Heat exchanger network synthesis exploits excess heat by integrating process hot and cold streams and improves energy efficiency by reducing utility usage. Determining provably good solutions to the minimum number of matches is a bottleneck of designing a heat recovery network using the sequential method. This subproblem is an N P-hard mixedinteger linear program exhibiting combinatorial explosion in the possible hot and cold stream configurations. We explore this challenging optimization problem from a graph theoretic perspective and correlate it with other special optimization problems such as cost flow network and packing problems. In the case of a single temperature interval, we develop a new optimization formulation without problematic big-M parameters. We develop heuristic methods with performance guarantees using three approaches: (i) relaxation rounding, (ii) water filling, and (iii) greedy packing. Numerical results from a collection of 51 instances substantiate the strength of the methods.This manuscript is dedicated, with deepest respect, to the memory of Professor C. A. Floudas.Professor Floudas showed that, given many provably-strong solutions to the minimum number of matches problem, he could design effective heat recovery networks. So the diverse solutions generated by this manuscript directly improve Professor Floudas' method for automatically generating heat exchanger network configurations. 1988, Furman and Sahinidis 2002, Escobar and Trierweiler 2013). Floudas et al. (2012) review the critical role of heat integration for energy systems producing liquid transportation * fuels (Niziolek et al. 2015). Other important applications of HENS include: refrigeration systems (Shelton and Grossmann 1986), batch semi-continuous processes (Zhao et al. 1998, Castro et al. 2015 and water utilization systems (Bagajewicz et al. 2002).Heat exchanger network design is a mixed-integer nonlinear optimization (MINLP) problem (Yee and Grossmann 1990, Ciric and Floudas 1991, Papalexandri and Pistikopoulos 1994, Hasan et al. 2010. Mistry and Misener (2016) recently showed that expressions incorporating logarithmic mean temperature difference, i.e. the nonlinear nature of heat exchange, may be reformulated to decrease the number of nonconvex nonlinear terms in the optimization problem. But HENS remains a difficult MINLP with many nonconvex nonlinearities. One way to generate good HENS solutions is to use the so-called sequential method (Furman and Sahinidis 2002). The sequential method decomposes the original HENS MINLP into three tasks: (i) minimizing utility cost, (ii) minimizing the number of matches, and (iii) minimizing the investment cost. The method optimizes the three mathematical models sequentially with: (i) a linear program (LP) (Cerda et al. 1983, Papoulias andGrossmann 1983), (ii) a mixed-integer linear program (MILP) (Cerda and Westerberg 1983, Papoulias and Grossmann 1983), and (iii) a nonlinear program (NLP) (Floudas et al. 1986). The sequential method may not return the global solution of the origi...