The degree sequence of a simple graph with
is the vector, where is the degree of vertex for all . If with is a subgraph of we also write for the degree sequence of . Degree sequences have been studied by many authors, starting from their characterization by Erdős and Gallai , see e.g.  and the references therein.
In this article we are interested in the following problem and some of its variants.
Degree Sequence Optimization. Given a graph , integer , and function , find a subgraph with edges maximizing .
We will also consider the unprescribed variant of the problem, where the number of edges is not specified, and the optimization is over any subgraph with vertex set .
Our subgraphs always consist of the entire original vertex set and a subset of the edges. Scaling up rational values if needed we assume the function takes on integer values. Also, it is enough that the function is defined only on its domain , and properties of the function such as convexity are with respect to this domain only. We assume that the function is presented either by an oracle that, queried on , returns , or by some compact presentation as will be clear from the context.
As an example, consider the case where the function is linear and given by the inner product for some . Then for every subset of edges we have that and so the problem can be easily solved by sorting the edges of by the value and taking to consist of the edges with maximum values.
However, the problem is generally much harder, even for convex functions, and when such functions are presented by oracles, any algorithm for the problem may need to make exponentially many queries and hence the problem is intractable, as explained in Section 4.
In contrast to this intractability, and as a natural extension of the linear case, we show that for the following convex multi-criteria objective the problem is polynomial time solvable. Let be given vectors. Each is interpreted as a linear criterion under which the value of subgraph is the inner product . These criteria are balanced by a convex function . The case of a single criterion and the identity on is the linear problem discussed above. Our first result is the efficient solution of this problem. Here and elsewhere an algorithm involving oracle presented objects is polynomial time if its running time in terms of the data, and the number of oracle queries it makes, are polynomial.
Fix any . Given a graph , , , and oracle presented convex , we can solve in polynomial time the multi-criteria problem
Next we consider separable functions, that is, of the form with each univariate. In this case several results are known in the literature. First, even in this case, the problem is generally NP-hard, see Section 4. On the positive side, the problem is polynomial time solvable in the following situations: over the complete graph when for all  and more generally when all functions are the same and arbitrary ; for any graph when all are concave for the prescribed problem  and independently for the unprescribed variant ; and for every fixed and any when of the functions are arbitrary and the rest are either all nondecreasing or all nonincreasing .
A further special case of the separable problem is the general factor problem , which is to decide, given a graph and subsets for , if there is a with for all , and find one if yes. Indeed, for each define a function by if and if . Then the optimal value of the degree sequence problem is zero if and only if there is a factor, in which case any optimal graph is one. An even more special case is the well studied -factor problem introduced by Lovász , where each is an interval. This reduces to the degree sequence problem even with concave functions, with if , if , and if . In particular, the perfect matching problem is that with for all .
We consider the following extension of the separable problem. We are now given, with the graph , a -partition , and integers for . We search for a subgraph maximizing the objective with the requirement that for all . We refer to the as colors and call the problem the colored degree optimization problem. Standard separable degree optimization is the special case of . Another special case is the notorious exact matching problem, where we are given a -partition of the edges of the complete bipartite graph and integers , and we need to decide if there is a perfect matching with for all . It is a special case of our problem with and for every vertex , where the optimal value is zero if and only if there is an exact matching.
The exact matching problem has randomized algorithms [12, 13] but its deterministic complexity is open already for . Moreover, standard separable degree optimization, with , is generally NP-hard, see Section 4. In contrast, we show that for graphs of bounded tree-depth (see Section 3), the colored problem can be solved in polynomial time.
Fix any . Given a graph of tree-depth at most and maximum degree at most , functions , -partition , and integers for , we can solve in polynomial time the colored problem
We conclude with some open problems. The complexity of degree sequence optimization is still wide open, even in the separable case. It would be interesting to determine it for various classes of graphs and various classes of functions . Particularly intriguing is the special case of the complete graph , where the optimization is over any graph on , for which the separable problem might be solvable in polynomial time for any functions . Also, while the separable problem over arbitrary was shown to be NP-hard in  already when for all , the complexity of the unprescribed variant, with no restriction on the number of edges, is open and might be solvable in polynomial time for any graph and any convex functions . In particular, what is the complexity of the following unprescribed separable problem over the complete graph with arbitrary convex functions,
2 Convex multi-criteria objectives
We need some terminology. Let be a finite set and let . Consider any edge (-dimensional face) of . A direction of is any nonzero multiple of . A linear optimization oracle for is one that, queried on , returns an element attaining . We make use of the following result [13, Theorem 2.16].
