Discrete Convexity and Polynomial Solvability
in Minimum 0-Extension Problems
May, 2014 (revised)
September, 2014 (final)
A -extension of graph is a metric on a set containing the vertex set of such that extends the shortest path metric of and for all there exists a vertex in with . The minimum -extension problem 0-Ext on is: given a set and a nonnegative cost function defined on the set of all pairs of , find a -extension of with minimum. The -extension problem generalizes a number of basic combinatorial optimization problems, such as minimum -cut problem and multiway cut problem.
Karzanov proved the polynomial solvability of 0-Ext for a certain large class of modular graphs , and raised the question: What are the graphs for which 0-Ext can be solved in polynomial time? He also proved that 0-Ext is NP-hard if is not modular or not orientable (in a certain sense).
In this paper, we prove the converse: if is orientable and modular, then 0-Ext can be solved in polynomial time. This completes the classification of graphs for which 0-Ext is tractable. To prove our main result, we develop a theory of discrete convex functions on orientable modular graphs, analogous to discrete convex analysis by Murota, and utilize a recent result of Thapper and Živný on valued CSP.
By a (semi)metric on a finite set we mean a nonnegative symmetric function on satisfying for all and the triangle inequalities for all . An extension of a metric space is a metric space with and for . An extension of is called a -extension if for all there exists with .
Let be a simple connected undirected graph with vertex set . Let denote the shortest path metric on with respect to the uniform unit edge-length of . The minimum -extension problem 0-Ext on is formulated as:
Given and ,
minimize over all -extensions of .
Here denotes the set of all pairs of . The minimum -extension problem is formulated by Karzanov , and is equivalent to the following classical facility location problem, known as multifacility location problem , where we let :
This problem can be interpreted as follows: We are going to locate new facilities on graph , where the facilities communicate each other and communicate existing facilities on . The cost of the communication is propositional to the distance. Our goal is to find a location of minimum total communication cost. This classic facility location problem arises in many practical situations such as the image segmentation in computer vision, and related clustering problems in machine learning; see . Also 0-Ext includes a number of basic combinatorial optimization problems. For example, take as the graph consisting of a single edge . Then 0-Ext is the minimum -cut problem. More generally, 0-Ext is the multiway cut problem on terminals. Therefore 0-Ext is solvable in polynomial time if and is NP-hard if .
What are the graphs for which 0-Ext is solvable in polynomial time?
Here such a graph is simply called tractable.
A classical result in location theory in the 1970’s is:
The tractability of graphs is preserved under taking Cartesian products. Therefore, cubes, grid graphs, and the Cartesian product of trees are tractable. Chepoi  extended this classical result to median graphs as follows.
A median of a triple of vertices is a vertex satisfying for . A median graph is a graph in which every triple of vertices has a unique median. Trees and their products are median graphs. See Figure 1 for illustration of the median concept.
Theorem 1.2 ().
If is a median graph, then 0-Ext is solvable in polynomial time.
Karzanov  introduced the following LP-relaxation of 0-Ext.
Given and ,
minimize over all extensions of .
This relaxation Ext is a linear program
with size polynomial in the input size.
Therefore, if for every input ,
Ext has an optimal solution that is a -extension,
then 0-Ext is solvable in polynomial time.
In this case
we say that Ext is exact.
In the same paper,
Karzanov gave a combinatorial
characterization of graphs
for which Ext is exact.
A graph is called a frame if
is bipartite, has no isometric cycle of length greater than , and has an orientation with the property that for every 4-cycle , one has if and only if .
Here an isometric cycle in means a cycle such that every pair of vertices in has a shortest path for in this cycle , and means that edge is oriented from to by .
Theorem 1.3 ().
Ext is exact if and only if is a frame.
Theorem 1.4 ().
If is a frame, then 0-Ext is solvable in polynomial time.
It is noted that the class of frames is not closed under taking Cartesian products, whereas the tractability of graphs is preserved under taking Cartesian products. Also it should be noted that Ext is the LP-dual to the -weighted maximum multiflow problem, and 0-Ext describes a combinatorial dual problem [32, 33]; see also [21, 22, 24, 23] for further elaboration of this duality.
Karzanov  also proved the following hardness result. For an undirected graph , an orientation with the property (1) (3) is said to be admissible. is said to be orientable if it has an admissible orientation. is said to be modular if every triple of vertices has a (not necessarily unique) median.
Theorem 1.5 ().
If is not orientable or not modular, then 0-Ext is NP-hard.
