Ranking tournaments with no errors II: Minimax relation

Abstract

A tournament $T=(V,A)$ is called cycle Mengerian (CM) if it satisfies the minimax relation on packing and covering cycles, for every nonnegative integral weight function defined on A. The purpose of this series of two papers is to show that a tournament is CM iff it contains none of four Möbius ladders as a subgraph; such a tournament is referred to as Möbius-free. In the first paper we have given a structural description of all Möbius-free tournaments, and have proved that every CM tournament is Möbius-free. In this second paper we establish the converse by using our structural theorems and linear programming approach.

Introduction

This is a follow-up of the paper by the same authors [5]. Let us first present the main results of our previous work.

Let $G=(V,A)$ be a digraph with a nonnegative integral weight $w(e)$ on each arc e. A subset F of arcs is called a feedback arc set (FAS) of G if $G\backslash F$ contains no cycles (directed). The FAS problem is to find an FAS in G with minimum total weight. A collection $\mathcal{C}$ of cycles (with repetition allowed) in G is called a cycle packing of G if each arc e is used at most $w(e)$ times by members of $\mathcal{C}$. The cycle packing problem consists in finding a cycle packing with maximum size. These two problems form a primal-dual pair. Let ${\tau }_w(G)$ be the minimum total weight of an FAS, and let ${\nu }_w(G)$ be the maximum size of a cycle packing. Clearly, ${\nu }_w(G)\le {\tau }_w(G)$. We call G cycle Mengerian (CM) if ${\nu }_w(G)={\tau }_w(G)$ for every nonnegative integral function w defined on A. As stated in [5], the study of CM digraphs has both great theoretical interest and practical value. Despite tremendous research efforts, only some special classes of CM digraphs [1], [2], [12], [14], [15] have been identified to date, and a complete characterization seems extremely hard to obtain. The interested reader is referred to [3], [4], [7], [8], [10], [12], [13], [14], [18], [19] for some related minimax theorems.

The purpose of this series of two papers is to give a complete characterization of all CM tournaments. We call a tournament Möbius-free if it contains none of $K_{3,3}$, $K_{3,3}^{\prime }$, $M_5$, and $M_5^{⁎}$ depicted in Fig. 1 as a subgraph. (Note that $M_5^{⁎}$ arises from $M_5$ by reversing the direction of each arc.) This class of tournaments is so named because the forbidden structures are all Möbius ladders.

Theorem 1.1

A tournament is CM iff it is Möbius-free.

In [5], we have given a structural description of all Möbius-free tournaments, and have proved that every CM tournament is Möbius-free. In this paper we establish the converse by using our structural theorems and linear programming approach.

Let us recall some terminology and notations introduced in [5]. Let $G=(V,A)$ be a digraph with a nonnegative integral weight $w(e)$ on each arc e. For each $v\in V$, we use $d_G^{+}(v)$ and $d_G^{-}(v)$ to denote the out-degree and in-degree of v, respectively. We call v a near-sink of G if its out-degree is one, and call v a near-source if its in-degree is one. For simplicity, an arc $e=(u,v)$ of G is also denoted by uv. Arc e is called special if u is a near-sink or v is a near-source of G. For each $U\subseteq V$, we use $G[U]$ denote the subgraph of G induced by U, and use $G/U$ to denote the digraph obtained from G by first deleting arcs between any two vertices in U, then identifying all vertices in U, and finally deleting the parallel arcs except one from each vertex to each other vertex; we say that $G/U$ is obtained from G by contracting U. Note that $G/U$ may contain pairs of opposite arcs but contains no parallel arcs. Let $u^{⁎}$ be the vertex in $G/U$ arising from contracting U and let $A^{\prime }$ be the arc set of $G/U$. The resulting weight function ${\mathit{w}}^{\prime }$ on $A^{\prime }$ is given by: $w^{\prime }(uv)=w(uv)$ if $u^{⁎}\notin \{u,v\}$, $w^{\prime }(u^{⁎}v)={\sum }_{uv\in A,u\in U}w(uv)$, and $w^{\prime }(v{u}^{⁎})={\sum }_{vu\in A,u\in U}w(vu)$. Moreover, U is called a homogeneous set of G if $|U|\ge 2$ and the arcs between U and any vertex v outside U are either all directed to U or all directed to v.

