Maximum matching on random graphs

– The maximum matching problem on random graphs is studied analytically by the cavity method of statistical physics. When the average vertex degree c is larger than 2.7183, groups of max-matching patterns which differ greatly from each other gradually emerge. An analytical expression for the max-matching size is also obtained, which agrees well with computer simulations. Discussion is made on this continuous glassy phase transition and the absence of such a glassy phase in the related minimum vertex covering problem. Introduction. – Studies on spin glasses focus on systems with random frustrations [1, 2]. The energy landscape of a spin glass is very rough. When environmental temperature is lower than certain critical value, the system gets trapped in one of many local regions of the whole configurational space (ergodicity breaking). The deep connection between frustrations in spin glasses and constraints in combinatorial optimization problems was noticed by many authors, and the replica method developed during the study of spin glass physics [2, 3] has been applied to hard combinatorial optimization problems including the k-satisfiability [4], the number partitioning [5], the Euclidean matching [6], the vertex covering [7], and many others. However, sophistication of replica method renders analytical discussion to be limited usually to the replica-symmetric (RS) level. Recently, considerable success has been attained in applying the cavity method [8, 9] to combinatorial optimization problems [10, 11, 12, 13]. The cavity formalism enables analytical calculations to be carried out to first-step replica-symmetry-breaking (RSB). For random 3satisfiability (3-Sat) problems it was discovered [10] that, between the Easy-SAT and UNSAT phase there is a Hard-SAT phase. The Hard-SAT phase is a glassy phase, with great many states of the same ground-state energy density which are separated by very high energy barriers. The Easy-SAT to Hard-SAT phase-transition is abrupt. Similar behavior was observed in random 3-XOR-Sat [12]. On the other hand, work performed on minimum vertex covering (min-covering) of random graphs [13] suggested there is no proliferation of ground-states in this model system even when replica symmetry is broken.