In fact, a frame is precisely an orientable modular graph with the hereditary property that every isometric subgraph is modular; see . A median graph is an orientable modular graph but the converse is not true. Moreover, a median graph is not necessarily a frame, and a frame is not necessarily a median graph. In , Karzanov proved a tractability theorem extending Theorem 1.2. He conjectured that 0-Ext is tractable for a certain proper subclass of orientable modular graphs including frames and median graphs. He also conjectured that 0-Ext is NP-hard for any graph not in this class.
The main result of this paper is the tractability theorem for all orientable modular graphs. Thus the class of tractable graphs is larger than his expectation.
If is orientable modular, then 0-Ext is solvable in polynomial time.
Combining this result with Theorem 1.5, we obtain a complete classification of the graphs for which 0-Ext is solvable in polynomial time.
In proving Theorem 1.6, we employ an axiomatic approach to optimization in orientable modular graphs. This approach is inspired by the theory of discrete convex analysis developed by Murota and his collaborators (including Fujishige, Shioura, and Tamura); see [17, 45, 48, 49, 47] and also [16, Chapter VII]. Discrete convex analysis is a theory of convex functions on integer lattice , with the goal of providing a unified framework for polynomially solvable combinatorial optimization problems including network flows, matroids, and submodular functions. The theory that we are going to develop here is, in a sense, a theory of discrete convex functions on orientable modular graphs, with the goal of providing a unified framework for polynomially solvable 0-extension problems and related multiflow problems. We believe that our theory establishes a new link between previously unrelated fields, broadens the scope of discrete convex analysis, and opens a new perspective and new research directions.
Let us start with a simple observation to illustrate our basic idea. Consider a path of length , and consider 0-Ext, where is trivially an orientable modular graph. Then 0-Ext for input can be regarded as an optimization problem on the integer lattice as follows. Suppose that , and and are adjacent for . Then , and 0-Ext is equivalent to the minimization of the function
over all . This function is a simple instance of L-convex functions, one of the fundamental classes of discrete convex functions. We do not give a formal definition of L-convex functions here. The only important facts for us are the following properties of L-convex functions in optimization:
Local optimality implies global optimality.
The local optimality can be checked by submodular function minimization.
An efficient descent algorithm can be designed based on successive application of submodular function minimization.
As is well-known, submodular functions can be minimized in polynomial time [20, 30, 54]. Actually the function (1.3) can be minimized by successive application of minimum-cut computation [37, 51], a special case of submodular function minimization.
Motivated by this observation, we regard 0-Ext as a minimization of a function defined on the vertex set of a product of , which is also orientable modular. We will introduce a class of functions, called L-convex functions, on an orientable modular graph. We show that our L-convex function satisfies analogues of (a), (b) and (c) above, and also that a multifacility location function, the objective function of 0-Ext, is an L-convex function, in our sense, on the product of . Theorem 1.6 is a consequence of these properties.
Let us briefly mention how to define L-convex functions, which constitutes the main body of this paper. Our definition is based on the Lovász extension , a well-known concept in submodular function theory , and a kind of construction of polyhedral complexes, due to Karzanov  and Chepoi , from a class of modular graphs. Let be an orientable modular graph with admissible orientation . We call a pair a modular complex. It turns out that can be viewed as a structure glued together from modular lattices, and gives rise to a simplicial complex as follows. Consider a cube subgraph of . The digraph oriented by coincides with the Hasse diagram of a Boolean lattice. Consider the simplicial complex whose simplices are sets of vertices forming a chain of the Boolean lattice corresponding to some cube subgraph of ; see Figure 2.
Each (abstract) simplex is naturally regarded as a simplex in the Euclidean space. is naturally regarded as a metrized simplicial complex. Then any function is extended to by interpolating on each simplex linearly; this is an analogue of the Lovász extension. The simplicial complex enables us to consider the neighborhood around each vertex , as well as the local behavior of in .
As in Figure 3, neighborhood can be described as a partially ordered set with the unique minimal element . Then, by restricting to , we obtain a function on associated with each vertex . In fact, the poset is a modular semilattice, a semilattice analogue of a modular lattice introduced by Bandelt, van de Vel, and Verheul . We first define submodular functions on modular semilattices, and next define L-convex functions on modular complex as functions on such that is submodular on neighborhood semilattice for each vertex .