A dicut of G is a partition $(X,Y)$ of V such that all arcs between X and Y are directed to Y. A dicut $(X,Y)$ is trivial if $|X|=1$ or $|Y|=1$. Recall that G is called weakly connected if its underlying undirected graph is connected, and is called strongly connected or strong if each vertex is reachable from every other vertex. Clearly, a weakly connected digraph G is strong iff G has no dicut. Furthermore, a weakly connected digraph G is called internally strong if every dicut of G is trivial, and is called internally 2-strong $(i2s)$ if G is strong and $G\backslash v$ is internally strong for every vertex v.

Let $T_i=(V_i,A_i)$ be a strong tournament, with $|V_i|\ge 3$ for $i=1,2$. Suppose that $(a_1,b_1)$ is a special arc of $T_1$ with $d_{T_1}^{+}(a_1)=1$ and $(b_2,a_2)$ is a special arc of $T_2$ with $d_{T_2}^{-}(a_2)=1$. The 1-sum of $T_1$ and $T_2$ over $(a_1,b_1)$ and $(b_2,a_2)$ is the tournament arising from the disjoint union of $T_1\backslash a_1$ and $T_2\backslash a_2$ by identifying $b_1$ with $b_2$ (the resulting vertex is denoted by b) and adding all arcs from $T_1\backslash \{a_1,b_1\}$ to $T_2\backslash \{a_2,b_2\}$. We call b the hub of the 1-sum. See Fig. 2 for an illustration. Note that if $|V_i|=3$ for $i=1$ or 2, then $T_i$ is a triangle (a directed cycle of length three), and thus $T=T_{3-i}$.

Let $C_3$ (resp. $F_0$) denote the strong tournament with three (resp. four) vertices (see Fig. 3), let $F_1,F_2,F_3,F_4,F_5$ be the five tournaments depicted in Fig. 4, and let $G_1,G_2,G_3$ be the three tournaments shown in Fig. 5. In these two papers, we reserve the symbols

$${\mathcal{T}}_0=\{C_3,F_0,F_1,F_2,F_3,F_4,G_1,G_2,G_3\}$$

and

$${\mathcal{T}}_1=\{C_3,F_0,F_2,F_3,F_4,G_2,G_3\}={\mathcal{T}}_0\backslash \{F_1,G_1\}.$$

The following are the structural theorems proved in [5].

Theorem 1.2

Let $T=(V,A)$ be an $i2s$ tournament with $|V|\ge 3$. Then T is Möbius-free iff $T\in {\mathcal{T}}_0$.

Theorem 1.3

Let $T=(V,A)$ be a strong Möbius-free tournament with $|V|\ge 3$. Then either $T\in \{F_1,G_1\}$ or T can be obtained by repeatedly taking 1-sums starting from the tournaments in ${\mathcal{T}}_1$.

While the proof methods of these two theorems are purely combinatorial, to show that every Möbius-free tournament is CM, we shall appeal to various optimization techniques.

Let $C\mathit{x}\ge \mathit{d}$, $\mathit{x}\ge \mathbf{0}$ be a rational linear system and let P denote the polyhedron $\{\mathit{x}:C\mathit{x}\ge \mathit{d},\mathit{x}\ge \mathbf{0}\}$. We call P integral if it is the convex hull of all integral vectors contained in P. As shown by Edmonds and Giles [9], P is integral iff the minimum in the LP-duality equation

$$\min \{{\mathit{w}}^T\mathit{x}:C\mathit{x}\ge \mathit{d},\mathit{x}\ge \mathbf{0}\}=\max \{{\mathit{y}}^T\mathit{d}:{\mathit{y}}^{T}C\le {\mathit{w}}^T,\mathit{y}\ge \mathbf{0}\}$$

has an integral optimal solution, for every integral vector w for which the optimum is finite. If, instead, the maximum in the equation has this property, then the system $C\mathit{x}\ge \mathit{d}$, $\mathit{x}\ge \mathbf{0}$ is called totally dual integral (TDI). It is well known that many combinatorial optimization problems can be naturally formulated as integer programs of the form $\min \{{\mathit{w}}^T\mathit{x}:\mathit{x}\in P,\mathrm{integral}\}$; if P is integral, then such a problem reduces to its LP-relaxation. Edmonds and Giles [9] proved that total dual integrality implies primal integrality: if $C\mathit{x}\ge \mathit{d},\mathit{x}\ge \mathbf{0}$ is TDI and d is integer-valued, then P is integral. Thus the model of TDI systems serves as a general framework for establishing many combinatorial min-max theorems. Over the past six decades, these two integrality properties have been the subjects of extensive research and the major concern of polyhedral combinatorics (see Schrijver [16], [17] for comprehensive accounts).

