The　0-1　linear　programming　problem　with　non-negative　constraint　matrix　and　objective　vector　e　origins　from　many　NP-hard　combinatorial　optimization　problems.　In　this　paper,　we　consider　under　what　condition　an　optimal　solution　of　the　0-1　problem　can　be　obtained　from　a　weighted　linear　programming.　To　this　end,　we　first　formulate　the　0-1　problem　as　a　sparse　minimization　problem.　Any　optimal　solution　of　the　0-1　linear　programming　problem　can　be　obtained　by　rounding　up　an　optimal　solution　of　the　sparse　minimization　problem.　Then,　we　establish　a　condition　under　which　the　sparse　minimization　problem　and　the　weighted　linear　programming　problem　have　the　same　optimal　solution.　The　condition　is　based　on　the　defined　non-negative　partial　s-goodness　of　the　constraint　matrix　and　the　weight　vector.　Further,　we　use　two　quantities　to　characterize　a　sufficient　condition　and　necessary　condition　for　the　non-negative　partial　s-goodness.　However,　the　two　quantities　are　difficult　to　calculate,　therefore,　we　provide　a　computable　upper　bound　for　one　of　the　two　quantities　to　verify　the　non-negative　partial　s-goodness.　Furthermore,　we　propose　two　operations　of　the　constraint　matrix　and　weight　vector　that　still　preserve　non-negative　partial　s-goodness.　Finally,　we　give　some　examples　to　illustrate　that　our　theory　is　effective　and　verifiable.　©　2023,　The　Author(s),　under　exclusive　licence　to　Springer　Science+Business　Media,　LLC,　part　of　Springer　Nature.