Fix any . Given any presented by a linear optimization oracle, , oracle presented convex function , and a set of directions of all edges of , we can solve in polynomial time the multi-criteria problem
We can now prove our first theorem.
Theorem 1.1 Fix any . Given a graph , , , and oracle presented convex , we can solve in polynomial time the multi-criteria problem
Proof. Recall that for any subset we write for the degree sequence of the subgraph with set of edges . In particular, if then is the sum of the unit vectors corresponding to the vertices of . Define
Assume else the problem is trivial. First we construct a set of directions of every edge of of size polynomial in . Consider any edge of and let be its vertices so with and . Let be such that the inner product is maximized over precisely at . Since there exist and . If then so and hence is a direction of the edge . Similarly if then is a direction of . Since directions are defined up to nonzero scalar multiplication, it follows that the set consisting of one representative of for each pair of distinct , is a set of vectors containing a direction of every edge of .
Next, we construct a linear optimization oracle for . Given , we need to solve
Now we observe that for every we have .
Thus, we let consist of the edges with largest values
and return . Using this oracle and the set in
Proposition 2.1 we obtain the claim of the theorem.
To solve the unprescribed variant of the problem where the number of edges is not specified we can simply solve the above problem for and pick the best solution. However, for the this variant there is a faster shortcut which we proceed to describe. Let now
Consider any edge of and let be such that is maximized over precisely at . Suppose first there exist . If then we have so and hence is a direction of the edge . Likewise if then is a direction of . A similar argument works if there exist . It follows that the set is a set of directions of every edge of . Next, given , we need to solve
Since we set and return .
So now the set has size instead of in the prescribed variant, and linear optimization can be done by checking signs in the set rather than sorting it in the prescribed variant. So applying Proposition 2.1 again we obtain a faster solution.
3 Colored bounded tree-depth graphs
We need some more terminology. The tree-depth of a graph is defined as follows. The height of a rooted tree is the maximum number of vertices on a path from the root to a leaf. A rooted tree on is valid for if for each edge one of lies on the path from the root to the other of . The tree-depth of is the smallest height of a rooted tree which is valid for . For instance, if is a perfect matching with then its tree-depth is where a tree validating it rooted at has edge set . Next, the graph of an matrix is the graph on where is an edge if and only if there is an such that . The tree-depth of is the tree-depth of its graph. We denote by the largest absolute value of an entry of . We use a recent result [6, 9, Theorems 5 and 6] on integer programs in variable dimension (as opposed to the classical result in fixed dimension ). It asserts that integer programming is solvable in fixed-parameter tractable time  when parameterized by the numeric measure and sparsity measure of the matrix defining the program. In particular, for any fixed it is solvable in time polynomial in .
Consider the following integer program in variable dimension , where , , , parameterized by and ,
It can be solved in time for some function of and some polynomial of .
We can now prove our second theorem.
Theorem 1.2 Fix any . Given a graph of tree-depth at most and maximum degree at most , functions , -partition , and integers for , we can solve in polynomial time the colored problem
Proof. We construct the following integer program, in binary variables, variable for each edge and variable for each vertex and each , where, as usual, is the set of edges in containing vertex ,
Suppose is a feasible solution of this program and let be the subgraph with . Consider any . Constraint (2) forces for exactly one . Then constraint (1) forces this to be . So the objective value of in the program is which is the objective value of in the degree optimization problem. Finally, constraint (3) forces for all .
It is easy to see that also, conversely, if is a feasible solution of the degree optimization problem then defined by if and otherwise, and if and otherwise, is a feasible solution to the program with the same objective value. So the degree optimization problem reduces to solving the integer program.
Consider the matrix expressing equations (1)–(3) and its transpose . The columns of are indexed by the variables. Let us index the equations and the rows of by corresponding to equations (1), corresponding to equations (2), and corresponding to equations (3). Let be a rooted tree on validating that and let be its root. We now use to obtain a rooted tree with vertices , rooted at , consisting of the edges of , the edges for , the edges for , and the edge .
We now show that is valid for . For this we need to show that if two equations share a variable then one lies on the path in from its root to the other. Consider any . They share a variable if only if and then, since is valid for , one of them lies on the path in from its root to the other, and hence also on the path in from its root to the other. Next, any distinct do not share a variable. Next, consider any . They share the variables if and only if in which case lies on the path in from its root to consisting of the path from to via and and the edge . Next, any distinct do not share a variable and any do not share a variable. Finally, for any we have that lies on the path from to . So is valid for .
