Total Dual Integrality of Triangle Covering

Abstract

This paper concerns weighted triangle covering in undirected graph $G=(V,E)$, where a nonnegative integral vector $\mathbf w=(w(e):e\in E)^T$ gives weights of edges. A subset S of E is a triangle cover in G if S intersects every triangle of G. The weight of a triangle cover is the sum of w(e) over all edges e in it. The characteristic vector $\mathbf x$ of each triangle cover in G is an integral solution of the linear system

$$\begin{aligned} \pi :A\mathbf x\ge \mathbf 1,\mathbf x\ge \mathbf 0, \end{aligned}$$

where A is the triangle-edge incidence matrix of G. System $\pi$ is totally dual integral if $\max \{\mathbf 1^T\mathbf y:A^{T}\mathbf y\le \mathbf w,\mathbf y\ge \mathbf 0\}$ has an integral optimum solution $\mathbf y$ for each integral vector $\mathbf w\in \mathbb Z_+^E$ for which the maximum is finite. The total dual integrality of $\pi$ implies the nice combinatorial min-max relation that the minimum weight of a triangle cover equals the maximize size of a triangle packing, i.e., a collection of triangles in G (repetitions allowed) such that each edge e is contained in at most w(e) of them. In this paper, we obtain graphical properties that are necessary for the total dual integrality of system $\pi$, as well as those for the (stronger) total unimodularity of matrix A and the (weaker) integrality of polyhedron $\{\mathbf x:A\mathbf x\ge \mathbf 1,\mathbf x\ge \mathbf 0\}$. These necessary conditions are shown to be sufficient when restricted to planar graphs. We prove that the three notions of integrality coincide, and are commonly characterized by excluding odd pseudo-wheels from the planar graphs.

Research supported in part by NNSF of China under Grant No. 11531014 and 11222109.

Appendix: A List of Mathematical Symbols

$(G,\mathbf w)$	Weighted graph $G=(V(G),E(G))$ with $\mathbf w\in \mathbb Z^{E(G)}_+$
$\tau _w(G)$	The minimum weight of an integral triangle cover in $(G,\mathbf w)$
$\nu _w(G)$	The maximum size of an integral triangle packing in $(G,\mathbf w)$
$\tau ^*_w(G)$	The minimum weight of a fractional triangle cover in $(G,\mathbf w)$
$\nu ^*_w(G)$	The maximum size of a fractional triangle packing in $(G,\mathbf w)$
$\tau (G)$	$\tau _w(G)$ when $\mathbf w=\mathbf 1$
$\nu (G)$	$\nu _w(G)$ when $\mathbf w=\mathbf 1$
$A_G$	The triangle-edge incidence matrix of graph G
$\varLambda (G)$	The set of triangles in graph G
$\mathfrak B$	The set of graphs G such that $A_G$ are TUM
$\mathfrak M$	The set of graphs G such that systems $A_Gx\ge 1, x\ge 0$ are TDI
$\mathfrak J$	The set of graphs G such that $\{\mathbf x: A_G\mathbf x\ge 1, \mathbf x\ge \mathbf 0\}$ are intergal
$\mathfrak N$	The set of minimal graphs not belonging to $\mathfrak B$
$\mathscr {T}_C$	The set of triangles in triangle-cycle $C\!=\!e_1\triangle _1e_2\!\cdots \!e_k\triangle _k e_1\!=\!\cup _{i\!=\!1}^k\triangle _i$
$\mathscr {B}_C$	The set of basic triangles in triangle-cycle C, i.e., $\{\triangle _1, \cdots ,\triangle _k\}$
$J_C$	The set of join edges in triangle-cycle C, i.e., $\{e_1,\cdots ,e_k\}$
$N_C$	The set of nonjoin edges in triangle-cycle C, i.e., $E(C)\backslash J_C$
$\mathscr {T}_{C,i}$	$\{\triangle \in \mathscr {T}_C: |\triangle \cap J_C|=i \}$, $i=0,1,2,3$