The minimum cut problem is one of the most fundamental problems in computer science and has numerous applications in other fields. Consequently, there has been a vast amount of research on the topic (see Chekuri et al. (1997) for an overview). In this problem you are given an undirected graph on nodes and an undirected non-negative cost function on every pair of nodes denoting the capacity of the edges in the graph. In the global cut problem the goal is to find a non-empty set where such that the sum of the edge weights crossing the cut is minimized. In the cut problem it must be the case that and where are given. In the fixed sized partition minimum cut problem, the set must have size where is an input parameter. The cut and global minimum cut problems are polynomially time solvable, but the fixed size partition minimum cut problem is known to be NP-Hard.
It is well known that the minimum cut problem yields polynomial time algorithms and a variety of efficient algorithms are known Chekuri et al. (1997); Karger (2000) as well as efficient parallel algorithms Karger and Stein (1996); Lattanzi et al. (2011). One widely used technique is mathematical programming. A few integer linear programs are known for the minimum cut problems Chekuri et al. (1997); Carr et al. (2009) and for several of them the LP relaxation gives exact solutions to the minimum cut problem (global or ). Like most problems, finding a linear programming relaxation for a given problem that is different than the obvious relaxation is generally a challenging task. The smallest known linear programming relaxation of the global mincut problem, called the -LP, was given in Carr et al. (2009). This formulation has variables and constraints. From a theoretical viewpoint, it is interesting to determine the bounds of the smallest linear program for a given problem, since small linear programs can lead to more efficient algorithms in practice. For the minimum cut problem, finding a small relaxation not only can improve the performance for finding the minimum cut of a graph, but can be also instrumental to the solution of other fundamental problems.
It was shown by in Yannakakis (1991) that no symmetric LP can have polynomial size for TSP or perfect matching. Roughly speaking, an LP is said to be symmetric if the nodes in a graph can be permuted without changing the feasible region. Further, it is known that certain classes of LP’s can be lifted to give symmetry Yannakakis (1991). This was originally thought to be a way to show that not every problem in P has a polynomial size LP. In particular, Yannakakis proved also that there are no compact LP formulations to perfect matching that have symmetricity, and he argued that the lack of existence of a compact LP formulation to perfect matching that has symmetricity is strong evidence there is no compact LP formulation to perfect matching (even asymmetric). Indeed, the underlying perfect matching problem has symmetricity. Thus, if there were such an asymmetric LP formulation then intuitively one could likely lift it to a slightly larger LP formulation that does have symmetricity. But this contradicts Yannakakis’ result. It turns out that Yannakakis was correct, and recently, Rothvoß (2017) showed that the matching polytope does not have a polynomial size linear program.
Motivation: There is a considerable amount of work on linear programming formulations. For problems which are in P, typically non-exact linear programming formulations are only of interest if they are very small. Small non-exact linear programs can be interesting because, in many cases, they can be used to give fast approximate guarantees on the optimal solution’s value.
In the Traveling Salesperson Problem (TSP), a salesperson wants to visit a set of cities and return home Jünger et al. (1995); Goemans (1995). There is a cost of traveling from city to city , which is the same in either direction for the Symmetric TSP. The objective is to visit each city exactly once, minimizing total travel costs. The subtour elimination linear program is a natural and easily solvable linear programming relaxation of TSP:
There is no natural compact LP relaxation for TSP which is stronger than the subtour elmination linear program relaxation. The TSP uses the minimum cut as a subroutine Carr et al. (2009).
Despite most work focusing on finding small non-exact linear programming formulations, there is another property that non-exact linear programming formulations can have for problems in P which is of interest to study. Consider the minimum cut problem. Say that one finds a linear programming formulation which returns a non-exact solution, yet one can prove that the LP’s solution is exact when the cost function (input graph) is a Hamiltonian cycle. Although the LP is non-exact for general graphs, we actually can use the LP to improve the subtour relaxation for the TSP problem. Indeed, we show in this paper, if the LP relaxation were not an exact formulation of the minimum cut problem (but exact on Hamiltonian cycles), it can be used to strengthen the subtour LP relaxation for the TSP problem.