4 Some limitations
Here we discuss some situations where the degree sequence optimization problem is hard.
Oracle presented convex functions
Given a graph consider the degree sequence polytope of defined as
When maximizing a convex there will be an optimal graph with a vertex of . If is presented by an oracle then we claim that any algorithm for the problem may need to make exponentially many queries and hence the problem is intractable. To see this, consider a perfect matching graph with vertices and . Then is isomorphic to the unit cube via the map . Any integer values on the vertices of extend to a convex function and hence any algorithm for the problem must query the oracle on each of the vertices of .
Concave-convex separable functions
We now show the hardness of the unprescribed variant of the problem with separable functions. The NP-complete cubic subgraph problem is to decide if a given graph has a subgraph where each vertex has degree or . Defining and for for , the optimal objective value is zero if and only if has a cubic subgraph, and so the corresponding degree optimization problem is NP-hard.
The problem remains NP-hard with bipartite and concave for and convex for . Indeed, the general factor problem is NP-complete for bipartite with maximum degree and for and for , see . Define
Then the optimal value of the degree sequence problem is zero if and only if there is a factor.
It remains hard moreover for bipartite graphs with a single concave function and all others convex. Recall that the prescribed problem on with specified number of edges and all functions is NP-hard . Define a bipartite graph by subdividing each edge of and denoting the new vertex by , and adding a new vertex connected to all vertices. Define for all original vertices, and, for a sufficiently large positive integer , let and for all new vertices . Then in any optimal subgraph for the unprescribed problem on , of the vertices have degree and the rest have degree , and the subgraph of with edge set is optimal for the prescribed problem on , reducing the latter to the former.
Weighted degree optimization
Another extension is the following problem. With the graph we are given an edge weighting . The problem is to find a subgraph maximizing
where is the set of edges in containing . So the standard separable degree sequence optimization problem is the special case with for all .
However, this is NP-hard already for with concave and all zero. Recall that the NP-complete partition problem is to decide, given positive integers , if there is a with . Given such integers, define the weight function by for each and , and define the functions as above. Then clearly there is a partition if and only if the optimal objective value of the degree sequence problem is zero, showing its hardness.
S. Onn was supported by a grant from the Israel Science Foundation and the Dresner chair.
-  Apollonio, N., Sebő, A.: Minconvex factors of prescribed size in graphs. SIAM Journal on Discrete Mathematics 23:1297–1310 (2009)
-  Cornuéjols, G.: General factors of graphs. Journal of Combinatorial Theory Series B 45:185–198 (1988)
-  Deza, A., Levin, A., Meesum, S.M., Onn, S.: Optimization over degree sequences. SIAM Journal on Discrete Mathematics 32:2067–2079 (2018)
-  Deza, G., Onn S.: Optimization over degree sequences of graphs. Discrete Applied Mathematics (2019). http://doi.org/10.1016/j.dam.2019.12.016
-  Downey, R.G., Fellows, M.R.: Fundamentals of parameterized complexity. Springer (2013)
-  Eisenbrand, F., Hunkenschröder, C., Klein, K.M., Koutecký, M., Levin, A., Onn, S.: An algorithmic theory of integer programming. ArXiv:1904.01361 1–63 (2019)
-  Erdős, P., Gallai, T.: Graphs with prescribed degrees of vertices (in Hungarian). Matematikai Lopak 11:264–274 (1960)
Erdős, P., Király, Z., Miklós, I.: On the swap-distances of different realizations of a graphical degree sequence. Combinatorics, Probability and Computing 22:366–383 (2013)
-  Koutecký, M., Levin, A., Onn, S.: A parameterized strongly polynomial algorithm for block structured integer programs. Proceedings of ICALP 2018, Leibniz International Proceedings in Informatics, 107-85:1–14 (2018)
-  Lenstra, H.W., Jr.: Integer programming with a fixed number of variables. Mathematics of Operations Research 8:538–548 (1983)
-  Lovász, L.: Subgraphs with prescribed valencies. Journal of Combinatorial Theory 8:391–416 (1970)
-  Mulmuley, K., Vazirani, U.V., Vazirani, V.V.: Matching is as easy as matrix inversion. Combinatorica 7:105–113 (1987)
-  Onn, S.: Nonlinear Discrete Optimization. Zurich Lectures in Advanced Mathematics, European Mathematical Society (2010). Available online at: http://ie.technion.ac.il/onn/Book/NDO.pdf
-  Peled, U.N., Petreschi, R., Sterbini, A.: -graphs with maximum sum of squares of degrees. Journal of Graph Theory 31:283–295 (1999)