Experiments in quadratic 0-1 programming

where x e {0, 1} n, Q c ~( , , , i is an u p p e r t r i angu la r mat r ix and c c R" is a co lumn vector. Since x~-x i we can assume that the d i agona l of Q is zero. Our p r o b l e m is to find the m i n i m u m o f f This is an N P h a r d p rob lem, cf. G a r e y and Johnson (1979). It is po lynomia l ly so lvable if all the e lements of Q are nonpos i t ive , cf. P icard and Ratliff (1974), and when the g raph def ined by Q is ser ies-para l le l , cf. Ba rahona (1986). A b ranch and b o u n d a lgor i thm has been proposed by Car te r (1984), and po lynomia l a lgor i thms for f inding lower b o u n d s have been p resen ted in H a m m e r , Hansen and S imeone (1984). A lgor i thms for a more genera l p r o b l e m have been p r o p o s e d by Balas and Mazzo la (1984). Our a p p r o a c h consists in reduc ing this p r o b l e m to a max-cu t p rob l em and then using a pa r t i a l desc r ip t ion of the cut p o l y t o p e to a t tack the p r o b l e m with l inear p r o g r a m m i n g techniques . We use b ranch and b o u n d i f no in teger so lu t ion has been found by our cut t ing p lane a lgor i thm. In Sect ion 2, we exp la in the reduc t ion to a max-cu t p r o b l e m and men t ion some results abou t the max-cu t po ly tope . In Sect ion 3, we p resen t the a lgor i thm. The compu ta t i ona l results are in Sect ion 4.