We call this feature of the LP interesting. Interesting LPs are not limited to the relationship between minimum cut and TSP. For example, suppose we had a compact relaxation of the minimum directed cut problem, but not necessarily an exact formulation. Further assume that this relaxation gives the exact answer of 1 when the cost function is an arborescence. Then, either this is an exact formulation of directed cut or we can strengthen the LP relaxation for the directed arborescence problem.
This leads to a somewhat unintuitive situation. One would want to find an LP formulation that is exact for computing a minimum cut when the graph is a Hamiltonian cycle, but simultaneously as loose as possible for computing the minimum cut for any other graph. In this case, one can strengthen the subtour LP relaxation for the TSP problem by the largest amount. Indeed, any time the minimum cut LP gives a non-exact solution this is a certificate that the objective function is outside the TSP polytope, even if this solution were in the subtour TSP relaxation. We can then cut off this point in a new TSP relaxation, thus making it better than the subtour relaxation.
Results. In this work, we consider linear programming relaxations for the fixed size partition and global minimum cut problems. We begin by giving a symmetric relaxation assuming one knows , the size of one side of the cut. We will call this the -LP. Note that one can assume since one side of the cut must have at most vertices. Our main results are the following:
The -LP gives the optimal solution for the fixed size partition minimum cut problem when the input graph is an unweighted Hamiltonian cycle for any (has the Hamiltonian cycle property).
The -LP is a relaxation of the -LP which also has the Hamiltonian cycle property.
The -LP is a further relaxation of the -LP which is within of the -LP, but creates a smaller set of LPs which need to be solved to approximate the minimum cut.
Now, we do not know for general graphs if the -LP returns at least that of the minimum global cut. However, if this is the case, then by extending the -LP to include all possible values of this will be the smallest known LP for the global minimum cut problem that is symmetric. The extension of the -LP to include all values of has variables and constraints where is the size of the support of the cost function. This compares to the previous best known formulation with variables and constraints which can be obtained by extending the standard formulation of minimum cut. Beyond being interesting from a theoretical viewpoint, symmetric LPs are also typically easier to analyze than other LPs.
Alternatively, say that the -LP admits feasible solutions of value smaller than the minimum global cut. In this case, we can use the -LP to obtain new valid inequalities for the TSP. Here, by leveraging the fact that the -LP is optimal for Hamiltonian cycle graphs, we can show that this LP can be exploited to strengthen the standard subtour relaxation for TSP. We show this by using compact optimization methods. Thus, we have a favorable situation, i.e., this new formulation either gives one of two interesting conclusions.
We then relax the -LP further and create a new model we call the -LP. While the -LP assumes is an integer, in the -LP we relax this assumption by allowing to be the convex combination of two consecutive integers. We show that the -LP has the same Hamiltonian cycle property as the -LP even though the -LP is a relaxation. We then show that in some special cases that the -LP and -LP are lower bounded by the minimum cut.
Lastly, we extend the -LP relaxation and give another relaxation which is -approximation for the minimum cut problem on Hamiltonian cycle graphs and a -approximation for the minimum cut problem when . This new relaxation has variables and constraints for any . We obtain this by extending the -LP to express as the convex combination of two integers, and . The -LP has size variables by constraints. By allowing variable sizes in the partitions where is in the range , we can geometrically choose a logarithmic number of values for . For these values, we bound the objective of the -LP by on Hamiltonian cycle graphs. Although this LP is not exact for the minimum cut problem on Hamiltonian cycle graphs, one can still use this relaxation to potentially improve the solution for the TSP relaxation, while yielding a significantly smaller LP.
We are going to construct symmetric LPs that solve or approximate fixed sized partition and (global) minimum cut. A cut in an undirected graph partitions the vertex set of
vertices into a set of top nodes and a set of bottom nodes, with the edges in the cut going between these two sets. Given a vectorof capacities and edge variables indicating whether edge is in a cut for we want to minimize . Minimum - cut where is specified to be on the bottom and is specified to be on the top is the most usual type of minimum cut problem (without such a specified pair of nodes it is the minimum global cut problem).