Then the multifacility location function, the objective of 0-Ext (see (1.1)), is indeed an L-convex function on the -fold product of , and the optimal solution of 0-Ext can be obtained by successive application of submodular function minimization on the product of modular semilattices. Thus our problem reduces to the problem of minimizing submodular function on the product of modular semilattices , where the input of the problem is , and an evaluating oracle of . We do not know whether this problem in general is tractable in the oracle model, but the submodular functions arising from 0-Ext take a special form; they are the sum of submodular functions with arity . Here the arity of a function is the number of variables of . Namely, if a function on is represented as
for some function on with , then the arity of is (at most) . See (1.1); our objective function is a weighted sum of distance functions, which have arity . This type of optimization problem with bounded arity is well-studied in the literature of valued CSP (valued constraint satisfaction problem) [7, 42, 53, 59]. Valued CSP deals with minimization of a sum of functions , where the arity of each is a part of the input; namely the input consists of all values of all functions . Valued CSP admits an integer programming formulation, and its natural LP relaxation is called the basic LP-relaxation. Recently, Thapper and Živný  discovered a surprising criterion for the basic LP-relaxation of valued CSP to exactly solve the original valued CSP instance. They proved that if the class of valued CSP (the class of input objective functions) has a certain nice fractional polymorphism (a certain set of linear inequalities which any input function satisfies), then the basic LP-relaxation is exact. We prove that the class of submodular functions on modular semilattice admits such a fractional polymorphism. Then the sum of submodular functions with bounded arity can be minimized in polynomial time. Consequently we can solve 0-Ext in polynomial time.
We believe that our classes of functions deserve to be called submodular and L-convex. Indeed, they include not only (ordinary) submodular/L-convex functions but also other submodular/L-convex-type functions. Examples are bisubmodular functions [9, 50, 52] (see [16, Section 3.5]), multimatroid rank functions by Bouchet , submodular functions on trees by Kolmogorov , -submodular functions by Huber and Kolmogorov  (also see ), and skew-bisubmodular functions by Huber, Krokhin and Powell  (also see [19, 29, 28]). Moreover, combinatorial dual problems arising from a large class of (well-behaved) multicommodity flow problems, discussed in [21, 22, 24, 23, 31, 32, 33], fall into submodular/L-convex function minimization in our sense. This can be understood as a multiflow analogue of a fundamental fact in network flow theory: the minimum cut problem, the dual of maxflow problem, is a submodular function minimization. The detailed discussion on these topics will be given in a separate paper ; some of the results were announced by .
In Section 2, we first explain basic notions of valued CSP and the Thapper-Živný criterion (Theorem 2.1) on the exactness of the basic LP relaxation. We then describe basic facts on modular graphs and modular lattices. In Section 3, we develop a theory of submodular functions on modular semilattices. We show that our submodular function satisfies the Thapper-Živný criterion, and that a sum of submodular functions with bounded arity can be minimized in polynomial time. In Section 4, we first explore several structural properties of orientable modular graphs. Based on the above mentioned idea, we define L-convex functions, and prove that our L-convex functions indeed have properties analogous to (a), (b) and (c) above. In Section 5, we formulate 0-Ext as an optimization problem on a modular complex. We show that a multifacility location function, the objective function of 0-Ext, is indeed an L-convex function, and we prove Theorem 1.6. Our framework is applicable to a certain weighted version of 0-Ext. As a corollary, we give a generalization of Theorem 1.6 to general metrics, which completes classification of metrics for which the 0-extension problem on is polynomial time solvable (Theorem 5.9). In the last section (Section 6), we discuss a connection to a dichotomy theorem of finite-valued CSP obtained by Thapper and Živný  after the first submission of this paper. In fact, the complexity dichotomy (of form “either P or NP-hard”) of 0-Ext, established in this paper, can be viewed as a special case of their dichotomy theorem of finite-valued CSP.
Let , and denote the sets of integers, rationals, and reals, respectively. Let and , where is an infinity element and is treated as: , , , . Let , and denote the sets of nonnegative integers, nonnegative rationals, and nonnegative reals, respectively. For a function on a set , let denote the set of elements with .
For a graph , the vertex set and the edge set are denoted by and , respectively. For a vertex subset , denotes the subgraph of induced by . For a nonnegative edge-length , denotes the shortest path metric on with respect to the edge-length . When for every edge , is denoted by . A path is represented by a chain of vertices with . The Cartesian product of graphs and is the graph with vertex set and edge set given as: and are connected by an edge if and only if and or and . The -fold Cartesian product of is denoted by . In this paper, graphs and posets (partially ordered sets) are supposed to be finite.
