The negative cycles polyhedron and hardness of checking some polyhedral properties
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Given a graph G=(V,E) and a weight function on the edges w:E→ℝ, we consider the polyhedron P(G,w) of negative-weight flows on G, and get a complete characterization of the vertices and extreme directions of P(G,w). Based on this characterization, and using a construction developed in Khachiyan et al. (Discrete Comput. Geom. 39(1–3):174–190, 2008), we show that, unless P=NP, there is no output polynomial-time algorithm to generate all the vertices of a 0/1-polyhedron. This strengthens the NP-hardness result of Khachiyan et al. (Discrete Comput. Geom. 39(1–3):174–190, 2008) for non 0/1-polyhedra, and comes in contrast with the polynomiality of vertex enumeration for 0/1-polytopes (Bussiech and Lübbecke in Comput. Geom., Theory Appl. 11(2):103–109, 1998). As further applications, we show that it is NP-hard to check if a given integral polyhedron is 0/1, or if a given polyhedron is half-integral. Finally, we also show that it is NP-hard to approximate the maximum support of a vertex of a polyhedron in ℝ n within a factor of 12/n.
KeywordsFlow polytope 0/1-polyhedron Vertex Extreme direction Enumeration problem Negative cycles Directed graph Half-integral polyhedra Maximum support Hardness of approximation
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- Abdullahi, S. D., Dyer, M. E., & Proll, L. G. (2003). Listing vertices of simple polyhedra associated with dual LI(2) systems. In DMTCS: discrete mathematics and theoretical computer science, 4th international conference, DMTCS 2003, proceedings (pp. 89–96). Google Scholar
- Avis, D., & Fukuda, K. (1992). A pivoting algorithm for convex hulls and vertex enumeration of arrangements and polyhedra. Discrete and Computational Geometry, 8(3), 295–313. Google Scholar
- Avis, D., Bremner, B., & Seidel, R. (1997). How good are convex hull algorithms. Computational Geometry: Theory and Applications, 7, 265–302. Google Scholar
- Björklund, A., Husfeldt, T., & Khanna, S. (2004). Approximating longest directed paths and cycles. In Automata, languages and programming: 31st international colloquium, ICALP (pp. 222–233). Google Scholar
- Boros, E., Elbassioni, K., Gurvich, V., & Makino, K. (2008). Generating vertices of polyhedra and related monotone generation problems. In Avis, D., Bremner, D., & Deza, A. (Eds.), Special issue on polyhedral computation : Vol. 48. CRM proceedings & lecture notes, centre de recherches mathématiques at the Université de Montréal (pp. 15–39). Providence: American Mathematical Society. Google Scholar
- Bussieck, M. R., & Lübbecke, M. E. (1998). The vertex set of a 0/1 polytope is strongly ℘-enumerable. Computational Geometry: Theory and Applications, 11(2), 103–109. Google Scholar
- Lovász, L. (1992). Combinatorial optimization: some problems and trends (DIMACS Technical Report 92-53). Rutgers University. Google Scholar
- Read, R. C., & Tarjan, R. E. (1975). Bounds on backtrack algorithms for listing cycles, paths, and spanning trees. Networks, 5, 237–252. Google Scholar
- Schrijver, A. (1986). Theory of linear and integer programming. New York: Wiley. Google Scholar
- Vazirani, V. V. (2001). Approximation algorithms. Berlin: Springer. Google Scholar