For the definition of a symmetric linear program, we use the definition given in Yannakakis (1991). Let denote a permutation of the nodes. In the complete graph, a permutation of the nodes defines a mapping of the costs of the edges of the graph such that each cost is mapped into . Let denote a polytope over the variables on the edges of a graph and other variables . A polytope of a linear program is said to be symmetric if for all permutations of the nodes the new variables can be extended so remains invariant. A linear program can be seen to yield a symmetric polytope if renaming the variables for the nodes yields the same linear program. Note that, however, a linear program need not necessarily have this property to be symmetric.
Minimum - cut has an easy (naturally integer) LP formulation, shown below. Let node variables indicate whether is on top () or on bottom () in a cut. Furthermore, let be edge variables indicating which edges are in the cut.
One can also formulate the NP-hard maximum cut problem. A naive IP formulation and LP relaxation is
However, setting shows that this is a weak LP relaxation.
Suppose one constrained minimum cut so that it has a fixed sized partition with exactly nodes on top. Note that one can always assume since one side of the cut always at most nodes. Call this the minimum -cut problem, which is also an NP-hard problem Garey and Johnson (1979). We note that when this is equivalent to the minimum edge bisection problem. We will now construct an LP relaxation for this problem. Let be the edges adjacent to in and . For consider . Suppose . Then there are nodes opposite , so . Now, suppose . Then there are nodes opposite , so . Hence, it is always the case that
Also, for distinct we have the (metric) triangle inequality property that .
Here is the first LP relaxation:
The above relaxation is stronger than we need for the results in this paper, we will refer to it as the old -LP. It can be relaxed to the following, which we call the (new) -LP:
We only need the triangle inequalities when in our proof of Theorem 3.10. The remaining triangle inequalities are unused. We obtain constraint (4) by summing the constraints giving in the previous LP. We emphasize that this -LP has size as we only need a subset of the triangle inqualities.
Denote the -LP relaxation’s feasible region by
In the next section we will bound the objective value of this LP. Before we do that, we introduce the -LP of Carr et al. (2009) which the LP can be compared to. In this linear program when is the last node (numerically) on top in the cut (where, w.l.o.g., we can assume that node is always on bottom).
While the -LP is asymmetric, the -LP is symmetric.
3 Hamiltonian Cycle Property
Denote the convex hull of the set of the incidence vectors of all the cuts of a complete graph by . An LP whose feasible region that minimizes or maximizes an objective function given by edge coefficients is a relaxation of the cut problem. If the LP is meant to bound the minimum cut of it is said to be a relaxation of the minimum cut problem. We introduce the dominant polyhedron of a polytope. The polyhedron is a set of points such that there exists an such that . An LP relaxation of the minimum cut problem is an LP mincut formulation if its feasible region .
We now introduce a surprising principle for creating LP relaxations of the TSP. Consider any linear program
on the edge variables of a complete graph (and possibly other variables). If for any Hamilton cycle, when its incidence vector is plugged into the minimum of this LP is a constant , then this LP is said to satisfy the Hamilton cycle property. If this LP is symmetric on the edge variables and the nodes, its minimum will automatically be a constant for all Hamilton cycles. If this constant is strictly positive, the LP will then satisfy the Hamilton cycle property. An LP that satisfies the Hamilton cycle property gives rise to an LP relaxation of the TSP through duality methods. Let be the costs for the TSP. Let be the variables of the TSP relaxation, which are so named to relate to the cost function of the LP satisfying the Hamilton cycle property. The TSP relaxation is given by
With respect to the subtour elimination relaxation of the LP, the idea of LP (9) is to keep the degree constraints, but replace the subtour elimination constraints with the dual of LP (8). Indeed, if the objective function plugged into LP (8) satisfying the Hamilton cycle property has a minimizer with objective value strictly less than , this LP solution is a certificate that is not in the TSP polytope, and we say is certified by LP (8) not to be in the TSP polytope. Moreover, is not feasible for any in the above LP relaxation of the TSP. Also, violates the valid TSP inequality .
Suppose that the LP satisfying the Hamilton cycle property has the triangle inequalities as constraints either explicitly or implicitly. That is,
for all distinct triples of nodes (where the order of the 2 endpoint nodes for an edge is arbitrary in the undirected graph).
