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Parameterized Algorithms for MILPs with Small Treedepth [article]

Cornelius Brand, Martin Koutecký, Sebastian Ordyniak
2019 arXiv   pre-print
efficient algorithms, and all of these special cases are subsumed by the class of ILPs of small treedepth.  ...  In this paper, we extend this line of work to the mixed case, by showing an algorithm solving MILP in time f(a,d) poly(n), where a is the largest coefficient of the constraint matrix, d is its treedepth  ...  if P is a polytope described by inequalities with small treedepth, then a projection of P also has an inequality description of small treedepth.  ... 
arXiv:1912.03501v1 fatcat:hibymvb4jfbvxoawiznog5flry

Parameterized Algorithms for MILPs with Small Treedepth

Cornelius Brand, Martin Koutecký, Sebastian Ordyniak
2021 PROCEEDINGS OF THE THIRTIETH AAAI CONFERENCE ON ARTIFICIAL INTELLIGENCE AND THE TWENTY-EIGHTH INNOVATIVE APPLICATIONS OF ARTIFICIAL INTELLIGENCE CONFERENCE  
efficient algorithms, and all of these special cases are subsumed by the class of ILPs of small treedepth.  ...  In this paper, we extend this line of work to the mixed case, by showing an algorithm solving MILP in time f(a,d)poly(n), where a is the largest coefficient of the constraint matrix, d is its treedepth  ...  We also explore the limits of approaching the problem by bounding the fractionality of inverses: Other ILP classes with parameterized algorithms involve constraint matrices with small primal treewidth  ... 
doi:10.1609/aaai.v35i14.17454 fatcat:i4jf5r4kbzb5ddukng4wm6bo3e

Integer Programming in Parameterized Complexity: Three Miniatures [article]

Tomáš Gavenčiak and Dušan Knop and Martin Koutecký
2018 arXiv   pre-print
However, our perception is that, except for Lenstra's algorithm for solving integer linear programming in fixed dimension, there is still little understanding in the parameterized complexity community  ...  Then, we show their applications in three case studies, obtaining FPT algorithms with runtime f(k)poly(n).  ...  They also combine primal treewidth with Lenstra's algorithm to obtain a new structural parameter called torso-width, and give an FPT algorithm for this parameterization.  ... 
arXiv:1711.02032v3 fatcat:zzn6cumyqrcifpk7kevo5kaafm

Evaluating and Tuning n-fold Integer Programming

Kateřina Altmanová, Dušan Knop, Martin Koutecký
2019 ACM Journal of Experimental Algorithmics  
the IP; thus, an algorithm efficient if the blocks are of small size and with small coefficients.  ...  ACM Subject Classification Software and its engineering → Software design engineering, Theory of computation → Parameterized complexity and exact algorithms  ...  are "natural" instances with small tree-depth?  ... 
doi:10.1145/3330137 fatcat:mrnjo3uikvbvrdemvvinkmo5dm

Evaluating and Tuning n-fold Integer Programming [article]

Kateřina Altmanová and Dušan Knop and Martin Koutecký
2018 arXiv   pre-print
the IP; thus, an algorithm efficient if the blocks are of small size and with small coefficients.  ...  However, surprisingly small values of the parameter already exhibit good behavior. Second, our new strategy for finding "approximatelly best" steps wildly outperforms the original construction.  ...  [23] have studied parameterized strongly polynomial algorithms for various block-structured ILPs, not just N -fold IP. Eisenbrand et al.  ... 
arXiv:1802.09007v1 fatcat:ps6brca3cbgopmnv5qkp642pay