ADVANCED ALGORITHMS Lecture 21: LPs for matching, covering � 1
ANNOUNCEMENTS ➤ HW 5 will be out tonight — due in two weeks ➤ Project “mid-way” report due next Friday � 2
LAST CLASS Nn k N i ➤ Linear programming a b ain AIN E be aina's bm � 3
LAST CLASS ➤ Geometry of linear programs ➤ what is a “corner point”? (intersection of n of the planes) ➤ neighboring corners for u (directions given by A -1 , where A defines u ) ➤ no neighbor gives improvement => optimum! ➤ main idea behind simplex algorithm globo f a local opt ➤ Can solve in polynomial time via Ellipsoid, interior point methods � 4
USING LP FOR COMBINATORIAL PROBLEMS discrete optimization g nu ➤ Continuous formulations … matching? j ng for edges i j3 variables i C foe o o Constraints T.fi eneigiiimcn lrL.repiAa B 1 Vi v j nij Voenij 4 0 max Wijnij � 5
0 USING LP FOR COMBINATORIAL PROBLEMS ➤ Continuous formulations … matching? ➤ Main problem: what if the solutions are fractions? Main question: are the “corner points” integral? � 6
MATCHING ➤ Theorem: all the corner points of the “matching polytope” are integral. a set of feasible solutions ➤ Thus, solving the LP gives a 0/1 solution! to the linear a System of edges variables m m qq.gs constraints q irreg s solve we m of the constraints If we take any integer point get an � 7
PROOF 1 (OUTLINE) : TOTAL UNIMODULARITY 2mi 2n constraints m of the Take Claim any 0 1 values The intersection point has only 1 1 values 0 ok N12 I 7h22 Z N f i i i EE m ni El 1 1,1 1 1 � 8
PROOF 1 (OUTLINE) : TOTAL UNIMODULARITY Em can be obtained by solving corner pt a Any B comprises where B n term z linear Sys is the RHS of rows of M and z m precisely Nu the corresponding constraint a n n ITI B EH Fit o dit 113 Nij � 9
PROOF 2: UNDERSTANDING CORNER POINTS … z f z Alternate characterization z corner ptiff u u is a For any perturbation direction Z z fl Z p and u z one of Utz at most is in the feasible set u w Not a corner we need to show that u if Given utz and 2 sit ive it suffices to inhibit then are both feasible 2 u � 10
PROOF 2 lae any feasible non integral point Let u Claim z both Sati utz and Then F zt o.it u satisfying oU ijs.t.ocuig.cl E o o a i i i i 8 0 i e i 0 Observation rO Entry vertex has degree � 11 O O
I g Any such graph has a cycle Exercised there is a cycle using only I e the edges of E Ptu no fading o 8 if a be can pt oo.us this will still be For small enough 8 feasible solution a we want z that This gives the perturbation
FLOWS IN NETWORKS ➤ Theorem: all the corner points of the “flow polytope” are integral. linear use can simply we finding max flow for programming � 12
WARNINGS ➤ This is a very special phenomenon! ➤ Polytopes usually have fractional corners … � 13
MONITORING EDGES (A.K.A. VERTEX COVER) ➤ Problem. given undirected graph G = (V , E), find a small set of nodes S such that every edge has at least one of its neighbors chosen . find the smallest 0 of nodes so that Lo monitored edges are all Vats p 7 01411 I Nu C o relaxation � 14
LP FOR VERTEX COVER i j ni taj 31 Hedges Constants 312 7 ni min ni I l O E U Nv Nw E 1 0 Of Nu Now nut nu 31 O v Nr t Kw 71 Nu Nu Nw 42 I Nwt Nu � 15
BAD CORNERS � 16
“ROUNDING” SOLUTION � 17
APPROXIMATION ALGORITHM � 18
OTHER PROBLEMS — INDEPENDENT SET ➤ Problem. given undirected graph G = (V , E), find the largest possible set of nodes S such that has no edges within. � 19
LP AND ITS LIMITATIONS � 20
IN SUMMARY ➤ LP (“continuous”) formulations for discrete problems ➤ can get lucky — all corners are integral (i.e., discrete) ➤ corners can be “somewhat integral” — approximation algorithms ➤ corners can be totally useless — means better LP is needed! ➤ Next time … randomized rounding � 21
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