Suppose an LP satisfies the Hamilton cycle property and includes the triangle inequalities explicitly (or satisfies them at optimality). Then if is certified by the LP to not be in the TSP polytope, either the minimum cut of is strictly less than or from one can construct a that is in the subtour relaxation of the TSP but is also certified to not be in the TSP polytope.
Proof: Suppose the minimizer for this LP (8) is and that the minimum cut of is at least . Normalize so that the minimum cut of is exactly 2. By assumption, .
Define splitting off by at a node with nodes and to be the operation of taking nodes and decreasing and by while increasing by for some fixed . In Lovász splitting off, this is done only on nodes such that the minimum global cut stays the same after the operation. It is known Lovasz (1976) that if the cut is not a minimum cut, then there exist nodes and and , such that we can perform Lovász splitting off by at with nodes and . Thus, we iteratively perform Lovász splitting off at nodes such that is not a minimum cut, until every node has fractional degree 2. During this process, we obtain intermediate vectors and eventually reach a final vector that has fractional degree at every node.
Since Lovász splitting off keeps the minimum global cut the same, then is a feasible subtour point. Due to the triangle inequalities in (8), at each step because and the objective decreases at each step, specifically, . In the end , and without loss of generality one can choose to be a subtour extreme point.
Now suppose the LP (8) is a relaxation of minimum global cut.
Proof: Suppose is feasible for (9) but that is not in the subtour polytope, i.e., the minimum global cut of is less than . Plug into (8). Since LP (8) is a relaxation of minimum global cut, its minimizer satisfies . Thus, in (9) is violated since , by duality, and is not feasible for (9) after all.
Now suppose the LP satisfying the Hamilton cycle property is a relaxation of minimum cut and also includes the triangle inequalities. Then either this LP is an exact minimum cut formulation or it leads to a TSP relaxation that is strictly stronger than the subtour relaxation of the TSP.
Suppose an LP (8) is a relaxation of minimum global cut. Take the case where it satisfies the Hamilton cycle property with , and includes the triangle inequality constraints explicitly (or satisfies them at optimality). Then either this LP is an exact minimum cut formulation or it leads to a TSP relaxation (9) that is strictly stronger than the subtour relaxation of the TSP.
Suppose (8) is not an exact minimum cut formulation. Then there exists an objective such that but for a minimizer of (8). By the reasoning of Lemma 3.1 there exists a in the subtour polytope such that . By the reasoning of Lemma 3.2, is then not in the relaxation (9).
We provide three examples of LPs satisfying the Hamilton cycle property here and more examples in the next section.
Consider the following linear program where :
If the integrality constraint is added to (10), the resulting integer program is an IP formulation of minimum cut. Hence (10) is an LP relaxation of minimum cut. It is readily shown here to satisfy the Hamilton cycle property.
Let be the incidence vector of a Hamilton cycle. Then the minimum objective value for (10) is 2.
Proof: Let be an arbitrary Hamilton cycle and be its incidence vector. The cycle can be decomposed into two vertex disjoint (except at ends) paths and . Consider first. Let the vertices in order in the path starting at be . For we have
Summing these up over the edges of the path yields a telescoping sum
Therefore we have
It is easy to show the bound of is attained.
If is a minimizer of (10) then without loss of generality in what follows, we may set
The feasible solutions to (10) (where the variables attain their mimimum values given the values of the variables) satisfy the triangle inequalities in the variables.
For what follows next, we introduce the concept of a disjunctive mathematical program Balas (1974).
Consider the feasible regions that arise from (10) when binary integrality constraints are placed on and one sets and varies to be in the set . This is an IP formulation of minimum cut. Consider the disjunctive program of these minimum cut integer programs. The feasible region of this disjunctive program is the convex hull of . This disjunctive program can be seen to be an IP formulation of the minimum global cut problem. Now remove the binary integrality constraints on the variables. The resulting feasible regions are . The feasible region is a relaxation of minimum cut for each . The disjunctive program of these LP relaxations has a feasible region which is the convex hull of . Hence, it is a relaxation of minimum global cut.
Either is an exact LP formulation of minimum global cut or it leads to a relaxation of the TSP that is strictly stronger than the subtour relaxation.
