SIAM J. DISCRETE MATH. \bigcircc 2019 Society for Industrial and Applied Mathematics Vol. 33, No. 2, pp. 1061{1094 COMPLETE DESCRIPTION OF MATCHING POLYTOPES WITH ONE LINEARIZED QUADRATIC TERM FOR BIPARTITE GRAPHS˚ : MATTHIAS WALTER Abstract. We consider, for complete bipartite graphs, the convex hulls of characteristic vectors of all matchings, extended by a binary entry indicating whether the matching contains two specific edges. This polytope is associated with the quadratic matching problem with a single linearized quadratic term. We provide a complete irredundant inequality description, which settles a conjec- ture by Klein [Combinatorial Optimization with One Quadratic Term, Ph.D. thesis, TU Dortmund, Dortmund, Germany]. In addition, we also derive facetness and separation results for the polytopes. The completeness proof is based on a geometric relationship to a matching polytope of a nonbipartite graph. Using standard techniques, we finally extend the result to capacitated b-matchings. Key words. quadratic matching problem, matching polytope, bipartite matching AMS subject classifications. 90C57, 90C20, 90C35 DOI. 10.1137/16M1089691 1. Introduction. Let Km;n “ pV; Eq be the complete bipartite graph with the node partition V “ UY9 W , |U| “ m, and |W | “ n for m; n ¥ 2. The maximum weight matching problem is to maximize the sum cpMq :“ c over all matchings M ePM e (i.e., M E and no two edges of M share a node) in K for given edge weights Ď m;n E ř c P Q . Note that we generally abbreviate jPJ vj as vpJ) for vectors v and subsets J of their index sets. Following the usual approach in polyhedralř combinatorics, we identify the match- ings M with their characteristic vectors \chip Mq P t0; 1u E, which satisfy \chip Mq “ 1 if e and only if e P M. The maximum weight matching problem is then equivalent to the problem of maximizing the linear objective c over the matching polytope, i.e., the con- vex hull of all characteristic vectors of matchings. In order to use linear programming techniques, one requires a description of that polytope in terms of linear inequalities. Such a description is well known [1] and consists of the constraints (1.1) xe ¥ 0 for all e P E; (1.2) xp\delta pvqq ¤ 1 for all v P U Y9 W; where \delta pvq denotes the set of edges incident to v. For general (nonbipartite) graphs, Edmonds [4, 5] proved that adding the following Blossom inequalities is sufficient to describe the matching polytope: 1 xpErSsq ¤ p|S| ´ 1q for all S Ď V , |S| odd, 2 where ErSs :“ ttu; vu P E : u; v P Su. His result is based on a primal-dual optimiza- tion algorithm, which also proved that the weighted matching problem can be solved in polynomial time. Later, Schrijver [23] gave a direct (and more geometric) proof of the polyhedral result. Note that one also often considers the special case of per- fect matchings, which are those matchings covering every node of the graph. The ˚Received by the editors August 16, 2016; accepted for publication (in revised form) April 8, 2019; published electronically June 27, 2019. Downloaded 07/16/19 to 130.89.46.16. Redistribution subject SIAM license or copyright; see http://www.siam.org/journals/ojsa.php http://www.siam.org/journals/sidma/33-2/M108969.html :University of Twente, Department of Applied Mathematics, 7500 AE Enschede, The Netherlands ([email protected]). 1061 Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 1062 MATTHIAS WALTER associated perfect matching polytope is the face of the matching polytope obtained by requiring that all inequalities (1.2) are satisfied with equality (1.3) xp\delta pvqq “ 1 for all v P U Y9 W: For more background on matchings and the matching polytopes we refer to volumes B and C of Schrijver's book [25]. For a basic introduction on polytopes and linear programming, we recommend [24]. In this paper, we consider the more general quadratic matching problem for which E we have, in addition to c, a set Q Ď 2 and weights p : Q Ñ Q for the edge-pairs in Q. The objective is now to maximize cpMq ` qPQ;qĎ M pq, again over all matchings M. Before we discuss the case |Q| “`1˘ in detail, we focus on the more general case. By requiring the matchings to be perfect, weř obtain as a special case the quadratic assignment problem, a problem that is not just NP-hard [22], but also hard to solve in practice (see [18] for a survey). A common strategy is then to linearize this quadratic objective