**Instant recognition of half integrality and 2-approximation**

## Abstract

We define a class of integer programs with constraints that involve
up to three variables each. A generic constraint in such integer
program is of the form ax + by z + c, where the variable z
appears only in that constraint. For such binary integer programs it is
possible to derive half integral superoptimal
solutions in polynomial time. The scheme is also applicable with
few modifications to
nonbinary integer problems. For some of these
problems it is possible to round the half integral solution to a
2-approximate solution. This extends the class of
integer programs with at most two variables per
constraint that were analyzed in HMNT93.
Interesting problems for which we can get superoptimal half
integral solutions include the sparsest cut problem, and minimum
weight edge deletion to obtain a bipartite graph.
Problems for which we can get 2-approximations include
minimum satisfiability, scheduling with precedence constraints,
minimum weight node deletion to obtain a complete bipartite graph,
several
boolean functions on two variables (generalized 2SAT) and the feasible cut problems.
The technique described here is used also to derive a
2-approximation algorithm for a complement of the maximum clique
problem.
In addition we cast a large class of problems as minimum
satisfiability problems and a recognition procedure for determining
when such problems are 2-approximable.
The approximation algorithms here provide an improvement
in running time and range of applicability compared to existing
2-approximations.
Furthermore, we conclude that problems in the framework are MAX
SNP-hard and at least as hard to approximate as vertex cover.

Problems that are amenable to the analysis provided here are easily
recognized.
The analysis itself is entirely technical and
involves manipulating the
constraints and transforming them to a totally unimodular system
while losing (for binary problems) no more than a factor of 2 in the integrality.

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*Instant recognition of half-integrality and 2 -approximation (13 pages, 211 KB)*
- Dorit S. Hochbaum.
*Proceedings of APPROX98*, Aalborg, Denmark, July 1998.

`dorit@hochbaum.ieor.berkeley.edu`
7/28/98