Let us return to the FAS problem. Let M be the cycle-arc incidence matrix of the input digraph G, and let $\pi (G)$ denote the linear system $M\mathit{x}\ge \mathbf{1},\mathit{x}\ge \mathbf{0}$. We call G cycle ideal (CI) if $\pi (G)$ defines an integral polyhedron. From the above Edmonds-Giles theorem, we see that every CM digraph is CI. Furthermore, G is cycle Mengerian (CM) iff $\pi (G)$ is a TDI system, which gives an equivalent definition of CM digraphs. To facilitate better understanding, we give an intuitive interpretation of these concepts. Let $\mathbb{P}(G,\mathit{w})$ stand for the linear program

$$\begin{gathered} \text{Minimize}{\mathit{w}}^T\mathit{x} \\ \text{Subject to}M\mathit{x}\ge \mathbf{1} \\ \mathit{x}\ge \mathbf{0}, \end{gathered}$$

and let $\mathbb{D}(G,\mathit{w})$ denote its dual

$$\begin{gathered} \text{Maximize}{\mathit{y}}^T\mathbf{1} \\ \text{Subject to}{\mathit{y}}^{T}M\le {\mathit{w}}^T \\ \mathit{y}\ge \mathbf{0}, \end{gathered}$$

where $\mathit{w}=(w(e):e\in A)$. Then $\mathbb{P}(G,\mathit{w})$ (resp. $\mathbb{D}(G,\mathit{w})$) is exactly the LP-relaxation of the FAS problem (resp. cycle packing problem), and thus is called the fractional FAS problem (resp. fractional cycle packing problem). Let ${\tau }_w^{⁎}(G)$ be the optimal value of $\mathbb{P}(G,\mathit{w})$, and let ${\nu }_w^{⁎}(G)$ be the optimal value of $\mathbb{D}(G,\mathit{w})$. Clearly,

$${\nu }_w(G)\le {\nu }_w^{⁎}(G)={\tau }_w^{⁎}(G)\le {\tau }_w(G);$$

these two inequalities, however, need not hold with equalities in general (as we shall see in Section 2). As is well known, G is CI iff $\mathbb{P}(G,\mathit{w})$ has an integral optimal solution for any nonnegative integral w iff ${\tau }_w^{⁎}(G)={\tau }_w(G)$ for any nonnegative integral w. Since the separation problem of $\mathbb{P}(G,\mathit{w})$ is the minimum-weight cycle problem, which admits a polynomial-time algorithm, it follows from a theorem of Grötschel, Lovász, and Schrijver [11] that $\mathbb{P}(G,\mathit{w})$ is always solvable in polynomial time. Therefore, the FAS problem can be solved in polynomial time for any nonnegative integral w, provided its input digraph G is CI.

We shall actually establish the following strengthening of Theorem 1.1 in this paper.

Theorem 1.4

For a tournament $T=(V,A)$, the following statements are equivalent:

• T is Möbius-free;

• T is cycle ideal; and

• T is cycle Mengerian.

Throughout we shall repeatedly use the following notations and terminology. As usual, ${\mathbb{R}}_{+}$ and ${\mathbb{Z}}_{+}$ stand for the sets of nonnegative real numbers and nonnegative integers, respectively. For any two sets Ω and K, where Ω is always a set of numbers and K is always finite, we use ${\mathrm{\Omega }}^K$ to denote the set of vectors $\mathit{x}=(x(k):k\in K)$ whose coordinates are members of Ω. If f is a function defined on a finite set S and $R\subseteq S$, then $f(R)$ denotes ${\sum }_{s\in R}f(s)$. An instance $(T,\mathit{w})$ consists of a Möbius-free tournament $T=(V,A)$ together with a weight function $\mathit{w}\in {\mathbb{Z}}_{+}^A$. We say that another instance $(T^{\prime },{\mathit{w}}^{\prime })$ is smaller than $(T,\mathit{w})$ if $|V^{\prime }|<|V|$ or if $|V^{\prime }|=|V|$ but $w^{\prime }(A^{\prime })<w(A)$, where $T^{\prime }=(V^{\prime },A^{\prime })$. As introduced in our first paper, we also say that a tournament $T_1=(V_1,A_1)$ is smaller than another tournament $T_2=(V_2,A_2)$ if $|V_1|<|V_2|$.