Proof: The disjunctive program above is a relaxation of minimum global cut. Its feasible region is the convex hull of . It satisfies the Hamilton cycle property with by Theorem 3.4. It also includes the triangle inequalities as constraints when at optimality by Lemma 3.5. Hence the result follows from Theorem 3.3.
Next consider the following linear program.
Let be the incidence vector of a Hamilton cycle. Then the minimum objective value for (11) is .
Proof: Let be the incidence vector of an arbitrary Hamilton cycle. Let . For each node , construct a Hamilton cycle vector on the set of nodes from a Hamilton cycle vector as follows. Let and be neighbors of in the Hamilton cycle of . Remove these two edges incident to and replace them with the edge . The result of this is the Hamilton cycle vector . We now derive a chain of inequalities for feasible solutions of (11).
Thus the Hamilton cycle property is satisfied.
The LP (11) is a subset of the LP (7). Hence (11) is an LP relaxation of minimum cut. However, Theorem 3.3 does not apply here unless the triangle inequalities are added to (11). So whether (11) leads to a stronger relaxation of the TSP than the subtour relaxation has not been resolved yet.
For proving the Hamilton cycle property of the -LP, we need the following lemma:
Let be a cost vector, and let be the support graph of . Consider we have the triangle constraints
Let be a path of edges in between and . Then is a valid inequality.
Make the inductive assumption that the theorem holds for paths in of length or less. Let be a path in of length where . Then we can sum two inqualities to get the desired result:
There is a symmetric optimal solution of the -LP when is the incidence vector of a Hamiltonian cycle.
Let be the incidence vector of Hamiltonian cycle . There is an optimal LP solution of the -LP with for each edge in and for each .
Without loss of generality, choose the canonical Hamiltonian cycle where for for each and . Let be an optimal LP solution to the -LP. We wish to rotate our Hamiltonian cycle by units for . Define for each . Define and for each . By symmetry, is also an optimal LP solution. Define . This is a symmetric optimal solution as desired. ∎
Now we show that the -LP (1) satisfies the Hamilton cycle property.
Let be the incidence vector of a Hamilton cycle. Then the minimum objective value for the old -LP (1) is .
Proof sketch: Because of the permutation symmetry of (1), we may make a remarkable assumption. That is, we may assume that the Hamilton cycle is simply going from to in order and back to again. But there is even another amazing consequence of this symmetry. We may also take a symmetric optimal solution where for all in the canonical Hamilton cycle and for all nodes . Since is an optimal solution, and 2 is an upper bound for the minimum objective value of the old -LP, we know that . We will show that must equal , by exploiting the fact that is a feasible solution to the old -LP, thus, yielding the theorem.
Combining these, we get
Hence we obtain
Since , we obtain
However, the degree constraint from the old -LP (1) says that thus, it must be the case that , and the theorem follows.
We only used the triangle inequalities when in the proof of Theorem 3.10. Thus, the old -LP has size as we only need a subset of the triangle inqualities.
4 Bounding the objective of -LP on Hamiltonian Cycles
In this section we show that when the input graph is a single cycle on all of the nodes of the graph and all edges of the graph have unit (1) capacity then the new -LP gives an exact solution. That is, we show the following theorem.
Let be the incidence vector of a Hamilton cycle. Then, the -LP is an exact formulation for any .
To show this for the -LP, we will in fact show that the Hamiltonian cycle property holds for a relaxation, called the -LP. We relax the -LP by expressing as the convex combination of two integers. This gives the -LP which has the following form:
We will use and for this section.
4.1 Removing the variables
The first thing we will consider about this linear program is how to remove the from the problem as they appear in very few constraints. We will project out these variables to better understand the linear program. Constraint (20) in the linear program guarantees that the sum of all the is . The other constraints involving are constraints (21) of the form . Let us consider that are fixed, then we can minimize the sum of the via the following linear program:
If we take the dual of this linear program, we get the following linear program:
This is just a linear programming formulation for a 2-factor scaled down by a factor of 2. Therefore, an optimal solution to this linear program is half the cost of maximum 2-factor where are the edge costs. So, we could project out the variables, by replacing them with the constraints that every 2-factor on the has weight at most .
