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But there is a redundancy here: the last constraint c x g is itself a linear inequality and can be absorbed into Ax b So, we de ne ILP to be following search problem: given A and b, nd a nonnegative integer vector x satisfying the inequalities Ax b, or report that none exists Despite the many crucial applications of this problem, and intense interest by researchers, no ef cient algorithm is known for it There is a particularly clean special case of ILP that is very hard in and of itself: the goal is to nd a vector x of 0 s and 1 s satisfying Ax = 1, where A is an m n matrix with 0 1 entries and 1 is the m-vector of all 1 s It should be apparent from the reductions in Section 714 that this is indeed a special case of ILP We call it ZERO - ONE EQUATIONS (ZOE) We have now introduced a number of important search problems, some of which are familiar from earlier chapters and for which there are ef cient algorithms, and others which are different in small but crucial ways that make them very hard computational problems To complete our story we will introduce a few more hard problems, which will play a role later in the chapter, when we relate the computational dif culty of all these problems The reader is invited to skip ahead to Section 82 and then return to the de nitions of these problems as required

Figure 26-2 The parasympathetic (craniosacral) division of the autonomic nervous system Preganglionic bers extend from nuclei of the brainstem and sacral segments of the spinal cord to peripheral ganglia Short postganglionic bers extend from the ganglia to the effector organs The lateral-posterior hypothalamus is part of the supranuclear mechanism for the regulation of parasympathetic activities The frontal and limbic parts of the supranuclear regulatory apparatus are not indicated in the diagram (see text) (Reproduced by permission from Noback CL, Demarest R: The Human Nervous System, 3rd ed New York, McGraw-Hill, 1981)

Hair follicle muscle (pilomotor)

Sweat glands (sudomotor)

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Recall the BIPARTITE MATCHING problem: given a bipartite graph with n nodes on each side (the boys and the girls), nd a set of n disjoint edges, or decide that no such set exists In Section 73, we saw how to ef ciently solve this problem by a reduction to maximum ow However, there is an interesting generalization, called 3D MATCHING, for which no polynomial algorithm is known In this new setting, there are n boys and n girls, but also n pets, and the compatibilities among them are speci ed by a set of triples, each containing a boy, a girl, and a pet Intuitively, a triple (b, g, p) means that boy b, girl g, and pet p get along well together We want to nd n disjoint triples and thereby create n harmonious households Can you spot a solution in Figure 84 236

S1 Vasomotor fibers to lower extremity CX Preganglionic neuron Postganglionic neuron Sympathetic trunk

Figure 26-3 The sympathetic (thoracolumbar) division of the autonomic nervous system Preganglionic bers extend from the intermediolateral nucleus of the spinal cord to the peripheral autonomic ganglia, and postganglionic bers extend from the peripheral ganglia to the effector organs, according to the scheme in Fig 26-1 (Reproduced by permission from Noback CL, Demarest R: The Human Nervous System, 3rd ed New York, McGraw-Hill, 1981)

via gray communicating rami to spinal nerves of T5 to L2; they supply blood vessels, sweat glands, and hair follicles and also form plexuses that supply the heart, bronchi, kidneys, intestines, pancreas, bladder, and sex organs The postganglionic bers of the prevertebral ganglia (located in the posterior abdomen rather than

Figure 84 A more elaborate matchmaking scenario Each triple is shown as a triangularshaped node joining boy, girl, and pet

paravertebrally) form the hypogastric, splanchnic, and mesenteric plexuses, which innervate the glands, smooth muscle, and blood vessels of the abdominal and pelvic viscera (Figs 26-1 and 26-4) The sympathetic innervation of the adrenal medulla is unique in that its secretory cells receive preganglionic bers directly, via

Figure 26-4 The principle of the sympathetic preganglionic innervation of ganglia that are placed beyond the limits of the usual preganglionic paravetebral sympathetic out ow from the spinal cord Preganglionic bers (heavy lines) emerging from a spinal segment do not synapse exclusively in the corresponding paravertebral ganglion Some pass as splanchnic nerves to prevertebral (posterior abdominal) ganglia; some bers also enter the sympathetic trunk, in which they pass up or down for a variable number of segments (Reproduced by permission from Pick)

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