

5 is like the ellipsoid method of Khachiyan as is well-known. It is also remarkable that Chubanov s LP algorithm terminates in a finite number of steps even for real-number input data, allowing basic arithmetic operations for reals appropriately. When (3.2) has a non-zero feasible solution x R V such that x(v ) = 0, define an index ν by ν = max{vol(w ) W : a parallelopiped formed by n 1 vectors w satisfying (3.2) with w(v ) = 0 and w 1}, (3.3) where vol(w ) is the volume of W in the Euclidean subspace x(v ) = 0. We can show that log(ν 1 ) = O(n 2 log nm), where M = max{ f(x) X V }. Put ν = n 2 log nm. Consequently, we have the following theorem due to [4, Theorem 2.1]. Theorem 3.3: Applying Chubanov s algorithm, we can discern whether 0 / Int(B(f (1) )) (i.e., 0 / B(f)) or not and obtain a set X V with f(x) < 0 if 0 / B(f) in O((n 5 + n 3 T + n 4 EO)ν time, where EO denotes the time required for the function evaluation oracle for f and T denotes the time required for computing a scalar γ(x) such that γ(x) 2 x 2 [1, 3], for extreme base x B(f (1) ). 2 (Proof) We should only show that when we decide 0 / Int(B(f (1) )), we get a set X V with f(x) < 0. So, suppose that we decide 0 / Int(B(f (1) )) and have a nonzero vector w R V with w(v ) = 0 satisfying (3.2). Then, letting b be the greedy solution that minimizes w, x over B(f (1) ), we have w, b 0. Let (v 1,, v n ) be a linear ordering of V that gives the greedy solution b, so that w(v 1 ) w(v n ). (3.4) Putting S i = {v 1,, v i } for i = 1,, n and S 0 =, we have w, b = w(v 1 )(f (1) (S 1 ) f (1) (S 0 )) + + w(v n )(f (1) (S n ) f (1) (S n 1 )) = w(v 1 )(f(s 1 ) f(s 0 )) + + w(v n )(f(s n ) f(s n 1 )) + w(v 1 ) w(v n ) = (w(v 1 ) w(v 2 ))f(s 1 ) + + (w(v n 1 ) w(v n ))f(s n 1 ) 0. +w(v 1 ) w(v n ) (3.5) Since at least one inequality in (3.4) holds with strict inequality and hence w(v 1 ) w(v n ) < 0, it follows from (3.4) and (3.5) that for some i {1,, n 1} we have f(s i ) < 0. So we can return X = S i as the output. It follows from Lemma 2.2 that T = O(log nm). It should also be noted that the argument for (3.4) and (3.5) is essentially the same as the proof in [6, Lemma 3.2]. Let us denote by Membership(f) the algorithm for testing membership 0 B(f) constructed above, which gives us a set X with f(x) < 0 when 0 / B(f). 5

6 4. Submodular Function Minimization We have shown how the recent result of Chubanov [4] for linear programming yields a weakly polynomial algorithm Membership(f) for testing membership 0 B(f). However, this does not give any minimizer of f in a direct way. Fujishige and Iwata [8] showed that by invoking Membership(f) as a subroutine O(n 2 ) times, we can find a minimizer of f. In this section we show another way of finding a minimizer of f by O(log M ) calls for the membership testing algorithm Membership(f), where M is defined by (4.1). First we make the following preprocessing. If f(v ) > 0, then put f(v ) 0 for set V alone. Then, function f remains submodular and the set of minimizers (except for V ) remains the same. If f(v ) < 0, then for all nonempty X V put f(x) f(x) f(v ). Then, function f remains submodular and the set of minimizers (except for ) remains the same. Put M = v V min{0, f(v ) f(v \ {v})}. (4.1) Note that 0 min{f(x) X V } M due to Lemma 2.2. Now we employ Procedure Membership(f) as follows. Algorithm SFM(f) Input: a submodular function f : 2 V Z. Output: a minimizer Z of f. (0) Let f be the submodular function after preprocessing f. (1) Perform Membership( f). If we decide 0 B( f), then if originally f(v ) 0, then return Z = (since is a minimizer of the original f), if originally f(v ) < 0, then return Z = V (since V is a minimizer of the original f). If 0 / B( f) and we get X with f(x) < 0, then put Z X, β 0, and β + M (defined for f). (2) While β + β 2, do the following (i) and (ii): (i) Put α (β + β + )/2. 6

7 (ii) Perform Membership( f (α) ). If we decide 0 B( f (α) ), then put β + α. If 0 / B( f (α) ) and we get X with f (α) (X) < 0, then put Z X and β α. (3) Return Z. We can show the following. Theorem 4.1: Procedure SFM(f) returns a minimizer Z of f after repeating Step (2) at most log M times. (Proof) The validity of Step (1) follows from the definition of preprocessing. While carrying out Step (2), we keep the property that 0 B( f (β +) ) and 0 / B( f (β ) ). Hence, when we reach Step (3), we have f (β ) (Z) < 0 f (β +) (X) ( X V ) and β + β = 1. This implies β + f(z) < β = β and β + f(x) for all X V. It follows that Z attains the minimum of f (and hence of f), which is equal to β +. Combining Fujishige and Iwata s method [8] with SFM(f), we can compute a minimizer of an integer-valued submodular function in time. O((n 5 + n 3 T + n 4 EO)ν min{n 2, log M }) Acknowledgments The author is very grateful to Tomonari Kitahara of Tokyo Institute of Technology for pointing out an error in an earlier version of this note. References [1] R. E. Bixby, W. H. Cunningham and D. M. Topkis: Partial order of a polymatroid extreme point. Mathematics of Operations Research 10 (1985) [2] S. Chubanov: A strongly polynomial algorithm for linear systems having a binary solution. Mathematical Programming, Ser. A 134 (2012) [3] S. Chubanov: A polynomial projection algorithm for linear feasibility problems. Mathematical Programming, Ser. A 153 (2015)

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