4.2 Hamiltonian cycle property
The Hamiltonian cycle property is a key property required of the -LP if we are to relate it to the TSP polytope. While the -LP is a strict relaxation of the -LP, it turns out the Hamiltonian cycle property still holds. Not only does the Hamiltonian cycle property hold for the -LP, but if the were relaxed any further then the Hamiltonian cycle property no longer holds.
Consider a modified -LP with a fixed and being the right hand side of constraint (19) and everything else the same. Call this modified LP the -LP. The Hamiltonian cycle property holds for this LP if and only if
Without loss of generality, consider the Hamiltonian cycle is . Let be a feasible solution to the -LP. By symmetry, we can consider all the cyclic rotations of on and these are all feasible. Similarly, by taking the convex combination of these rotations where each is weighted by then we also arrive at a feasible solution. Call this modified solution .
In the modified solution, for all by symmetry. Similarly, for all by symmetry. Now let . We will compute the maximum possible sum of the . Let denote the distance between nodes and in the Hamiltonian cycle . By the triangle inequality, given by constraint (22), we have the following bound:
By the max cut inequality, given by constraint (21), we have the following upper bound:
When we will use the first upper bound and otherwise we will use the second upper bound. Combining these upper bounds we get:
We know that the sum of the is . Let . We can first simplify this expression to:
Now we will use this to find a bound on :
Therefore, the Hamiltonian cycle property holds when is at least .
Now let , and consider the following assignment of .
This is a valid solution for the -LP with the given . This solution gives value less than 2 for where is the Hamiltonian cycle . Therefore, we have proven that is the smallest value of such that satisfies the Hamiltonian cycle property. ∎
The Hamiltonian cycle property holds for the -LP when .
The sum of the is always in the -LP. Therefore, the previous theorem holds. ∎
Proof of 4.1.
By making a constant equal to in the -LP, then we get back the -LP exactly. So, the Hamiltonian cycle property holds for the -LP as well with value .
The solution and provides a solution with cost exactly . ∎
Interestingly, this last property of the motivates the use of the single minimum cut constraint in the -LP. In particular,
5 Using the -LP to strengthen a TSP relaxation
For any the -LP achieves the optimal solution on the Hamiltonian cycle when as stated in Theorem 4.1. This leads to an interesting relaxation of the TSP using the technique relating compact separation to compact optimization Martin (1991); Carr and Lancia (2002). Let be the TSP objective function and be the TSP edge variables. Recall the -LP’s objective is . Consider the dual of the -LP with for the columns in corresponding to the variables and otherwise, which is
Note that and are set to on the columns corresponding to the variables. Then the -TSP relaxation is
How this compares to the subtour relaxation of the TSP is of interest. Compact subtour relaxations can be found in Arthanari and Usha (2000); Carr (1996); Martin (1991). We have the following surprising result:
If the -LP gives an answer of strictly less than for a subtour extreme point , then -TSP can be made stronger than the subtour relaxation (after adding a compact subtour relaxation to it).
Proof: One can see that and for a feasible -LP solution is contradicted by weak duality and . Therefore, is an infeasible value for .
The companion theorem that uses Lovász splitting Lovasz (1976) is:
If the -LP gives an answer of strictly less than the global minimum cut for some cost function , then the -LP gives an answer of strictly less than for some subtour extreme point .
Combining these two lemmas gives us the following theorem:
Either -LP always gives an answer of at least the minimum global cut or we have produced, using -TSP, a compact relaxation stronger than the subtour relaxation.
6 Minimum Spanning Trees
In this subsection, we will examine the cost of the minimum spanning tree (MST) of a point which has the Hamiltonian cycle property to help determine if dominates a convex combination of cuts. We say a point satisfies the Hamiltonian cycle property if for any cost function which is on the edges of Hamiltonian cycle and 0 elsewhere and is a metric. The cost of the minimum Hamiltonian cycle is bounded above by twice the MST, so the value of the MST on is at least . For certain values of the MST we can determine that dominates a convex combination of cuts. In particular, when the value of the MST is exactly 1, or at least 2.
Let the Hamiltonian cycle property hold for and let the cost of the minimum spanning tree be exactly 1, then is exactly a convex combination of cuts.
Let have a minimum spanning tree of cost exactly , and call this MST . Let be the path in