In this section, we give preliminary arguments for valued CSP, and modular graphs and modular (semi)lattices. Our references are [41, 59] for valued CSP and [3, 5, 6, 13, 58] for modular graphs and lattices. A further discussion on valued CSP is given in Section 6.
2.1 Valued CSP and fractional polymorphism
Let be finite sets, and let . A constraint on is a function for some , where is called the scope of , and is called the arity of . Let , and for , let . The valued CSP (valued constraint satisfaction problem) is:
Given a set of constraints on ,
minimize over all .
The input of VCSP is the set of all values of all constraints in , and hence its size is estimated by , where , , and is the bit size to represent constraints in . By a constraint language we mean a (possibly infinite) set of constraints. A constraint in is called a -constraint. Let VCSP denote the subclass of VCSP such that the input is restricted to a set of -constraints.
The minimum 0-extension problem 0-Ext is formulated as an instance of VCSP. Let for . Define constraints and by
Define the input of VCSP by
Notice that the size of is polynomial in , , and the bit size representing . Hence 0-Ext is a particular subclass of VCSP.
VCSP admits the following integer programming formulation:
Indeed, for each there uniquely exists with . Also for there uniquely exists such that and for . Define by . Then if and only if . Therefore we obtain a solution of VCSP with the same objective value. Conversely, for a solution of VCSP, define if , and if . The other variables are defined as zero. Then we obtain a solution of (2.2) with the same objective value.
Observe that there are variables and constraints. Therefore the size of this IP is bounded by a polynomial of the input size. The basic LP relaxation (BLP) is the linear problem obtained by relaxing the 0-1 constraints and into and , respectively. In particular BLP can be solved in (strongly) polynomial time.
Recently Thapper and Živný  discovered a surprisingly powerful criterion for which BLP solves VCSP. To describe their result, let us introduce some notions. For a constraint language , BLP is said be exact for if for every input , the optimal value of BLP coincides with the optimal value of VCSP. An operation on is a function . A (separable) operation on is a function such that is represented as
for some operations on for . A fractional operation is a function from the set of all operations to such that the total sum over all operations is . We denote a fractional operation by the form of a formal convex combination of operations . The support of is the set of operations with . For a constraint language , a fractional polymorphism is a fractional operation on such that it satisfies
where is regarded an operation on by for . For example, if is a lattice for each , then is nothing but a fractional polymorphism for submodular functions, i.e., functions satisfying for .
Theorem 2.1 (Special case of [56, Theorem 5.1 ]).
If a constraint language admits a fractional polymorphism such that the support of contains a semilattice operation, then BLP is exact for , and hence VCSP can be solved in polynomial time.
Here a semilattice operation is an operation satisfying , , and for . Although the feasible region of BLP is not necessarily an integral polytope, we can check whether there exists an optimal solution with by comparing the optimal values of BLP for the input and for , which is the set of cost functions obtained by fixing variable to for each cost function on . Necessarily BLP is exact for if there is an optimal solution with . Hence, after fixing procedures, we obtain an optimal solution .
In the setting in , is the same set for all . Our setting reduces to this case by taking the disjoint union of as , and extending each cost function to by for . Without such a reduction, their proof also works for our setting in a straightforward way.
2.2 Modular metric spaces and modular graphs
For a metric space , the (metric) interval of is defined as
For two subsets , denotes the infimum of distances between and , i.e.,
For , an element in is called a median of , , and . A metric space is said to be modular if every triple of elements in has a median. In particular, a graph is modular if and only if the shortest path metric space is modular. We will often use the following characterization of modular graphs.
A connected graph is modular if and only if
is bipartite, and
for vertices and neighbors of with , there exists a common neighbor of with .
For a modular graph, every admissible orientation is acyclic.
Suppose indirectly that the statement is false. Take a vertex belonging to a directed cycle, and take a directed cycle containing with minimum. The length of is at least four (by simpleness and bipartiteness). By the definition of admissible orientation, is impossible. Hence . Take a vertex in with maximum. Take two neighbors of in . Then (by the maximality of and the bipartiteness of ). By the quadrangle condition, there is a common neighbor of with . Here the cycle obtained from by replacing by is a directed cycle, since the orientation is admissible. Then we have . This contradicts the minimality of . ∎
2.2.1 Orbits and orbit-invariant functions
Let be a modular graph. Edges and are said to be projective if there is a sequence of edges such that and belong to a common 4-cycle and share no common vertex. We will use the following criterion for two edges to belong to a common orbit.
Let be a modular graph. For edges and , suppose that and .
and are projective.
