# Parameterized Graph Connectivity and Polynomial-Time Sub-Linear-Space Short Reductions

## Abstract

We are focused on the solvability/insolvability of the directed *s*-*t* connectivity problem (DSTCON) parameterized by suitable size parameters *m*(*x*) on multi-tape deterministic Turing machines working on instances *x* to DSTCON by consuming simultaneously polynomial time and sub-linear space, where the informal term “sub-linear” refers to a function of the form \(m(x)^{\varepsilon } \ell (|x|)\) on instances *x* for a certain absolute constant \(\varepsilon \in (0,1)\) and a certain polylogarithmic function \(\ell (n)\). As natural size parameters, we take the numbers \(m_{ver}(x)\) of vertices and of edges \(m_{edg}(x)\) of a graph cited in *x*. Parameterized problems solvable simultaneously in polynomial time using sub-linear space form a complexity class \(\mathrm {PsubLIN}\) and it is unknown whether \(\mathrm {DSTCON}\) parameterized by \(m_{ver}\) belongs to \(\mathrm {PsubLIN}\). Toward this open question, we wish to investigate the relative complexity of \(\mathrm {DSTCON}\) and its natural variants and classify them according to a restricted form of many-one and Turing reductions, known as “short reductions,” which preserve the polynomial-time sub-linear-space complexity. As variants of \(\mathrm {DSTCON}\), we consider the breadth-first search problem, the minimal path problem, and the topological sorting problem. Certain restricted forms of them fall into \(\mathrm {PsubLIN}\). We also consider a stronger version of “sub-linear,” called “hypo-linear.” Additionally, we refer to a relationship to a practical working hypothesis known as the linear space hypothesis.

### Keywords

Sub-linear space Hypo-linear space Directed s-t-connectivity NL search NL optimization Short reduction Linear space hypothesis### References

- 1.Allender, E., Barrington, D.A.M., Chakraborty, T., Datta, S., Roy, S.: Planar and grid graph reachability problems. Theory Comput. Syst.
**45**, 675–723 (2009)MathSciNetCrossRefMATHGoogle Scholar - 2.Allender, E., Mahajan, M.: The complexity of planarity testing. Inf. Comput.
**189**, 117–134 (2004)MathSciNetCrossRefMATHGoogle Scholar - 3.Asano, T., Kirkpatrick, D., Nakagawa, K., Watanabe, O.: \(\widetilde{O}(\sqrt{n})\)-space and polynomial-time algorithm for planar directed graph reachability. In: Csuhaj-Varjú, E., Dietzfelbinger, M., Ésik, Z. (eds.) MFCS 2014. LNCS, vol. 8635, pp. 45–56. Springer, Heidelberg (2014). doi: 10.1007/978-3-662-44465-8_5 Google Scholar
- 4.Barnes, G., Buss, J.F., Ruzzo, W.L., Schieber, B.: A sublinear space, polynomial time algorithm for directed s-t connectivity. SIAM J. Comput.
**27**, 1273–1282 (1998)MathSciNetCrossRefMATHGoogle Scholar - 5.Bourke, C., Tewari, R., Vinodchandran, N.V.: Directed planar reachability is in unambiguous logspace. In: Proceedings CCC 2007, 217–221 (2007)Google Scholar
- 6.Chakraborty, D., Pavan, A., Tewari, R., Vinodchandran, N.V., Yang, L.F.: New time-space upperbounds for directed reachability in high-genus and H-minor-free graphs. In: The Proceedings of FSTTCS 2014, Leibniz International Proceedings in Informatics, pp. 585–595 (2014)Google Scholar
- 7.Gross, J.L., Yellen, J., Zhang, P.: Handbook of Graph Theory. CRC Press, Boca Raton (2014)MATHGoogle Scholar
- 8.Immerman, N.: Nondeterministic space is closed under complement. SIAM J. Comput.
**17**, 935–938 (1988)MathSciNetCrossRefMATHGoogle Scholar - 9.Jones, N.D.: Space-bounded reducibility among combinatorial problems. J. Comput. Syst. Sci.
**11**, 68–75 (1975)MathSciNetCrossRefMATHGoogle Scholar - 10.Kannan, S., Khanna, S., Roy, S.: STCON in directed unique-path graphs. In: Proceedings of FSTTCS 2008, UPIcs 2, pp. 256–267 (2008)Google Scholar
- 11.Reingold, O.: Undirected connectivity in log-space. J. ACM
**55**, 1–24 (2008)MathSciNetCrossRefMATHGoogle Scholar - 12.Reingold, O., Trevisan, L., Vadhan, S.P.: Pseudorandom walks on regular digraphs and the RL vs. L problem. In: The Proceedings of STOC 2006, pp. 457–466 (2006)Google Scholar
- 13.Savitch, W.J.: Relationships between nondeterministic and deterministic tape complexities. J. Comput. Syst. Sci.
**4**, 177–192 (1970)MathSciNetCrossRefMATHGoogle Scholar - 14.Szelepcsényi, R.: The method of forced enumeration for nondeterministic automata. Acta Inf.
**26**, 279–284 (1988)MathSciNetCrossRefMATHGoogle Scholar - 15.Tantau, T.: Logspace optimization problems and their approximation properties. Theory Comput. Syst.
**41**, 327–350 (2007)MathSciNetCrossRefMATHGoogle Scholar - 16.Yamakami, T.: Uniform-circuit and logarithmic-space approximations of refined combinatorial optimization problems. In: Widmayer, P., Xu, Y., Zhu, B. (eds.) COCOA 2013. LNCS, vol. 8287, pp. 318–329. Springer, Cham (2013). doi: 10.1007/978-3-319-03780-6_28. A complete version is available at arXiv:1601.01118v1 CrossRefGoogle Scholar
- 17.Yamakami, T.: The 2CNF Boolean formula satisfiability problem and the linear space hypothesis. In: The Proceedings of MFCS 2017, 7–11 August 2017. Leibniz International Proceedings in Informatics (2017, to appear)Google Scholar