function by in- troducing additional variables ye;f “ xe ¨ xf for all te; fu P Q. Usually, the straight- forward linearization of this product equation is very weak, and one seeks to find (strong) inequalities that are valid for the associated polytope. There were several polyhedral studies, in particular for the quadratic assignment problem, e.g., by Pad- berg and Rijal [21] and J\"ungerand Kaibel [13, 14, 15]. One way of finding such inequalities, recently suggested by Buchheim and Klein [2], is the so-called one term linearization technique. The idea is to consider the special case of |Q| “ 1 in which the optimization problem is still polynomially solvable. By the polynomial-time equivalence of separation and optimization [11, 16, 19], one can thus hope to characterize all (irredundant) valid inequalities and develop separation algorithms. These inequalities remain valid when more than one monomial is present, and hence one can use the results of this special case in the more general setting. Buchheim and Klein suggested this for the quadratic spanning-tree problem and con- jectured a complete description of the associated polytope. This conjecture was later confirmed by Fischer and Fischer [7] and Buchheim and Klein [3]. Fischer, Fischer, and McCormick [9] recently generalized this result to matroids and multiple monomi- als, which must be nested in a certain way. In her dissertation [17], Klein considered several other combinatorial polytopes, in particular the quadratic assignment poly- tope. Hupp, Klein, and Liers [12] generalized these results, in particular proofs for certain inequality classes to be facet-defining, to nonbipartite matchings. They car- ried out a computational study on the practical strength of this approach, using these inequalities during branch-and-cut. The main goal of this paper is to prove that the description for bipartite graphs conjectured by Klein [17] is indeed complete. Moreover, we extend the theoretical work of Klein to nonperfect matchings. Our setup is as follows: Consider two disjoint edges e1 “ tu1; w1u and e2 “ tu2; w2u (with ui P U and wi P W for i “ 1; 2) in Km;n and denote by V ˚ :“ tu ; u ; w ; w u the union of their node sets. Our polytopes of 1 2 1 2 interest are the convex hulls of all vectors p\chip Mq; yq for which M is a matching in Km;n, y P t0; 1u and one of the relationships between M and y holds: 1QÓ 1QÓ ‚ Pmatch :“ PmatchpKm;n; e1; e2q: y “ 1 implies e1; e2 P M. P 1QÒ : P 1QÒ K ; e ; e : y 0 implies e M or e M. Downloaded 07/16/19 to 130.89.46.16. Redistribution subject SIAM license or copyright; see http://www.siam.org/journals/ojsa.php ‚ match “ matchp m;n 1 2q “ 1 R 2 R 1Q 1Q ‚ Pmatch :“ PmatchpKm;n; e1; e2q: y “ 1 if and only if e1; e2 P M. Note that P 1QÓ (resp., P 1QÒ ) is the downward (resp., upward) monotonization of match match Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. MATCHING POLYTOPES WITH ONE QUADRATIC TERM 1063 1Q Pmatch with respect to the y-variable, and that 1Q 1QÓ 1QÒ E Pmatch “ convpPmatch X Pmatch X pZ ˆ Zqq: Clearly, constraints (1.1) and (1.2) as well as the bound constraints 0 y 1(1.4) ¤ ¤ are valid for all three polytopes. Additionally, the two inequalities (1.5) y ¤ xei i “ 1; 2; are also valid for P 1Q and P 1QÓ (and belong to the standard linearization of match match y “ x ¨ x ). Klein [17] introduced two more inequality classes, and proved them to e1 e2 be facet-defining (see Theorems 6.2.2 and 6.2.3 in [17]). They are indexed by subsets SÓ and SÒ of nodes (see Figure 1), defined via SÓ :“ tS Ď UY9 W : |S| odd and either S V ˚ u ; u and S U S W 1 or X “ t 1 2u | X | “ | X | ` ˚ S X V “ tw1; w2u and |S X W | “ |S X U| ` 1u; Ò ˚ ˚ S :“ tS Ď UY9 W : |S X U| “ |S X W | and either S X V “ tu1; w2u or S X V “ tu ; w uu; 2 1 and read 1 (1.6) xpErSsq ` y ¤ p|S| ´ 1q for all S P SÓ; 2 1 Ò (1.7) xpErSsq ` xe1 ` xe2 ´ y ¤ |S| for all S P S : 2 e1 u1 w1 u1 w1 e1 e2 e2 u2 w2 u2 w2 u3 w3 u3 w3 u4 w4 u4 w4 u5 w5 u5 w5 (A) A set S P SÓ indexing inequal- (B) A set S P SÒ indexing inequal- Downloaded 07/16/19 to 130.89.46.16. Redistribution subject SIAM license or copyright; see http://www.siam.org/journals/ojsa.php ity (1.6). ity (1.7). Fig. 1. Node sets indexing additional facets. Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 1064 MATTHIAS WALTER Klein [17] even conjectured that constraints (1.1) and (1.3){(1.7) completely de- 1Q scribe the mentioned face of Pmatch.