In addition, if has an admissible orientation , then implies .
We use the induction on . The case of is obvious. Take a neighbor of with . Then . By the quadrangle condition for , there is a common neighbor of with . Also . Obviously and are projective, and implies . Apply the induction for and . ∎
An orbit is an equivalence class of the projectivity relation. The (disjoint) union of several orbits is called an orbit-union. For an orbit-union , is the graph obtained by contracting all edges not in and by identifying multiple edges. The vertex in corresponding to is denoted by . The graph is also modular, and any shortest path in induces a shortest path in as follows.
Let be a modular graph, and an orbit-union.
is a modular graph.
For every , every shortest -path , and every -path , we have .
For every and every shortest -path , the image of is a shortest -path in .
In particular, for any partition of into orbit-unions, we have
A function on edge set is called orbit-invariant if provided and belong to the same orbit. For an orbit , let denote the value of on . An orbit-invariant function is said to be nonnegative if for , and is said to be positive if for . For a constant , if for all edges , then is simply denoted by ; in particular . By taking the value of of the preimage, we can define a function on the edge set of for any orbit-union , which is also orbit-invariant in and is denoted by . By Lemma 2.6 (2), the shortest path structures of and are the same in the following sense:
If an orbit-invariant function is nonnegative, then implies , where
is a shortest -path with respect to ,
is a shortest -path with respect to .
If is positive, then the converse also holds.
2.2.2 Convex sets and gated sets
Let be a metric space. A subset is called convex if for every . A subset is called gated if for every there is , called a gate of at , such that holds for every . One can easily see that gate is uniquely determined for each [15, p. 112]. Therefore we obtain a map by defining to be the gate of at .
Theorem 2.8 ().
Let and be gated subsets of and let and .
and induce isometries, inverse to each other, between and .
For and , the following conditions are equivalent:
and are gated, and and .
As remarked in , every gated set is convex (see the proof of Lemma 2.9 below). The converse is not true in general, but is true for modular graphs. The following useful characterization of convex (gated) sets in a modular graph is due to Chepoi . Here, for a graph , a subset of vertices is said to be convex (resp. gated) if is convex (resp. gated) in .
Lemma 2.9 ().
Let be a modular graph. For , the following conditions are equivalent:
is connected and holds for every with .
We give a proof for the convenience of readers as the original paper is in Russian.
is denoted by . (1) (3) is obvious. We show (3) (1). Take , and take . We are going to show . Since is connected, we can take a path with . Take such a path with minimum. If for some , then, by the quadrangle condition in Lemma 2.3, there is a common neighbor of with . Since by (3), belongs to . Then we can replace by in to obtain another path connecting with ; a contradiction to the minimality. Therefore there is no index with . Thus there is a unique index with minimum. Then we have and . By , we have . Hence we must have , implying .
We show (2) (1). As already mentioned, any gated set is convex. Indeed, suppose that is gated. Take , and take . Consider the gate of in . Then and . Since , we have , implying and . Thus we get (2) (1).
Finally we show (1) (2). Suppose that is convex. Let be an arbitrary vertex. Let be a point in satisfying . We show that is a gate of at . Take arbitrary . Consider a median of . By convexity, belongs to , and also . By definition of , it must hold . Thus holds for every . This means that is the gate of , and therefore is gated. ∎
2.3 Modular lattices and modular semilattices
Let be a partially ordered set (poset) with partial order . For , the (unique) minimum common upper bound, if it exists, is denoted by , and the (unique) maximum common lower bound, if it exists, is denoted by . is said to be a lattice if both and exist for every , and said to be a (meet-)semilattice if exists for every . In a semilattice, if and have a common upper bound, then exists. Such is said to be bounded. By the expression“” we mean that exists. A pair is said to be comparable if or , and incomparable otherwise. We say “ covers ” if and there is no with , where means and . The maximum element (universal upper bound) and the minimum element (universal lower bound), if they exist, are denoted by and , respectively. For , the interval is denoted by . A chain from to is a sequence with for ; the number is the length of the chain. The length of the interval is defined as the maximum length of a chain from to . The rank of an element is defined by . An atom is an element of rank . The covering graph of a poset is the underlying undirected graph of the Hasse diagram of .
A lattice is called modular if for every with . Modular lattices are also characterized by the modular equality of the rank function.
Lemma 2.10 (See [6, Chapter III, Corollary 1]).
A lattice is modular if and only if
A lattice is called complemented if for every there is an element , called a complement of , such that and , and relatively complemented if is complemented for every