Recall the fractional problems introduced above. In view of the equivalent definition, to show that every Möbius-free tournament is CM, we shall turn to proving that $\mathbb{D}(T,\mathit{w})$ has an integral optimal solution for every instance $(T,\mathit{w})$. To this end, we apply the double induction on V and $w(A)$, where $T=(V,A)$. Since the desired statement holds trivially when $|V|=1$, we proceed to the induction step, and propose to establish the following statement.

Theorem 1.5

Let $(T,\mathit{w})$ be an instance, such that $\mathbb{D}(T^{\prime },{\mathit{w}}^{\prime })$ has an integral optimal solution for any smaller instance $(T^{\prime },{\mathit{w}}^{\prime })$ than $(T,\mathit{w})$. Then $\mathbb{D}(T,\mathit{w})$ also has an integral optimal solution.

Note that $C_3$ is CM, because both ${\nu }_w(C_3)$ and ${\tau }_w(C_3)$ are equal to the minimum weight of an arc. We shall present a computer-assisted proof that $G_1$ is CM, which consists of analyses of a finite number of cases (see Lemma 6.2 for a characterization of TDI systems). Since $F_1$ is isomorphic to $G_1\backslash v_6$, it is also CM.

For $T\notin \{C_3,F_1,G_1\}$, the proof strategy of Theorem 1.5 is described below.

Obviously, we may assume that T is strong and ${\tau }_w(T)>0$. We shall prove that T can be expressed as the 1-sum of two strong Möbius-free tournaments $T_1$ and $T_2$ over two special arcs $(a_1,b_1)$ and $(b_2,a_2)$, such that one of the following three cases occurs (see Lemma 2.5):

• ${\tau }_w(T_2\backslash a_2)>0$ and $T_2\in {\mathcal{T}}_2$, where ${\mathcal{T}}_2=({\mathcal{T}}_1\backslash \{C_3\})\cup \{F_6\}$ for some tournament $F_6$ to be introduced in Section 2;

• ${\tau }_w(T_2\backslash a_2)>0$ and there exists a vertex subset S of $T_2\backslash \{a_2,b_2\}$ with $|S|\ge 2$, such that $T[S]$ is acyclic, $T_2/S\in {\mathcal{T}}_3$, and the vertex $s^{⁎}$ arising from contracting S is a near-sink in $T/S$, where ${\mathcal{T}}_3=({\mathcal{T}}_2\backslash F_2)\cup \{G_4,G_5,G_6\}$ for some tournaments $G_4,G_5,G_6$ to be introduced in Section 2; and

• every positive cycle in T crosses the hub b, where a cycle C in T is called positive if $w(e)>0$ for each arc e on C.

In the first two cases, we shall prove that $\mathbb{D}(T,\mathit{w})$ has an optimal solution y such that $y(C)$ is a positive integer for some cycle C contained in $T_2\backslash a_2$. Define $w^{\prime }(e)=w(e)$ if $e\notin C$ and $w^{\prime }(e)=w(e)-y(C)$ for each $e\in C$. By the induction hypothesis, $\mathbb{D}(T^{\prime },{\mathit{w}}^{\prime })$ has an integral optimal solution ${\mathit{y}}^{\prime }$. We can thus obtain an integral optimal solution to $\mathbb{D}(T,\mathit{w})$ by combining ${\mathit{y}}^{\prime }$ with $y(C)$ (the details can be found in Lemma 3.2(iii)). As we shall see, this strategy is carried out by performing various reductions. The detailed analyses of these two cases are given in Section 4 and Section 5, respectively.

In the third case, by splitting the hub b into two vertices s and t, we can apply the max-flow min-cut theorem to show that T is CM. This step is established in Lemma 6.1.

The remainder of this paper is organized as follows. In Section 2, we show that every cycle ideal tournament is Möbius-free. We also prove that one of the three cases described in the above proof strategy of Theorem 1.5 occurs (see Lemma 2.5). In Section 3, we make technical preparations for the proof of Theorem 1.5, and derive properties enjoyed by optimal solutions to $\mathbb{P}(T,\mathit{w})$ and $\mathbb{D}(T,\mathit{w})$. In Section 4, we prove Theorem 1.5, in the case of (i) exhibited in Lemma 2.5, by performing a series of basic reduction operations. In Section 5, we prove Theorem 1.5, in the case of $(ii)$ exhibited in Lemma 2.5, by performing a series of composite reduction operations. In Section 6, we accomplish the last step of our proof of Theorem 1.5 and hence the proof of Theorem 1.4. In Section 7, we conclude this paper with some remarks.